The correct answer is, [–5, 3]. In the other words, the interval in which the function [tex]f(x) = \sqrt{x^2 + 2x - 15}[/tex] is increasing is [–5, 3].
To determine the interval in which the radical function [tex]f(x) = \sqrt{x^2 + 2x - 15}[/tex] is increasing, we need to find the interval(s) where the derivative of the function is positive.
Let's first find the derivative of f(x):
[tex]f'(x) = (1/2) * (x^2 + 2x - 15)^(-1/2) * (2x + 2)[/tex]
To find where f'(x) > 0, we set f'(x) = 0 and solve for x:
[tex](1/2) * (x^2 + 2x - 15)^(-1/2) * (2x + 2) = 0[/tex]
Since the derivative is never equal to zero (since the denominator (x^2 + 2x - 15)^(-1/2) is never equal to zero), there are no critical points.
To determine the intervals of increase, we can evaluate f'(x) at test points in each interval. We'll consider the intervals defined by the given answer choices:
[3, ∞):
Choose a test point x > 3, let's say x = 4.
Evaluate [tex]f'(4) = (1/2) * (4^2 + 24 - 15)^{(-1/2)} * (24 + 2)[/tex]
[tex]= (1/2) * (16 + 8 - 15)^{(-1/2)} * 10[/tex]
[tex]= (1/2) * (9)^{(-1/2)} * 10[/tex]
= (1/2) * (1/3) * 10
= 5/3
Since f'(4) > 0, the function is increasing in the interval [3, ∞).
(4, ∞):
Choose a test point x > 4, let's say x = 5.
Evaluate f'(5) = (1/2) * (5^2 + 25 - 15)^(-1/2) * (25 + 2)
= (1/2) * (25 + 10 - 15)^(-1/2) * 12
= (1/2) * (20)^(-1/2) * 12
Since f'(5) = 0, the function is not increasing in the interval (4, ∞).
[–5, 3]:
Choose a test point x in the interval, let's say x = 0.
Evaluate [tex]f'(0) = (1/2) * (0^2 + 20 - 15)^{(-1/2)} * (20 + 2)[/tex]
[tex]= (1/2) * (-15)^{-1/2} * 2[/tex]
[tex]= (1/2) * (1/\sqrt{15}) * 2[/tex]
[tex]= 1/\sqrt{15}[/tex]
Since f'(0) > 0, the function is increasing in the interval [–5, 3].
(–∞, –5] ∪ [3, ∞):
Since we have already determined the function is increasing in [–5, 3] and [3, ∞), this interval is valid.
Therefore, the correct answer is, [–5, 3]. In the other words, the interval in which the function [tex]f(x) = \sqrt{x^2 + 2x - 15}[/tex] is increasing is [–5, 3].
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which of the following key characteristics is not true of a quadratic function
Here are the characteristics of Quadratic Functions
Axis of symmetry
x and y-intercepts
Zeroes
vertex
Point symmetric to y-intercept
use the chain rule to find dz/dt. z = xy7 − x2y, x = t2 1, y = t2 − 1
If z = xy⁷ − x²y, x = t² + 1, y = t² − 1, then using chain rule the value of dz/dt = 2t(y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)) + 2t(xy⁷ + 7xy⁶ - x²)
To find dz/dt using the chain rule, we need to differentiate z with respect to t while considering the relationship between z, x, y, and t.
Given:
z = xy⁷ − x²y
x = t² + 1
y = t² − 1
We can start by finding dz/dx and dz/dy separately and then use the chain rule to combine them to find dz/dt.
First, let's find dz/dx:
Differentiating z with respect to x, keeping y constant:
dz/dx = (d/dx)(xy⁷) - (d/dx)(x²y)
Using the product rule for differentiation:
dz/dx = y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)
Next, let's find dz/dy:
Differentiating z with respect to y, keeping x constant:
dz/dy = (d/dy)(xy⁷) - (d/dy)(x²y)
Using the product rule again:
dz/dy = x(y⁷ + 7y⁶(dy/dy)) - x²
Since dy/dy = 1, we can simplify dz/dy as:
dz/dy = xy⁷ + 7xy⁶ - x²
Now, let's use the chain rule to find dz/dt:
dz/dt = (dz/dx)(dx/dt) + (dz/dy)(dy/dt)
Substituting the expressions we found earlier for dz/dx and dz/dy, and using dx/dt = 2t and dy/dt = 2t:
dz/dt = (y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx))(2t) + (xy⁷ + 7xy⁶ - x²)(2t)
Simplifying this expression gives us dz/dt in terms of t, x, and y:
dz/dt = 2t(y⁷ + xy⁶(dy/dx) - 2xy - x²(dy/dx)) + 2t(xy⁷ + 7xy⁶ - x²)
This is the complete calculation for finding dz/dt using the chain rule.
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Complete question is:
Use the chain rule to find dz/dt. z = xy⁷ − x²y, x = t² + 1, y = t² − 1
Find the Inverse of the following function.
Answer: A
Step-by-step explanation:
According to the Chronicles of Contrived Statistics (March, 2016), the probability that Elon Musk will offer you a free trip into space in the next month is 41%. The probability that scientists from Area 51 will start selling pet aliens next month is 27%. Finally, the probability that either Elon Musk will offer you a free trip to space or that scientists will begin sales of pet aliens next month is 24%. What is the probability that both Elon Musk will offer you a free trip into space and scientists from Area 51 will start selling pet aliens in the next month?
Answer:
[tex]P(A\ and\ B) = 44\%[/tex]
Step-by-step explanation:
Given
Represents the event as follows:
A = [tex]Elon\ Musk[/tex] will offer you a [tex]free\ trip[/tex] into space in the next month\
B = Scientists from [tex]Area\ 51[/tex] will [tex]start\ selling[/tex] pet aliens next month
So, we have:
[tex]P(A) = 41\%[/tex]
[tex]P(B) = 27\%[/tex]
[tex]P(A\ or\ B) = 24\%[/tex]
Required
Determine [tex]P(A\ and\ B)[/tex]
This is calculated as:
[tex]P(A\ and\ B) = P(A) + P(B) - P(A\ or\ B)[/tex]
So, we have:
[tex]P(A\ and\ B) = 41\% + 27\% - 24\%[/tex]
[tex]P(A\ and\ B) = 44\%[/tex]
A researcher interested in the age at which women have their first child surveyed a simple random sample of 250 women who have one child and found an approximately normal distribution with a mean age of 27.3 and a standard deviation of 5.4. According to the Empirical rule, approximately 95% of women had their first child between the ages of a. 11.1 years and 43.5 years b. 16.5 years and 38.1 years C. 21.9 years and 32.7 years d. 25.0 years and 29.6 years
According to the Empirical rule, approximately 95% of women had their first child between the ages of 16.5 years and 38.1 years. The correct option is b. 16.5 years and 38.1 years.
The given question states that a researcher interested in the age at which women have their first child surveyed a simple random sample of 250 women who have one child and found an approximately normal distribution with a mean age of 27.3 and a standard deviation of 5.4. We are to find the range of age at which approximately 95% of women had their first child, according to the empirical rule.
There are three ranges for the empirical rule as follows:
Approximately 68% of the observations fall within the first standard deviation from the mean.
Approximately 95% of the observations fall within the first two standard deviations from the mean.
Approximately 99.7% of the observations fall within the first three standard deviations from the mean.
Now, we will apply the empirical rule to find the age range at which approximately 95% of women had their first child. The mean age is 27.3 years and the standard deviation is 5.4 years, hence:
First, find the age at which 2.5% of women had their first child:
µ - 2σ = 27.3 - (2 × 5.4) = 16.5
Then, find the age at which 97.5% of women had their first child:
µ + 2σ = 27.3 + (2 × 5.4) = 38.1
Therefore, approximately 95% of women had their first child between the ages of 16.5 years and 38.1 years. Hence, the correct answer is option b. 16.5 years and 38.1 years.
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Point S is on line segment RT. Given ST=3 and RT=20, determine the length RS.
Answer:
RS = 17
Step-by-step explanation:
Given S lies on RT , then
RS + ST = RT , that is
RS + 3 = 20 ( subtract 3 from both sides )
RS = 17
Which of the equations below could be the equation of this parabola?
(0,0)
Vertex
O A. y = 2x2
B. x = -2y2
O c. x = 2y2
O D. y= -2
Answer:
Y=2x^2
Step-by-step explanation:
The following equations represent a parabola with vertex (0,0): y=2x², x=-2y² and x=2y².
Quadratic functionThe quadratic function can be represented by a quadratic equation in the Standard form: ax²+bx+c=0 where: a, b and c are your respective coefficients. In the quadratic function the coefficient "a" must be different than zero (a≠0) and the degree of the function must be equal to 2.
A parabola also can be represented by a quadratic equation. The vertex of an up-down facing parabola of the form ax²+bx+c is [tex]x_v=\frac{-b}{2a}[/tex] . Knowing the x-coordinate of vertex, you can find the y-coordinate of vertex.
The another form for describing a parabola is [tex]4p\left(x-h\right)=\left(y-k\right)^2[/tex], where h and k are the vertex coordinates.
You should analyse each one of the options, considering the equations that can be represented a parabola.
Letter A - y=2x²
The coefficients of the quadratic equation are:
a=2, b=0, c=0
Then,
[tex]x_v=\frac{-b}{2a}\\ \\ x_v=\frac{-0}{2*2}=0[/tex].
If x-coordinate of vertex is equal to 0, from y=2x²you can:
[tex]y_v=2x^2\\ \\ y_v=2*0^2=0[/tex]
Therefore, the given equation ( y=2x²) represents a parabola with vertex (0,0).
Letter B - x=-2y²
From equation [tex]4p\left(x-h\right)=\left(y-k\right)^2[/tex], you can rewrite the given equation parabola for vertex (0,0) in:
[tex]4p(x-0)=(y-0)^2\\ \\ 4*\frac{-1}{8} x=y^2\\ \\ \frac{-1}{2} x=y^2\\ \\ x=-2y^2[/tex]
Therefore, the given equation ( x=-2y²) represents a parabola with vertex (0,0).
Letter C - x=2y²
From equation [tex]4p\left(x-h\right)=\left(y-k\right)^2[/tex], you can rewrite the given equation parabola for vertex (0,0) in:
[tex]4p(x-0)=(y-0)^2\\ \\ 4*\frac{1}{8} x=y^2\\ \\ \frac{1}{2} x=y^2\\ \\ x=2y^2[/tex]
Therefore, the given equation ( x=2y²) represents a parabola with vertex (0,0).
Letter D - y=-2
The degree of equation is not equal 2. Therefore, it does not represent a parabola.
Only the equations of letter A, B and C represent a parabola with vertex (0,0). See the attached image.
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Determine the range of the function [tex]\displaystyle f(x)=a\sqrt[3]{bx-c}+d[/tex].
Hi there!
[tex]\large\boxed{(-\infty, \infty)}}[/tex]
We are given the function:
[tex]f(x) = a\sqrt[3]{bx-c}+d[/tex]
This is the transformation form of a cubic root function. Recall these properties of cubic root functions:
Domain: -∞ < x < ∞ (all real numbers)
Range: -∞ < x < ∞ (all real numbers)
Therefore, the range of the given function is all real numbers, or on (-∞, ∞).
Let X be a Markov chain with transition probability matrix 0 1 2 0 0.7 0.2 0.1 P= 1 0.3 0.5 0.2 2 0 0 1 The Markov chain starts at time zero in state Xo = 0. Let T = min{n > 0: X = 2} be the first time that the process reaches state 2. Eventually, the process will reach and be absorbed into state 2. If in some experiment we observed such a process and noted that absorption had not yet taken place, we might be interested in the conditional probability that the process is in state 0 (or 1), given that absorption had not yet taken place. Determine P(X3 = 0 T > 3).
The conditional probability that the process is in state 0 (or 1), given that absorption had not yet taken place. Therefore, P(X3 = 0, T > 3) = 0.075.
The Markov chain starts at time zero in state Xo = 0 and the transition probability matrix of Markov Chain is as follows: 0 1 2 0 0.7 0.2 0.1 P= 1 0.3 0.5 0.2 2 0 0 1
P(X3 = 0, T > 3).We know that the probability of moving from state i to state j in two steps is given by P2(i, j).
Thus, the probability of moving from state i to state j in three steps is given by P3(i, j). We have P2(i, j) = P(i, ·)P(j, ·) = Σk P(i, k)P(k, j).
For a 3-step transition probability, we use the equation P3 = P2P = P2 (P2) and so on. Therefore,P3(1, 2) = P2(1, 1)P(1, 2) + P2(1, 2)P(2, 2) + P2(1, 3)P(3, 2) = (0.7)(0.3) + (0.2)(0.5) + (0.1)(0) = 0.235
Similarly,P3(1, 0) = P2(1, 0)P(0, 0) + P2(1, 1)P(1, 0) + P2(1, 2)P(2, 0) = (0)(0.7) + (0.3)(0) + (0.235)(0.2) = 0.047
Since we are interested in finding P(X3 = 0, T > 3), we need to find the probability that absorption had not yet taken place at time 3 and that the process is in state 0 at time 3, which can be expressed as:
P(X3 = 0, T > 3) = P(X3 = 0, X4 ≠ 2) = P(X3 = 0, X4 = 0) + P(X3 = 0, X4 = 1)
We know that T is the first time that the process reaches state 2 and the process will reach and be absorbed into state 2.
Thus, T is the absorption time for state 2 and it has a geometric distribution with parameter P2(2, 2) = 1, which implies that P(T = t) = (1 – 1)P2(2, 2) = 0 for all t < 1.
Therefore, P(X3 = 0, X4 = 0) = P(X3 = 0, X4 = 0, T > 3)
= P(X3 = 0, T > 3)P(X4 = 0 | X3 = 0, T > 3) = P(X3 = 0, T > 3)P(0, 0) / P(X3 = 0, T > 3)P(0, 0) + P(X3 = 1, T > 3)
P(1, 0) + P(X3 = 2, T > 3)
P(2, 0) = (0.047)(1) / [(0.047)(1) + (0.235)(0.7) + (0)(0.2)]
= 0.067
Therefore, P(X3 = 0, T > 3) = P(X3 = 0, X4 = 0) + P(X3 = 0, X4 = 1) = (0.067)(0.7) + (0.235)(0.2) = 0.075.
Therefore, P(X3 = 0, T > 3) = 0.075.
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A cylindrical canister has a radius of 7 cm and a height of 16 cm. Find the volume of
the canister.
pls show your work thx pls help me will give brainly.
Answer:
7/16 = 0.4375
Step-by-step explanation:
sry
A recent survey of the alumni of a university indicated that the average salary of 10,000 of its 200,000 graduates was $130,000. The $130,000 would be considered a: a. Population. b. Parameter. c. Sample. d. Statistic.
The $130,000 would be considered as Statistic.
In statistics, a population refers to the entire group of individuals or items of interest, while a sample is a subset of the population. A parameter is a numerical value that describes a characteristic of a population.
In this scenario, the survey results are based on a sample of 10,000 graduates out of a total population of 200,000 graduates. The average salary of $130,000 is calculated from the data collected within this sample. Since it is derived from the sample, it is considered a statistic.
A parameter would be used to describe the average salary of the entire population of 200,000 graduates if data were collected from all of them. However, in this case, the given information only pertains to the subset of the sample.
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2
When a set of data has an even member the median is found by:
(1 Point)
O adding the two middle numbers
finding the mean of the two numbers at the end.
O finding the mean of the middle members
O choosing the number in the middle
Answer:
It is 1, or A: adding the two middle numbers
Step-by-step explanation:
When you think of it, here's an example:
15
23
55
34
Add 23 and 55:
23+55=78
We've got this now:
15
78
34
Now, all you've gotta do is find the mediam.
Hope this helped, and please mark as Brainliest <3
Which measure is equivalent to 65 kilograms?
659
6509
65000
65,0000
Answer:
65,000 grams
Step-by-step explanation:
Because 1000 grams is in 1 kilogram, you would multiply the 65 kilograms by 1,000 to convert it to grams.
So to convert 65 kilograms to grams you would do:
65 * 1,000 = 65,000 grams
A pentagon has angles that measure
80°, 90°, 130°, 120°, and t. What is t?
t=
Answer:
120°
Step-by-step explanation:
To find the measure of angle t in the pentagon, we can use the property that the sum of the interior angles of a pentagon is equal to 540 degrees.
So, we can set up an equation:
80° + 90° + 130° + 120° + t = 540°
Simplifying the equation:
420° + t = 540°
Next, we can isolate the variable t by subtracting 420° from both sides of the equation:
t = 540° - 420°
t = 120°
Therefore, the measure of angle t in the pentagon is 120 degrees.
Hope this helps!
Expert Answer Please Find The Area Of A Circle With R=13.8 Explain Answer!
Answer:
if r= to 13.8 so the answer is 598.28
When you find volume of a 3-D shape, you are finding the measurement of the outside of the shape.
Group of answer choices
True
False
Find the length of side x in simplest radical form with a rational denominator
Answer:
x = 8/√3
Step-by-step explanation:
We solve that above question using the trigonometric function of sin
Sin theta = Opposite/Hypotenuse
Theta = 60°
Opposite = 4
Hypotenuse = x
Hence,
Sin 60 = 4/x
Cross Multiply
sin 60 × x = 4
x= 4/sin 60
in rational form
sin 60 = √3/2
Hence, x = 4/ √3/2
= 4 ÷ √3/2
= 4 × 2/√3
= 8/√3
help me please it’s math
Answer:
1) 345 2) 62 3) 36 4) 51 5) 96 6) 142
Step-by-step explanation:
I hope you and your 3rd or 4th grader self enjoy the answers i gave you.
HELPP PLSSSS NO BOTS OR I WILL REPORT!!!
Answer:
B. false
Step-by-step explanation:
Which situation would not be represented by the integer -5?
There is a temperature of 5°.
You owe $5.
There is a hole 5 feet deep.
There is a 5-yard penalty.
Answer:
A
Step-by-step explanation:
It is talking about a positive number
Answer:
a
Step-by-step explanation:
Which of the following is a solution to the differential equation xy′−3y=6xy′−3y=6 ?
A solution to the differential equation xy' - 3y = 6 is y = -2/x. This is a particular solution that satisfies the given differential equation. Therefore, y = -2/x is a solution to the differential equation xy' - 3y = 6.
To find a solution to the differential equation xy' - 3y = 6, we need to solve the equation and find a function that satisfies it. We can begin by rearranging the equation:
xy' - 3y = 6
To solve this linear first-order ordinary differential equation, we can use the method of integrating factors. The integrating factor is given by the exponential of the integral of the coefficient of y, which in this case is -3:
IF = e^(-3x)
Multiplying both sides of the equation by the integrating factor, we have:
e^(-3x)xy' - 3e^(-3x)y = 6e^(-3x)
This can be rewritten as:
(d/dx)(e^(-3x)y) = 6e^(-3x)
Integrating both sides with respect to x, we get:
e^(-3x)y = ∫(6e^(-3x))dx
Simplifying the integral and applying the constant of integration, we have:
e^(-3x)y = -2e^(-3x) + C
Dividing both sides by e^(-3x), we obtain:
y = -2 + Ce^(3x)
The constant C can take any value. By choosing C = 0, we have y = -2/x, which is a particular solution to the given differential equation. Therefore, y = -2/x is a solution to the differential equation xy' - 3y = 6.
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I need help pls is due today
Answer:
a) 1/2, 0.5, 50% b)3/4, 0.75, 75% c) 1/12, 0.08333, 8.333% d) 5/12, 41.6667%, 0.416
Step-by-step explanation:
Answer:
a] 1/2 b] 5/6 c] 1/6 d] 5/4
Step-by-step explanation:
no of even no present/total numbers and so on
try it later
Which of the following is not determined in the case of multiple linear regression?
Question 4 options:
1)Potential confounding effects from "lurking" variables included in the analysis
2)Observed relationship between predictor and outcome variables adjusting for confounding relationships
3)Causal relationships between predictor and outcome variables adjusting for confounding relationships
4) Potential observed interaction effects between variables Save
It does not determine the observed interaction effects between variables.
In the case of multiple linear regression, the option that is not determined is:
Variable interactions that have been observed to have effects.
Option 1: Multiple linear regression can estimate the observed relationship between predictor and outcome variables after adjusting for confounding relationships and option 2: it can determine the potential confounding effects from lurking variables. Furthermore, numerous direct relapse can evaluate the causal connections among indicator and result factors adapting to jumbling connections (choice 3).
However, the potential observed interaction effects between variables cannot be determined by multiple linear regression on its own. Association impacts happen when the connection between two indicators and the result variable isn't added substance however relies upon the mix of the indicator values. In order to evaluate the effects of interactions, additional analysis methods like including interaction terms and carrying out specific interaction tests are required.
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PLEASE HELP ASAP!!!!!!!!!!!!!!!
Answer:
the answer is D
Step-by-step explanation:
y=[tex]x^{2}[/tex]+1
half of 1/2 cake = _____ cake.
Answer:
well
Step-by-step explanation:
please add the image
Applying the knowledge of fractions, half of 1/2 cake = 1/4 cake.
Multiplying FractionsWe are asked to find how many parts of are there in half of one-half of a cake.
Thus:
half = 1/2
1/2 of 1/2 cake = 1/2 × 1/2 (of = multiplication)
= (1 × 1)/(2 × 2)
= 1/4
Therefore, applying the knowledge of fractions, half of 1/2 cake = 1/4 cake.
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Please please help please
Yikes i dont know this but thanks for the coins C:
PLZZZ HELP MEE BIG BRAIN PLP PLZZ
Represent each linear situation with an equation in slope-intercept form.
A family bucket meal at Chicken Deluxe costs twenty-six dollars plus $1.50 for every extra piece of chicken added to the bucket.
Answer:
y = 1.5x + 26Step-by-step explanation:
Total cost is $26 add x-pieces each $1.50:
y = 1.5x + 26Find the flux of the field F=axi-3yj F outward across the ellipse: x-cost, y - 4 sint ostaan Use Green's thm. to find the area enclosed by the ellipse; x = a cose, yabsine.
Using Green's theorem the flux of the field F across the ellipse is 0, indicating that there is no net flow across the ellipse.
To find the flux of the field F = a × i - 3y × j outward across the ellipse, we can use Green's theorem. Green's theorem relates the flux of a vector field across a closed curve to the circulation of the field around the curve.
Let's denote the ellipse as C, which is parameterized by x = a cos(t) and y = b sin(t), where a and b are the semi-major and semi-minor axes of the ellipse, and t varies from 0 to 2π.
Calculate the curl of the vector field F:
∇ × F = (∂Fₓ/∂y - ∂Fᵧ/∂x) k
= (-3)k
Determine the area enclosed by the ellipse using Green's theorem:
The flux of F across the ellipse is equal to the circulation of F around the ellipse:
∮C F · dr = ∬R (∇ × F) · dA
Since the curl of F is -3k, the flux simplifies to:
∮C F · dr = ∬R (-3k) · dA
= -3 ∬R dA
= -3A
Therefore, the flux of F across the ellipse is -3 times the area enclosed by the ellipse.
Find the area enclosed by the ellipse:
The equation of the ellipse is given as x = a cos(t) and y = b sin(t).
To find the limits of integration, we note that t varies from 0 to 2π, which represents one complete revolution around the ellipse.
∬R dA = ∫₀²π ∫₀²π (a cos(t))(b) dt
= ab ∫₀²π cos(t) dt
= ab [sin(t)]₀²π
= ab (sin(2π) - sin(0))
= 0
Therefore, the area enclosed by the ellipse is 0.
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find an equation of the tangent plane to the given parametric surface at the specified point. x = u + v, y = 5u², z = u − v; (2, 5, 0)
The equation of the tangent plane to the parametric surface at the point (2, 5, 0) is x + 20y + z - 102 = 0.
To find the equation of the tangent plane to the given parametric surface at the point (2, 5, 0), we need to compute the partial derivatives and evaluate them at the given point.
The parametric surface is defined by the equations:
x = u + v
y = 5u^2
z = u - v
First, we find the partial derivatives with respect to u and v:
∂x/∂u = 1
∂x/∂v = 1
∂y/∂u = 10u
∂y/∂v = 0
∂z/∂u = 1
∂z/∂v = -1
Next, we evaluate the partial derivatives at the given point (2, 5, 0):
∂x/∂u = 1
∂x/∂v = 1
∂y/∂u = 10u = 10(2) = 20
∂y/∂v = 0
∂z/∂u = 1
∂z/∂v = -1
At the point (2, 5, 0), the partial derivatives are:
∂x/∂u = 1
∂x/∂v = 1
∂y/∂u = 20
∂y/∂v = 0
∂z/∂u = 1
∂z/∂v = -1
The equation of the tangent plane can be written as:
(x - x₀) (∂x/∂u) + (y - y₀) (∂y/∂u) + (z - z₀) (∂z/∂u) = 0,
where (x₀, y₀, z₀) is the given point.
Substituting the values, we have:
(x - 2)(1) + (y - 5)(20) + (z - 0)(1) = 0.
Simplifying further, we get:
x - 2 + 20(y - 5) + z = 0.
Expanding and rearranging the terms, the equation of the tangent plane to the parametric surface at the point (2, 5, 0) is:
x + 20y + z - 102 = 0.
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Suppose that the mean score for a critical reading test is 580 with a population standard deviation of 115 points. What is the probability that a random sample of 500 students will have a mean score of more than 590? Less than 575? Solve using Excel.
the mean score for a critical reading test, using Excel, the probability that a random sample of 500 students will have a mean score of more than 590 can be calculated to be approximately 0.408.
To calculate the probabilities using Excel, we can utilize the standard normal distribution. First, we need to convert the sample means to z-scores by using the formula: z = (sample mean - population mean) / (population standard deviation / sqrt(sample size)). For the sample mean of more than 590, we can calculate the probability of z being greater than the corresponding z-score using the formula "=1-NORM.S.DIST(z-score,TRUE)". In this case, the z-score is (590 - 580) / (115 / sqrt(500)), which gives approximately 0.408.
Similarly, for the sample mean of less than 575, we calculate the probability of z being less than the corresponding z-score using the formula "=NORM.S.DIST(z-score,TRUE)". The z-score is (575 - 580) / (115 / sqrt(500)), which gives approximately 0.084.
Therefore, the probability that a random sample of 500 students will have a mean score of more than 590 is approximately 0.408, and the probability that the sample mean is less than 575 is approximately 0.084.
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