The joint PDF of X and Y is f(x, y) = 1/2 for 0 < x < 2 and 0 < y < 1.
The joint PDF of two random variables X, Y given by f(x, y) = C, where 0 < x < 2 and 0 < y < 1, is a uniform distribution. To find C, you can use the property that the total probability should equal 1.
1. Recognize that the problem describes a uniform distribution.
2. Determine the range of the variables: X ranges from 0 to 2, and Y ranges from 0 to 1.
3. Calculate the area of the rectangle formed by these ranges: Area = (2 - 0) * (1 - 0) = 2.
4. Use the property that the total probability should equal 1: ∫∫f(x, y)dxdy = 1.
5. Since the distribution is uniform, f(x, y) = C, and the integral becomes ∫∫Cdxdy = C * Area.
6. Solve for C: C * Area = C * 2 = 1, therefore C = 1/2.
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complete question:
Consider the joint PDF of two random variables X, Y given by f x,y (x, y) = C, calculate the joint PDF of X and Y .
help someone with these two questions
The shapes involved in the first figure is a triangle and a trapezium, with an area of 139.5. The shapes involved in the second figure is a triangle and a rectangle, with an area of 22 square units.
How to calculate for the area of the figuresThe first figure can be observed to be made up of a triangle and a trapezium. While the second is a triangle and a rectangle, so we shall calculate for the area and sum the results to get the total area of the composite figures as follows:
First figure:
area of the triangle = 1/2 × 9 × 6 = 27 square units
area of the trapezium = 1/2 × (6 + 9) × 15 = 112.5 square units
area of the first figure = 27 + 112.5 = 139.5 square units
Second figure:
area of the triangle = 1/2 × 4 × 2 = 4 square units
area of the rectangle = 9 × 2 = 18 square
area of the second figure = 4 + 18 = 22 square units.
Therefore, the shapes involved in the first figure is a triangle and a trapezium, with an area of 139.5. The shapes involved in the second figure is a triangle and a rectangle, with an area of 22 square units.
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the game of four square on a 12 foot by 12-foot court you square is a 6foot by 6 foot what is the area if four square not including you court
Answer:
108 ft ^2
Step-by-step explanation:
12^2 - 6^2 = 108
Given P(x) = x^3 + 2x^2 + 9x + 18. Write P in factored form (as a product of linear factors). Be sure to write the full equation, including P(x) = ______.
The factored form of polynomial P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]
To factor [tex]P(x) = x^3 + 2x^2 + 9x + 18,[/tex]we need to first look for any common factors that we can factor out. In this case, we can factor out a 1, so:
[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)[/tex]
Next, we can try to find the roots of the polynomial by using the Rational Root Theorem, which states that if a polynomial has integer coefficients, then any rational root of the polynomial must have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is 18 and the leading coefficient is 1, so the possible rational roots are:
±1, ±2, ±3, ±6, ±9, ±18
We can try these roots by using synthetic division or long division to see if they are roots of the polynomial. After trying a few of these roots, we find that -2 is a root of the polynomial, so we can factor out (x + 2):
[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)\\ = 1(x + 2)(x^2 + ax + b)[/tex]
where a and b are coefficients that we need to find. To find a and b, we can use the fact that the coefficient of x^2 in the factored form should be equal to the coefficient of x^2 in the original polynomial. That is,
2 + 2a = 2
Solving for a, we get a = -1. Next, we can expand the factor (x^2 - x + b) and equate the coefficients of x and the constant term to the corresponding coefficients in the original polynomial. That is,
2a + b = 9
2b = 18
Solving for b, we get b = 9. Therefore, we have:
[tex]P(x) = 1(x + 2)(x^2 - x + 9)[/tex]
So the factored form of P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]
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The factored form of polynomial P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]
To factor [tex]P(x) = x^3 + 2x^2 + 9x + 18,[/tex]we need to first look for any common factors that we can factor out. In this case, we can factor out a 1, so:
[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)[/tex]
Next, we can try to find the roots of the polynomial by using the Rational Root Theorem, which states that if a polynomial has integer coefficients, then any rational root of the polynomial must have the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. In this case, the constant term is 18 and the leading coefficient is 1, so the possible rational roots are:
±1, ±2, ±3, ±6, ±9, ±18
We can try these roots by using synthetic division or long division to see if they are roots of the polynomial. After trying a few of these roots, we find that -2 is a root of the polynomial, so we can factor out (x + 2):
[tex]P(x) = 1(x^3 + 2x^2 + 9x + 18)\\ = 1(x + 2)(x^2 + ax + b)[/tex]
where a and b are coefficients that we need to find. To find a and b, we can use the fact that the coefficient of x^2 in the factored form should be equal to the coefficient of x^2 in the original polynomial. That is,
2 + 2a = 2
Solving for a, we get a = -1. Next, we can expand the factor (x^2 - x + b) and equate the coefficients of x and the constant term to the corresponding coefficients in the original polynomial. That is,
2a + b = 9
2b = 18
Solving for b, we get b = 9. Therefore, we have:
[tex]P(x) = 1(x + 2)(x^2 - x + 9)[/tex]
So the factored form of P(x) is [tex]P(x) = 1(x + 2)(x^2 - x + 9).[/tex]
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Discuss the existence and uniqueness of a solution to the differential equation 3+ 2)y"y-y-tant that satisfies the initial conditions y(3)- Yo.y(8)-Y, where Yo and Y1 are real constants. Select the correct choice below and fill in any answer boxes to complete your choice A. A solution is guaranteed on the interval___< t < because its contains the point T0 =___ and the function p(t)= ___ q(t)___ and gt ___ are equal on the interval B. A solution is guaranteed on the interval___< t < because its contains the point T0 =___ and the function p(t)= ___ q(t)___ and gt ___ are simultaneously countionous on the interval C. A solution is guaranteed only at the pouint T0 =___ and the function p(t)= ___ q(t)___ and gt ___ are simultaneously defined at the point
The solution to the differential equation that satisfies the initial conditions y(3) = y0 and y(8) = y1 is:
y(t) = (2/3)t - (1/3)cos(t) + (1/3)sin(t) + y1 + (1/3)sin(3) - (2
The given differential equation is:
3y''+2y'y-y-tan(t)=0
To check the existence and uniqueness of a solution, we need to verify if the function p(t) and q(t) satisfy the conditions of the Existence and Uniqueness Theorem.
The Existence and Uniqueness Theorem states that if the functions p(t) and q(t) are continuous on an interval containing a point t0 and if p(t) is not equal to zero at t0, then there exists a unique solution to the differential equation y'' + p(t) y' + q(t) y = g(t) that satisfies the initial conditions y(t0) = y0 and y'(t0) = y1.
Comparing the given differential equation with the standard form of the Existence and Uniqueness Theorem, we get:
p(t) = 2y(t)
q(t) = -t - tan(t)
g(t) = 0
To find the interval of existence, we need to check the continuity of p(t) and q(t) and also the value of p(t) at t0.
Here, p(t) is continuous everywhere and q(t) is continuous on the interval (3, 8). To check the value of p(t) at t0, we need to find y(t) that satisfies the initial conditions y(3) = y0 and y(8) = y1.
Let's assume that y(t) = A(t) + B(t), where A(t) satisfies y(3) = y0 and A'(3) = 0 and B(t) satisfies y(8) = y1 and B'(8) = 0.
Solving the differential equation for A(t), we get:
A(t) = c1 cos(sqrt(3)(t-3)) + c2 sin(sqrt(3)(t-3)) + (2/3)t - (1/3)cos(t) + (1/3)sin(t) + (1/3)sin(3)
Using the initial conditions y(3) = y0 and A'(3) = 0, we get:
A(t) = (2/3)t - (1/3)cos(t) + (1/3)sin(t) + (1/3)sin(3) - (2/3)cos(3) - y0
Solving the differential equation for B(t), we get:
B(t) = c3 cos(sqrt(3)(t-8)) + c4 sin(sqrt(3)(t-8)) + (2/3)t - (1/3)cos(t) + (1/3)sin(t) - (1/3)sin(3)
Using the initial conditions y(8) = y1 and B'(8) = 0, we get:
B(t) = (2/3)t - (1/3)cos(t) + (1/3)sin(t) - (1/3)sin(3) + (2/3)cos(3) + y1
Therefore, the solution to the differential equation that satisfies the initial conditions y(3) = y0 and y(8) = y1 is:
y(t) = (2/3)t - (1/3)cos(t) + (1/3)sin(t) + y1 + (1/3)sin(3) - (2)
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The positions of a particle moving in the xy-plane is given by the parametric equations x=t3−3t2 and y=2t3−3t2−12t. For what values of t is the particle at rest?
The particle is at rest when the velocity is zero.
To find the values of t, you need to calculate the first derivatives of the parametric equations and set them equal to zero.
Main answer: The particle is at rest for t = 0 and t = 2.
1. Calculate the first derivatives of x(t) and y(t):
dx/dt = 3t² - 6t
dy/dt = 6t² - 6t - 12
2. Set the derivatives equal to zero and solve for t:
3t² - 6t = 0
6t² - 6t - 12 = 0
3. Factor the equations:
t(3t - 6) = 0
6(t² - t - 2) = 0
4. Solve for t:
t = 0, (3t - 6) = 0
t² - t - 2 = 0
5. From the first equation, t = 0 or t = 2.
From the second equation, use the quadratic formula:
t = (1 ± √(1 + 8))/2
t ≈ 1.41, -1.41
6. The particle is at rest for t = 0 and t = 2. The other values do not correspond to a stationary point.
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an athlete can run 6 miles in 51 minutes . at this rate , how many miles could the athlete run in 1.5 hours ?
At the given rate, the athlete could run 10.584 miles in 1.5 hours.
To determine how many miles the athlete could run in 1.5 hours at the given rate, follow these steps:
Step 1: Calculate the athlete's speed in miles per minute.
The athlete can run 6 miles in 51 minutes, so their speed is:
Speed = Distance ÷ Time = 6 miles ÷ 51 minutes ≈ 0.1176 miles per minute.
Step 2: Convert 1.5 hours to minutes.
1.5 hours = 1.5 × 60 = 90 minutes.
Step 3: Calculate the distance the athlete can run in 1.5 hours.
Distance = Speed × Time = 0.1176 miles per minute × 90 minutes ≈ 10.584 miles.
Therefore, at the given rate, the athlete could run approximately 10.584 miles in 1.5 hours.
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You invest $2,000 in a Certificate of Deposit (CD) with an APR 2.25% for 3 years
that compounds annually. What is the balance after 3 years?
The balance after 3 years on a Certificate of Deposit with an APR of 2.25% that compounds annually is $2,163.05.
What is meant by balance?Balance refers to the equality between two expressions or equations, where both sides have the same value. It is often used in solving equations or evaluating algebraic expressions.
What is meant by compounds?A compound refers to a combination of two or more simple mathematical statements or propositions, connected by logical operators such as "and", "or", or "not". It is used in logic and boolean algebra.
According to the given information:To calculate the balance after 3 years on a Certificate of Deposit with an APR of 2.25% that compounds annually, we can use the formula:
A = P(1 + r/n)^{n*t}
Where:
A = the balance after t years,
P = the principal amount invested,
r = the annual interest rate as a decimal,
n = the number of times the interest is compounded per year,
t = the number of years
Plugging in the given values, we get:
P = $2,000r = 0.0225 (2.25% expressed as a decimal)
n = 1 (compounded annually)
t = 3 years,
[tex]A = 2,000(1 + 0.0225/1)^{1*3}[/tex]
[tex]A = 2,000(1 + 0.0225)^3[/tex]
[tex]A = 2,000(1.0225)^3[/tex]
A = $2,163.05 (rounded to the nearest cent)
Therefore, the balance after 3 years on a Certificate of Deposit with an APR of 2.25% that compounds annually is $2,163.05.
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What is the area of this composite figure
The composite figure has an area of 24 square units.
How to determine the area of a composite figure
In this question we find the representation of a composite figure formed by the combination of four figures, a triangle and three rectangles, whose area formulas are listed below:
Rectangle
A = b · h
Triangle
A = 0.5 · b · h
Where:
A - Areab - Widthh - HeightNow we proceed to determine the area of the composite figure:
A = 2 · 3 + 0.5 · 2 · 1 + 7 · 2 + 1 · 3
A = 6 + 1 + 14 + 3
A = 24
The area of the composite figure is equal to 24 square units.
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express the general solution of the given differential equation on the interval (0,[infinity]) in termsof bessel functions:(a) 4x2y′′ 4xy′ (64x2−9)y= 0(b)x2y′′ xy′−(36x2 9)y= 0
The following parts can be answered by the concept of Differential equation.
(a) For the differential equation 4x²y'' + 4xy' - (64x² - 9)y = 0, we can rewrite it as:
y'' + (1/x)y' - (16 - 9/x²)y = 0
This is a Bessel's equation of order ν = 3. The general solution is given by:
y(x) = c_1 J_3(2√2x) + c_2 Y_3(2√2x)
where c_1 and c_2 are constants, J_3 is the Bessel function of the first kind of order 3, and Y_3 is the Bessel function of the second kind of order 3.
(b) For the differential equation x²y'' + xy' - (36x² - 9)y = 0, we can rewrite it as:
y'' + (1/x)y' - (36 - 9/x²)y = 0
This is also a Bessel's equation, but with order ν = 3/2. The general solution is given by:
y(x) = c_1 J_(3/2)(6x) + c_2 Y_(3/2)(6x)
where c_1 and c_2 are constants, J_(3/2) is the Bessel function of the first kind of order 3/2, and Y_(3/2) is the Bessel function of the second kind of order 3/2.
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Members of a softball team raised $1952. 50 to go to a tournament. They rented a bus
Eor $983. 50 and budgeted $57 per player for meals. Write and solve an equation
_which can be used to determine p, the number of players the team can bring to the
Cournament.
If the null space of a 7 times 9 matrix is 3-dimensional, find:
Rank A= DIm Row A, and Dim Col A.
Rank A = 4, Dim Row A = 4, DIm Col A = 4
Rank A = 6, Dim Row A = 3, Dim Col A = 3
Rank A = 6, Dim Row A = 6, Dim Col A = 6
Rank A = 6, Dim Row A = 6, Dim Col A = 3
By the rank-nullity theorem, we know that for any matrix A, the rank of A plus the dimension of the null space of A is equal to the number of columns of A. If the null space of a 7 times 9 matrix is 3-dimensional, Rank A = 6, Dim Row A = 6, Dim Col A = 6
we know that for any matrix A, the rank of A plus the dimension of the null space of A is equal to the number of columns of A. That is:
Rank A + Dim Null A = # of columns of A
In this case, we are given that the null space of the 7x9 matrix A is 3-dimensional. Therefore, we have:
Rank A + 3 = 9
Solving for Rank A, we get:
Rank A = 6
Now, we also know that the rank of a matrix is equal to the dimension of its row space and the dimension of its column space. That is:
Rank A = Dim Row A = Dim Col A
Therefore, we have:
Rank A = Dim Row A = Dim Col A = 6
So the correct option is: Rank A = 6, Dim Row A = 6, Dim Col A = 6
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Let W be the region bounded by the cylinders z= 1-y^2 and y=x^2, and the planes z=0 and y=1 . Calculate the volume of W as a triple integral in the three orders dzdydx, dxdzdy, and dydzdx.Im having trouble figuring out my parameters for which i am integrating. I do understand however that i should get the same volume for all three orders since the orders don't matter.
The order of integration does not affect the final answer, but may affect the complexity of the integrals.
To calculate the volume of the region W using triple integrals, we need to determine the bounds for each variable.
First, we can see that the planes z=0 and y=1 bound the region in the z and y directions, respectively.
Next, to find the bounds for x, we need to find the intersection of the two cylinders. Solving for y in the equation [tex]z=1-y^2[/tex], we get y = ±sqrt(1-z). Substituting this into the equation [tex]y=x^2[/tex], we get [tex]x^2[/tex] = ±sqrt(1-z), or x = ±sqrt(sqrt(1-z)). So the bounds for x are -sqrt(sqrt(1-z)) to sqrt(sqrt(1-z)).
Now we can set up the triple integrals in the three orders:
Note that the order of integration does not affect the final answer, but may affect the complexity of the integrals.
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I do not understand how to get b and what if i have to get c?
The value of b is given as follows:
b = 5.
How to define a linear function?The slope-intercept representation of a linear function is given by the equation presented as follows:
y = mx + b
The coefficients of the function and their meaning are described as follows:
m is the slope of the function, representing the change in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, which is the initial value of the function, i.e., the numeric value of the function when the input variable x assumes a value of 0. On a graph, it is the value of y when the graph of the function crosses the y-axis.When two lines are parallel, they have the same slope, hence:
4x + 5y = 1
5y = -4x + 1
y = -4x/5 + 1.
Hence:
y = -4x/5 + b.
When x = 4, y = 3, hence the intercept is given as follows:
3 = -16/5 + b
b = 31/5
Hence, in standard format, the equation will be given as follows:
y = -4x/5 + 31/5
4x/5 + y = 31/5
4x + 5y = 31
Meaning that the value of b is of 5.
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find the mean (i.e. expected value) of the random variable x associated with the probability density function over the indicated interval. f(x) = 1 72 x2; [0, 6]
The mean (expected value) of the random variable x associated with the probability density function f(x) = (1/72)x^2 over the interval [0, 6] is 4.5.
To find the mean (expected value) of the random variable x associated with the probability density function f(x) = 1/72 x^2 over the interval [0, 6], we use the formula:
E(x) = ∫[0,6] x f(x) dx
= ∫[0,6] x (1/72 x^2) dx
= (1/72) ∫[0,6] x^3 dx
= (1/72) [(1/4) x^4] [0,6]
= (1/72) [(1/4) (6^4 - 0^4)]
= (1/72) (6^4/4)
= (1/72) (324)
= 4.5
To find the mean (expected value) of the random variable x associated with the probability density function f(x) = (1/72)x^2 over the interval [0, 6], we need to integrate the product of x and the probability density function over the given interval.
Mean (expected value) = E(x) = ∫(x * f(x)) dx, over the interval [0, 6]
E(x) = ∫(x * (1/72)x^2) dx from 0 to 6
E(x) = (1/72) * ∫(x^3) dx from 0 to 6
Now, integrate x^3 with respect to x:
E(x) = (1/72) * (x^4 / 4) | from 0 to 6
Now, evaluate the integral at the limits:
E(x) = (1/72) * ((6^4 / 4) - (0^4 / 4))
E(x) = (1/72) * (1296 / 4)
E(x) = (1/72) * 324
Finally, multiply the result:
E(x) = 4.5
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jane is eight years older than amy. if amy is now twice as old as jane was at one-third jane's current age, how old is jane now?
Currently Jane is 24 years old. Let's start by using variables to represent the ages of Jane and Amy. Let j be Jane's current age and a be Amy's current age.
From the first sentence of the problem, we know that j = a + 8. Now, let's focus on the second sentence of the problem. It says that Amy is now twice as old as Jane was at one-third Jane's current age.
Let's break down this sentence into smaller pieces. "Jane was at one-third Jane's current age" means that Jane's age at that time was j/3. "Amy is now twice as old as Jane was at one-third Jane's current age" means that:
a = 2(j/3)
3/2 × a = j
Now we have two equations that relate the ages of Jane and Amy:
j = a + 8
3/2 × a = j
We can substitute the first equation into the second equation to get an equation that only has one variable:
1/2 × a = 8
a = 16
So Amy = 16 years old. We can use the first equation to find Jane's age:
j = a + 8
j = 16 + 8
j = 24
Currently Jane is 24 years old.
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Pls help! I need to find the angle measures for questions 14-17.
Answer:
3
Step-by-step explanation:
gd=14cm
dc=17cm
then,
gd-dc
14cm-17cm
0=14cm-17cm
0=-3
0+3
3
determine whether the series is absolutely convergent, conditionally convergent, or divergent. [infinity] ∑ sin(n) + 3^n
n = 1 a. absolutely convergent b. conditionally convergent c. divergent
The correct answer to the above question is Option C. divergent i.e., The series [infinity] ∑ sin(n) + 3^n is divergent.
To determine the convergence of the series, we need to check both the convergence of sin(n) and 3^n series.
Firstly, the sin(n) series is a divergent oscillating series, which means it does not converge. Secondly, the 3^n series is a divergent geometric series, which means it only converges when |r| < 1, where r is the common ratio. However, in this case, r = 3 which is greater than 1, so the series diverges.Since both series diverge, their sum will also diverge, and the given series is therefore divergent.
In summary, the given series [infinity] ∑ sin(n) + 3^n is divergent as both the sin(n) and 3^n series diverge.
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use the empirical rule to estimate the percentage of cold sufferers who experience symptoms for less than 9.9 days.
If Z is between -1 and 1, then the percentage is within the 68% range. If Z is between -2 and 2, then the percentage is within the 95% range. If Z is between -3 and 3, then the percentage is within the 99.7% range.
To use the empirical rule to estimate the percentage of cold sufferers who experience symptoms for less than 9.9 days, we first need to know the mean (average) and the standard deviation of the data.
Let's assume that the mean (µ) is X days and the standard deviation (σ) is Y days. The empirical rule states that for a normal distribution:
- Approximately 68% of the data falls within 1 standard deviation (σ) of the mean (µ)
- Approximately 95% of the data falls within 2 standard deviations (σ) of the mean (µ)
- Approximately 99.7% of the data falls within 3 standard deviations (σ) of the mean (µ)
Now, we want to estimate the percentage of cold sufferers who experience symptoms for less than 9.9 days. We need to determine how many standard deviations away 9.9 days is from the mean.
To do this, use the formula:
Z = (Observed Value - Mean) / Standard Deviation
Z = (9.9 - X) / Y
Once you calculate the Z score, refer to the empirical rule:
- If Z is between -1 and 1, then the percentage is within the 68% range.
- If Z is between -2 and 2, then the percentage is within the 95% range.
- If Z is between -3 and 3, then the percentage is within the 99.7% range.
Finally, based on the Z score and the empirical rule, you can estimate the percentage of cold sufferers who experience symptoms for less than 9.9 days.
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a = 2.7 cm, b = 12 cm and c = 9.2 cm. If m is the midpoint of SR Calculate the size of angle MwwT (correct to 1 d.p.)
The size of angle MWT is calculated to 1 d.p. to give
37.8 degrees
How to find angle MWTThe size of angle MWT is solved using trigonometry tan
tan (angle MWT) = (distance midpoint of a to edge w) / b
Where distance midpoint of a to edge w is calculated using Pythagoras theorem
(distance midpoint of a to edge w)² = (1/2 a)² + c²
(distance midpoint of a to edge w)² = (1.35)² + 9.2²
distance midpoint of a to edge w = 9.3
tan (angle MWT) = 9.3 / 12
angle MWT = arc tan (9.3/12) = 37.776
angle MWT = 37.8 degrees to 1 d.p.
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Which is equivalent to (x > 5), given that x is a numeric variable. A.(x < 5) B.!(x >= 5) C.!(x <= 5) D.!(x < 5)
The numeric variable equivalent to equivalent to (x > 5) is, !(x < 5). The answer is D.
The original statement is "x > 5". The negation of this statement is "not (x > 5)", which is equivalent to "x <= 5". However, option A is the opposite of the correct answer since it says "x < 5", not "x <= 5". Option B says "not (x >= 5)", which is equivalent to "x < 5", but again, it is not the correct answer since it uses the "not greater than or equal to" symbol.
Option C says "not (x <= 5)", which is equivalent to "x > 5", but this is the opposite of the original statement. Therefore, the correct answer is D. !(x < 5), which is equivalent to "not (x is less than 5)", or "x is greater than or equal to 5". Hence, option D is correct.
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use the alternative form of the derivative to find the derivative at x = c (if it exists). (if the derivative does not exist at c, enter undefined.) f(x) = x3 2x2 9, c = −2
The derivative of f(x) at x = c does not exist.
To find the derivative of f(x) at x = c using the alternative form of the derivative, we first need to calculate the derivative of f(x) with respect to x.
Given that f(x) = x^3 - 2x^2 + 9, we can find the derivative of f(x) using the power rule and the constant multiple rule. The power rule states that the derivative of x^n, where n is a constant, is n*x^(n-1). The constant multiple rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function.
Applying the power rule and constant multiple rule to f(x), we get:
f'(x) = 3x^2 - 4x
Now, we can evaluate f'(x) at x = c, which in this case is x = -2:
f'(-2) = 3(-2)^2 - 4(-2)
= 3(4) + 8
= 12 + 8
= 20
So, the derivative of f(x) at x = -2 is 20. However, we are asked to find the derivative at x = c = -2 using the alternative form of the derivative.
The alternative form of the derivative states that the derivative of a function at a specific point is equal to the limit of the difference quotient as x approaches the given point. In other words, the derivative at x = c is equal to the limit of (f(x) - f(c))/(x - c) as x approaches c.
Substituting c = -2 into the alternative form of the derivative, we get:
f'(-2) = lim(x->-2) (f(x) - f(-2))/(x - (-2))
However, if we try to evaluate this limit, we get an indeterminate form of 0/0. This means that the derivative of f(x) at x = -2 does not exist, as the limit of the difference quotient is undefined. Therefore, the main answer is that the derivative of f(x) at x = c does not exist.
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Prism A and prism B are similar.
Check the picture below.
[tex]\cfrac{1^2}{2^2}=\cfrac{110}{A}\implies \cfrac{1}{4}=\cfrac{110}{A}\implies A=440~in^2[/tex]
A stone is tossed into the air from ground level with an initial velocity of 34 m/s. Its height at time t is h(t) = 34t − 4.9t2 m. Compute the stone's average velocity over the time intervals [3, 3.01], [3, 3.001], [3, 3.0001],and[2.99, 3], [2.999, 3], [2.9999, 3]. (Round your answers to three decimal places.)T interval [3,3.01] [3,3.001] [3,3.0001]
Average Velocity ??? ???? ????
T interval [2.99,3] [2.999,3] [2.9999,3]
Average Velocity ???? ????? ????
Estimate the instataneous velocity v at t=3.
V= _____ m/s
To compute the average velocity over each time interval, we use the formula: average velocity = (h(t2) - h(t1))/(t2 - t1), where h(t) is the height function.
Using the given height function, h(t) = 34t - 4.9t^2, we calculate the average velocities:
1. [3, 3.01]:
Average Velocity = (h(3.01) - h(3))/(3.01 - 3) ≈ -17.147 m/s
2. [3, 3.001]:
Average Velocity = (h(3.001) - h(3))/(3.001 - 3) ≈ -17.194 m/s
3. [3, 3.0001]:
Average Velocity = (h(3.0001) - h(3))/(3.0001 - 3) ≈ -17.199 m/s
4. [2.99, 3]:
Average Velocity = (h(3) - h(2.99))/(3 - 2.99) ≈ -17.243 m/s
5. [2.999, 3]:
Average Velocity = (h(3) - h(2.999))/(3 - 2.999) ≈ -17.205 m/s
6. [2.9999, 3]:
Average Velocity = (h(3) - h(2.9999))/(3 - 2.9999) ≈ -17.200 m/s
To estimate the instantaneous velocity at t=3, observe the average velocities as the time intervals approach t=3:
As the intervals get closer to t=3, the average velocities appear to approach -17.2 m/s. Thus, the estimated instantaneous velocity at t=3 is:
V ≈ -17.2 m/s
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h(x)=3x-5 and g(x)=2x+1 find gh(x)
Required function g(h(x)) is 6 x - 9.
What is Functions?A function is a relationship between a set of outputs referred to as the range and a set of inputs referred to as the domain, with the condition that each input is contain to exactly one output. An input x corresponding to a function f output, which is represented by f(x).
What is Composite Function?We can combine two functions so that the outputs of one function become the inputs of the other if we have two functions is known as composite function . A composite function is defined by this action,that the function g f(x) = g(f(x)) is known as a composite function. This is occasionally referred to as a function of a function. g f can also be written as g o f instead.
We have, h(x)=3 x-5 and g(x)=2 x+1.
So, g(h(x)) = g(3 x - 5) = 2(3 x - 5) + 1 = 6 x - 9.
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1) 2( x + $3.60 ) = $19.40
2) 45.93 + 112 + (−61.24)
3) 20x + 2 > −98
4) 2/5 (4x - 8)
5) On a school field trip, the number of students (y) is always proportional to the number of adults (x). In one group there are 96 students and 8 adults. What is the constant of proportionality between this relationship?
Answer:
1. 2(x+3.60) = 19.40
Divide both sides by 2:
x+3.60 = 9.70
Subtract 3.60 from both sides:
x = 6.10
Answer: 6.10
2. 45.93 + 112 + (−61.24)
45.93 + 112 = 157.93
157.93 - 61.24 = 96.69
Answer: 96.69
3. 20x + 2 > −98
Subtract 2 from both sides:
20x > −100
Divide both sides by 20:
x > −5
Answer: x > −5
4. 2/5 (4x - 8)
= 8x/5 - 16/5
Answer: 8x/5 - 16/5
5. On a school field trip, the number of students (y) is always proportional to the number of adults (x). In one group there are 96 students and 8 adults. What is the constant of proportionality between this relationship?
The constant of proportionality is the number that, when multiplied by the number of adults, gives the number of students. In this case, the constant of proportionality is 96/8 = 12.
Answer: 12
solve the given initial-value problem. x' = 1 2 0 1 − 1 2 x, x(0) = 4 9 x(t)
The solution of the initial-value problem of x'=[1/2 0; 1 -1/2] x is x(t) = [4/3 * e^(t/2); 5/3 * e^t + 8/3 * e^(t/2)].
To solve the given initial value problem x'=[1/2 0; 1 -1/2] x with x(0)=[4;9], we need to find the solution of the system of differential equations.
The characteristic equation of the matrix [1/2 0; 1 -1/2] is λ^2 - (3/2)λ + (1/4) = 0, which has two distinct roots, λ_1 = 1/2 and λ_2 = 1.
The general solution of the system is x(t) = c_1 * [1; 2] * e^(λ_1t) + c_2 * [0; 1] * e^(λ_2t), where c_1 and c_2 are constants to be determined using the initial condition x(0) = [4; 9].
Substituting the values of λ_1, λ_2, and x(0) in the above equation, we get c_1 = 4/3 and c_2 = 5/3.
Therefore, the solution of the initial-value problem is x(t) = [4/3 * e^(t/2); 5/3 * e^t + 8/3 * e^(t/2)].
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--The given question is incomplete, the complete question is given
" Solve the given initial-value problem x' is matrix of 2x2 form, x' = [1/2 0 1 −1/2] x, x(0) = [4 9] of 2x1 matrix form. find x(t)"--
without solving for the de, describe the spring system y'' 8y' 16y=0
The given differential equation y'' + 8y' + 16y = 0 represents a damped spring system with a damping coefficient of 8 and a spring constant of 16.
To describe the spring system represented by the differential equation y'' + 8y' + 16y = 0, we will be using the given terms.
1. Differential equation (DE): The given DE is a second-order linear homogeneous differential equation with constant coefficients. It represents the motion of a damped spring system, where y'' denotes the acceleration, y' denotes the velocity, and y denotes the displacement of the mass.
2. Damping: The term 8y' represents the damping in the spring system. It is proportional to the velocity (y') of the mass, and acts to oppose the motion, thus slowing down the oscillation.
3. Spring constant: The term 16y represents the restoring force exerted by the spring, which is proportional to the displacement (y) of the mass. The spring constant is 16.
4. Natural frequency: The natural frequency of the spring system can be found by considering the undamped case (i.e., without the 8y' term). In this case, the DE becomes y'' + 16y = 0. The natural frequency (ω_n) can be calculated as the square root of the spring constant divided by the mass (ω_n = √(k/m)). We don't have the mass value, so we can only state that ω_n = √(16/m).
5. Damping coefficient: The damping coefficient is the constant proportionality factor for the damping term. In this case, it is 8.
6. Damped frequency: Damped frequency (ω_d) is the frequency of oscillation when damping is present. It can be found using the natural frequency and the damping ratio (ζ). However, we do not have enough information to calculate the damping ratio or the damped frequency in this case.
In summary, the given differential equation y'' + 8y' + 16y = 0 represents a damped spring system with a damping coefficient of 8 and a spring constant of 16. The natural frequency depends on the mass, but the damped frequency cannot be calculated without additional information.
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What is the value of n if the equation n*y^2+ 2y − 4 = 0 has exactly one root?
Answer:
0
Step-by-step explanation:
ny^2 + 2y - 4 = 0
ny^2 + 2y = 4
y(ny + 2) = 4
y = 4
ny + 2 = 4
ny = 2, 0 = 2
The only possible solution to make this expression incorrect is if 0 = 2, so n is equal to 0.
find the length of the curve y =x4 for 0≤ x ≤1. round your answer to 3 decimal places if needed.
Only use numerical characters and decimal point
where needed. i.e. Enter the number without any
units, commas, spaces or other characters.
The length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.
To find the length of the curve y = x^4 for 0≤ x ≤1, you'll need to use the arc length formula:
Arc length = ∫√(1 + (dy/dx)^2) dx from a to b, where a = 0 and b = 1.
First, find the derivative of y with respect to x:
y = x^4
dy/dx = 4x^3
Now, square the derivative and add 1:
(4x^3)^2 + 1 = 16x^6 + 1
Next, find the square root of the result:
√(16x^6 + 1)
Now, integrate the expression with respect to x from 0 to 1:
∫(√(16x^6 + 1)) dx from 0 to 1
Unfortunately, this integral doesn't have a closed-form solution, so we'll need to use numerical methods, such as Simpson's rule or a numerical integration calculator, to approximate the length.
Using a numerical integration calculator, the length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.
Your answer: 1.082
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The length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.
To find the length of the curve y = x^4 for 0≤ x ≤1, you'll need to use the arc length formula:
Arc length = ∫√(1 + (dy/dx)^2) dx from a to b, where a = 0 and b = 1.
First, find the derivative of y with respect to x:
y = x^4
dy/dx = 4x^3
Now, square the derivative and add 1:
(4x^3)^2 + 1 = 16x^6 + 1
Next, find the square root of the result:
√(16x^6 + 1)
Now, integrate the expression with respect to x from 0 to 1:
∫(√(16x^6 + 1)) dx from 0 to 1
Unfortunately, this integral doesn't have a closed-form solution, so we'll need to use numerical methods, such as Simpson's rule or a numerical integration calculator, to approximate the length.
Using a numerical integration calculator, the length of the curve y = x^4 for 0≤ x ≤1 is approximately 1.082.
Your answer: 1.082
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At the city museum, child admission is $6.10 and adult admission is $9.90. On Friday, four times as many adult tickets as child tickets were sold, for a total sales of $1188.20. How many child tickets were sold that day?
Answer: 26 child tickets were sold that day.
Step-by-step explanation:
Let's say the number of child tickets sold is "x".
According to the problem, the number of adult tickets sold is four times the number of child tickets sold. So, the number of adult tickets sold would be 4x.
6.10x + 9.90(4x) = 1188.20
6.10x + 39.60x = 1188.20
45.70x = 1188.20
x = 26