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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identify the domain and range of the inverse of f(x) = 0.5x.
The domain and range of the inverse of the function f(x) = 0.5x need to be determined. The domain and range of the inverse function g(x) = 2x are both all real numbers.
To find the domain and range of the inverse of a function, we can swap the roles of x and y in the original function and then solve for y.
The original function is f(x) = 0.5x. Swapping x and y, we get x = 0.5y. Solving this equation for y, we multiply both sides by 2, giving us y = 2x.
The domain of the inverse function, denoted as g(x), is the set of all possible x-values. In this case, the domain of g(x) is the same as the range of the original function f(x). The range of f(x) is all real numbers, so the domain of g(x) is also all real numbers.
Similarly, the range of the inverse function is the set of all possible y-values. In this case, the range of g(x) is the same as the domain of the original function f(x). The domain of f(x) is all real numbers, so the range of g(x) is also all real numbers.
Therefore, the domain and range of the inverse function g(x) = 2x are both all real numbers.
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Calculate the distance between the points K = (1, -1) and P=(9.-6) in the coordinate plane.
Give an exact answer (not a decimal approximation)
Answer:
√89
Step-by-step explanation:
√(x2 - x1)² + (y2 - y1)²
√(9 - 1)² + [-6 - (-1)]²
√(8)² + (-5)²
√64 + 25
√89
use a truth table to determine whether the symbolic form of the argument is valid or invalid ~p -> q
The symbolic argument ~p -> q is valid.
To determine the validity of the argument ~p -> q using a truth table, we need to consider all possible combinations of truth values for p and q and evaluate the truth value of the implication ~p -> q.
The symbolic form of the argument is ~p -> q, which can also be written as ¬p → q.
A truth table for this argument would have columns for p, ~p, q, and ~p -> q.
Let's construct the truth table:
| p | ~p | q | ~p -> q |
| True | False | True | True |
| True | False | False | False |
| False | True | True | True |
| False | True | False | True |
In the truth table, we consider all possible combinations of true (T) and false (F) for p and q. For ~p, we negate the value of p.
For the implication ~p -> q, it is true (T) if either ~p is false (F) or q is true (T). In all other cases, it is false (F).
Looking at the truth table, we can see that in all rows where ~p -> q is true (T), the corresponding conclusion q is true (T). Therefore, the argument ~p -> q is valid because whenever ~p is true (F), the conclusion q is also true (T).
In summary, the symbolic argument ~p -> q is valid.
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Find the square root of -i
that graphs in the second quadrant.
The square root of -i that graphs in the second quadrant,
⇒ z = (1/2)((√(2))(cos(135°) + i sin(135°))
To find the square root of -i,
we can start by writing -i in polar form.
⇒ i = 1(cos(270°) + i sin(270°))
Now, we can find the square root by taking the square root of the magnitude and dividing the angle by 2.
⇒ √(-i) = √(1) [cos(270°/2) + i sin(270°/2)]
⇒ [tex](1)^{(1/2)}[/tex] [cos(135°) + i sin(135°)]
⇒ (1/2)(√(2)) [cos(135°) + i sin(135°)]
⇒ (1/2)(√(2)) (-√(2)/2 + i(√(2))/2)
⇒ -1/2 + ((√t(2)/2)i
Therefore,
The square root of -i that graphs in the second quadrant in the form of
z = r(cosθ + i sinθ)
Hence,
z = (1/2)((√(2))(cos(135°) + i sin(135°)).
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I need somebody to choose a random amount of time in either minutes or hours. Doesn't matter how long. Thanks!
Answer:
69 minutes? i don't know haha
Step-by-step explanation:
Find the volume of the cylinder. Round your answer to the
nearest tenth.
WILL GIVE BRAINLY
Answer:
7 cm
Step-by-step explanation:
10 mm = 1cm
All you need to do is add 10mm + 6cm.
10mm = 1cm
So, 6cm + 1cm.
Therefore, it equals to 7cm.
Answer:
471.2 cm cubed
Step-by-step explanation:
The formula for the volume of a cylinder is:
π*radius squared*height
The diameter is the distance across the circle.
10 mm is the same as 1 cm
The radius is half of the diameter.
1/2=.5
Radius=.5 cm
Height=6 cm
Plug the numbers in.
.5 squared is .25
π*(.25)(6 cm)
Volume=4.71
Rounded to the nearest tenth:
Volume=4.7 cm cubed
Let P(x, y) means x +1> y. Let 2 € Z and y € N, select all the formulas below that are true in the domain. A. Vay P(x,y) B. ByVx P(x,y) C. 3xVy P(x,y) D. Vyc P(x,y) E. 3xVy - P(x, y) F. ByVx - P(x, y) G. Vzy -P(,y) H. -3xVy P(x,y) I. None of the above.
The formulas below that are true in the domain.
The correct answer is (A, B, C, E, F). A. Vay P(x,y) B. ByVx P(x,y) C. 3xVy P(x,y) E. 3xVy - P(x, y) F. ByVx - P(x, y)
Let's evaluate each formula to determine which ones are true in the given domain:
A. Vay P(x, y): This formula states that for all y, there exists an x such that x + 1 > y. Since there is no restriction on x, this formula is true in the given domain.
B. ByVx P(x, y): This formula states that for all x, there exists a y such that x + 1 > y. Since there is no restriction on y, this formula is true in the given domain.
C. 3xVy P(x, y): This formula states that there exists an x such that for all y, x + 1 > y. Since there is no restriction on y, this formula is true in the given domain.
D. Vyc P(x, y): This formula states that for all y, there exists a constant c such that x + 1 > y. However, there is no mention of c in the given domain, so this formula is not true.
E. 3xVy -P(x, y): This formula states that there exists an x such that for all y, x + 1 ≤ y. This is the negation of the original condition x + 1 > y. Since there is no restriction on y, this formula is true in the given domain.
F. ByVx -P(x, y): This formula states that for all x, there exists a y such that x + 1 ≤ y. This is again the negation of the original condition x + 1 > y. Since there is no restriction on x, this formula is true in the given domain.
G. Vzy -P(x, y): This formula states that for all z, there exists a y such that x + 1 ≤ y. However, there is no mention of z in the given domain, so this formula is not true.
H. -3xVy P(x, y): This formula states that there does not exist an x such that for all y, x + 1 > y. Since the original condition x + 1 > y is true for any value of x and y in the given domain, this formula is not true.
Based on the evaluations above, the formulas that are true in the given domain are:
A. Vay P(x, y)
B. ByVx P(x, y)
C. 3xVy P(x, y)
E. 3xVy -P(x, y)
F. ByVx -P(x, y)
Therefore, the correct answer is (A, B, C, E, F).
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The car consumes 6 liters per 100 km. How many kilometers can you drive with this car if the tank has 42 liters?
Answer:
If 100km enables consumption of 6lts
what about 42lts
42/6 multiply by 100km
42/6*100= 700km
The car will consume 42lts in 700 km.
Thank you.
Step-by-step explanation:
A thin wire is bent into the shape of a semicircle
x^2 + y62 = 9, x ≥ 0.
If the linear density is a constant k, find the mass and center of mass of the wire.
The mass of the wire is given by the integral [tex]\int[0, R] k\sqrt{(1 + (-x/\sqrt{(9 - x^2}))^2}[/tex] dx, and the centre of mass is given by [tex]\int[0, R] x(k\sqrt{1 + (-x/\sqrt{9 - x^2})^2}[/tex] dx divided by the mass.
Find the mass and centre of mass of the wire?
To find the mass and center of mass of the wire, we need to integrate the linear density function along the curve of the wire.
The linear density function is given as a constant k, which means the mass per unit length is constant.
To find the mass of the wire, we integrate the linear density function over the length of the wire. The length of the semicircle can be found using the arc length formula:
[tex]s = \int[0, R] \sqrt{(1 + (dy/dx)^2} dx[/tex]
In this case, the equation of the semicircle is x² + y² = 9, so y = √(9 - x²). Taking the derivative with respect to x, we have dy/dx = -x/√(9 - x²).
Substituting this into the arc length formula, we have:
s = ∫[0, R] √(1 + (-x/√(9 - x²))²) dx
To find the centre of mass, we need to find the weighted average of the x-coordinate of the wire. The weight function is the linear density function, which is a constant k.
Therefore, the mass of the wire is given by the integral [tex]\int[0, R] k\sqrt{(1 + (-x/\sqrt{(9 - x^2}))^2}[/tex] dx, and the center of mass is given by [tex]\int[0, R] x(k\sqrt{1 + (-x/\sqrt{9 - x^2})^2}[/tex] dx divided by the mass.
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evaluate the exponent expression for a = –2 and b = 3. question 15 options: a) –9∕8 b) –2∕5 c) –6 d) 3
The correct option is A) 9∕8, evaluating the exponent expression with a = -2 and b = 3, we find that the value is -8.
We are given the expression a^b, where a = -2 and b = 3. Substituting these values into the expression, we have (-2)^3.
To evaluate this expression, we raise -2 to the power of 3. When we raise a negative number to an odd power, the result will be a negative number.
So, (-2)^3 will yield a negative value.
Calculating (-2)^3, we multiply -2 by itself three times: (-2) × (-2) × (-2). This equals -8.
Therefore, the correct option is A) 9∕8 and the value of the exponent expression (-2)^3 is -8.
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Rewrite the decimals as
fractions and percents
.08
Answer:
8/10 or 4/5 as a fraction. the percentage is 80%
Step-by-step explanation:
I need help on this
Answer: 1/9
Step-by-step explanation: 9(1/3)^5-1, which is 9(1/3)^4.
Solve and you get 1/9.
8. Jack is going to paint the ceiling and four
walls of a room that is 10 feet wide, 12 feet
long, and 10 feet from floor to ceiling. How
many square feet will he paint?
(A) 120 square feet
(B) 560 square feet
(C) 680 square feet
(D) 1,200 square feet
Answer:
D
Step-by-step explanation:
I need help on this question it's confusing
Answer:
Well what's the question lol
Answer:
show the question
Step-by-step explanation:
The town of KnowWearSpatial, U.S.A. operates a rubbish waste disposal facility that is overloaded if its 4712 households discard waste with weights having a mean that exceeds 27.22 lb/wk. For many different weeks, it is found that the samples of 4712 households have weights that are normally distributed with a mean of 26.97 lb and a standard deviation of 12.29 lb. What is the proportion of weeks in which the waste disposal facility is overloaded? P(M> 27.22) = Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z- scores or Z-scores rounded to 3 decimal places are accepted. Is this an acceptable level, or should action be taken to correct a problem of an overloaded system? O No, this is not an acceptable level because it is not unusual for the system to be overloaded. O Yes, this is an acceptable level because it is unusual for the system to be overloaded.
The proportion of weeks in which the waste disposal facility is overloaded is approximately 0.4920. No, this is not an acceptable level because it is not unusual for the system to be overloaded.
To solve this problem, we need to find the proportion of weeks in which the waste disposal facility is overloaded, given that the weights of the samples of 4712 households are normally distributed with a mean of 26.97 lb and a standard deviation of 12.29 lb.
Let's denote X as the random variable representing the mean weight of the samples of 4712 households in a week. We want to find P(X > 27.22).
To calculate this probability, we can use the standard normal distribution. First, we need to standardize the random variable X using the z-score formula:
z = (X - μ) / σ
where μ is the mean and σ is the standard deviation.
Substituting the given values:
z = (27.22 - 26.97) / 12.29
z ≈ 0.0203
Next, we can use a standard normal distribution table or a calculator to find the proportion of weeks in which the waste disposal facility is overloaded:
P(X > 27.22) = P(Z > 0.0203)
Looking up the z-score 0.0203 in the standard normal distribution table, we find that the corresponding proportion is approximately 0.4920.
Therefore, the proportion of weeks in which the waste disposal facility is overloaded is approximately 0.4920. This means that it is not unusual for the system to be overloaded.
So the correct answer is:
No, this is not an acceptable level because it is not unusual for the system to be overloaded.
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if -3x + 7 = -8 what does X equal
Answer:
5
Step-by-step explanation:
3x-7=8
x=5
What is the quotient of 58,110 and 65?
Answer:
894
Step-by-step explanation:
58,110 divided by 65 is 894
The quotient of the given numbers 58,110 and 65 would be equal to 894.
What are the Quotients?Quotients are the number that is obtained by dividing one number by another number.
Dividend ÷ Divisor = Quotients
The quotient of the given numbers is 58,110 and 65.
58,110 divided by 65
58,110 /65
= 894
Thus, the quotient of the given numbers is 894.
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Nationwide 13.7% of employed wage and salary workers are union members. At random sample of 200 local wage and salary workers showed that 30 belonged to a union. At 0.01 level of significance, is there sufficient evidence to conclude that the proportion of union members differs from 13.7%?
There is not sufficient evidence to conclude that the proportion of union members differs from 13.7% at the 0.01 level of significance.
To determine if there is sufficient evidence to conclude that the proportion of union members differs from 13.7%, we can perform a hypothesis test using the given sample data.
Let's define the hypotheses:
Null Hypothesis (H0): The proportion of union members is equal to 13.7%.
Alternative Hypothesis (H1): The proportion of union members differs from 13.7%.
We can set up the test using the z-test for proportions. The test statistic is calculated as:
z = (p(cap) - p) / √(p × (1 - p) / n)
where p(cap) is the sample proportion, p is the hypothesized proportion, and n is the sample size.
Given that the sample size is 200 and 30 workers belonged to a union, the sample proportion is p(cap) = 30/200 = 0.15.
The hypothesized proportion is p = 0.137.
Let's calculate the test statistic:
z = (0.15 - 0.137) / √(0.137 × (1 - 0.137) / 200)
z ≈ 1.073
To determine if there is sufficient evidence to conclude that the proportion differs from 13.7%, we compare the test statistic to the critical value.
At a significance level of 0.01, the critical value for a two-tailed test is approximately ±2.576 (obtained from a standard normal distribution table).
Since |1.073| < 2.576, the test statistic does not fall in the rejection region. Therefore, we fail to reject the null hypothesis.
Conclusion: Based on the given sample data, there is not sufficient evidence to conclude that the proportion of union members differs from 13.7% at the 0.01 level of significance.
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g (b) find the amount of salt in the tank after 1.5 hours.a tank contains 90 kg of salt and 1000 l of water. a solution of a concentration 0.045 kg of salt per liter enters a tank at the rate 8 l/min. the solution is mixed and drains from the tank at the same rate what is the concentration of our solution in the tank initially?
The initial concentration of the solution in the tank is 0.036 kg of salt per liter.
Initially, the tank contains 90 kg of salt and 1000 liters of water, resulting in a total volume of 1000 liters. A solution with a concentration of 0.045 kg of salt per liter enters the tank at a rate of 8 liters per minute. Since the solution is mixed and drains from the tank at the same rate, the concentration remains constant throughout the process.
To find the initial concentration, we can calculate the amount of salt in the tank after a certain time period. After 1.5 hours, the solution has been entering and draining from the tank for 90 minutes (1.5 hours * 60 minutes/hour). During this time, the total volume of the solution that has entered and drained is 90 minutes * 8 liters/minute = 720 liters.
The amount of salt that has entered the tank is 720 liters * 0.045 kg/liter = 32.4 kg. Since the initial amount of salt in the tank was 90 kg, the amount of salt remaining after 1.5 hours is 90 kg - 32.4 kg = 57.6 kg.
To find the concentration, we divide the remaining amount of salt (57.6 kg) by the remaining volume of the solution (1000 liters - 720 liters = 280 liters). The concentration of the initial solution in the tank is 57.6 kg / 280 liters ≈ 0.206 kg of salt per liter.
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Simplify the expression e3 x e9
Answer:
e^12
Step-by-step explanation:
Write the problem as a mathematical expression.
e^3 ⋅ e^9
Use the power rule and combine the exponents.
e^3 + 9^e
Add 3 and 9.
e^12
The result can be shown in multiple forms.
Exact Form:
e^12
If not the answer
The Decimal Form:
162754.79141900…
What is the slope of the line containing points
A(4, -1) and B(0, 2)?
A.3/4
B. 4/3
C. -3/4
D. -4/3
Gabriel has these cans of soup in his kitchen cabinet.
• 2 cans of tomato soup
• 3 cans of chicken soup
• 2 cans of cheese soup
• 2 cans of potato soup
• 1 can of beef soup
Gabriel will randomly choose one can of soup. Then he will put it back and randomly choose another can of soup. What is the probability that he will choose a can of tomato soup and then a can of cheese soup?
Answer:
the answer is the cheese soup has a good change of being picked but not as good as the chicken soup there would so 3:2 to 2. so if they picked the cheese soup the first time then there is not a good change of the cheese souo to be picked again i hope that helps if not let me know
ABCD is a quadrilateral that has a four right angle What is the most accurate way to classify this quadrilateral
Answer:
A rectangle
Step-by-step explanation:
A square can be a rectangle but a rectangle can't always be a square.
Convert:
35 pounds, 24 ounces=_ ounces
Answer:
Step-by-step explanation:
1 pound = 16 ounces
35 pounds = 560 ounces + 24 ounce = 584ounces
35 pounds,24ounces = 16 ounces
El largo total de una correa transportadora debe ser de 4,5 km para poder llevar el mineral hasta la planta- Si la correa mide 3200 m ¿Cuántos Km faltan para completar el largo requerido
Answer:
Distancia restante = 1,3 kilómetroslp
Step-by-step explanation:
Dados los siguientes datos;
Distancia total = 4,5 km
Distancia recorrida = 3200 metros a kilómetros = 3200/1000 = 3,2 km
Para encontrar la distancia restante para cubrir la longitud requerida;
Distancia total = distancia recorrida + distancia a la izquierda
4.5 = 3.2 + distancia a la izquierda
Distancia a la izquierda = 4.5 - 3.2 Distancia restante = 1,3 kilómetros
A fifth-grade class is earning points for a pizza party by reading books.
For every book they read, they earn 6 points. Complete the table, where b represents the number of books read. How many points will they earn if b = 24?
Answer:
144
Step-by-step explanation:
24 books at 6 points each would be 24 x 6 which equals 144
Answer:
They'll have 144 points if they read 24 books.
Step-by-step explanation:
6 x 24 = 144
simplify (3x 2y)2 using the square of a binomial formula. question 18 options: a) 9x2 4y2 b) 9x2 5xy 4y2 c) 9x2 6xy 4y2 d) 9x2 12xy 4y2
The correct answer is option d) 9x² + 12xy + 4y². To simplify the expression (3x + 2y)² using the square of a binomial formula, we need to apply the formula (a + b)² = a² + 2ab + b². In this case, a = 3x and b = 2y.
Using the formula, we have:
(3x + 2y)² = (3x)² + 2(3x)(2y) + (2y)²
= 9x² + 12xy + 4y²
So the simplified form of (3x + 2y)² is 9x² + 12xy + 4y².
Now let's analyze the given options:
a) 9x² + 4y²: This option is incorrect because it is missing the term 12xy.
b) 9x² + 5xy + 4y²: This option is also incorrect because it contains an additional term, 5xy, which is not present in the simplified expression.
c) 9x² + 6xy + 4y²: This option is incorrect because it also contains an additional term, 6xy, which is not present in the simplified expression.
d) 9x² + 12xy + 4y²: This option is correct because it matches the simplified form we obtained using the square of a binomial formula.
Therefore, the correct answer is option d) 9x² + 12xy + 4y².
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verify that the following function is a cumulative distribution function. f(x)={0x<10.51≤x<313≤x round your answers to 1 decimal place (e.g. 98.7).
The probability distribution of X is as follows:P(X = x) = 0 for x < 1P(X = x) = 0 for 1 ≤ x < 10.5P(X = x) = 1 for 10.5 ≤ x < 31P(X = x) = 0 for x ≥ 31.
Given function: f(x)={0x<10.51≤x<313≤xTo verify that the given function is a cumulative distribution function, we have to check the following conditions:1. f(x) is non-negative for all x.2. f(x) is continuous from the right for all x.3. f(x) is a non-decreasing function for all x.4. lim{x → -∞} f(x) = 0 and lim{x → ∞} f(x) = 1. Now, let's verify these conditions one by one.1. f(x) is non-negative for all x. This condition is satisfied, as the given function f(x) is defined only for x ≥ 1, and is non-negative for all x in this domain. Therefore, f(x) is non-negative for all x.2. f(x) is continuous from the right for all x. This condition is also satisfied, as the given function f(x) is defined piecewise as a continuous function for all x. Therefore, f(x) is continuous from the right for all x.3. f(x) is a non-decreasing function for all x. This condition is satisfied, as the given function f(x) is non-decreasing for all x in its domain.4. lim{x → -∞} f(x) = 0 and lim{x → ∞} f(x) = 1.
Using the definition of f(x) for x < 10, we have f(x) = 0 for x < 10. Therefore, lim{x → -∞} f(x) = 0. Using the definition of f(x) for x ≥ 31, we have f(x) = 1 for x ≥ 31. Therefore, lim{x → ∞} f(x) = 1. Hence, all four conditions are satisfied. Therefore, the given function is a cumulative distribution function. To find the probabilities of the intervals, we use the formula: P(a ≤ X ≤ b) = F(b) - F(a)where P(a ≤ X ≤ b) is the probability that X lies between a and b, and F(x) is the cumulative distribution function of X. Furthermore, the probability of the interval (a, b] is: P(a < X ≤ b) = F(b) - F(a)Therefore, the probabilities of the intervals are: P(1 < X ≤ 10.5) = F(10.5) - F(1) = 0 - 0 = 0P(10.5 < X ≤ 31) = F(31) - F(10.5) = 1 - 0 = 1P(X > 31) = F(∞) - F(31) = 1 - 1 = 0Therefore, we have: P(1 < X ≤ 10.5) = 0P(10.5 < X ≤ 31) = 1P(X > 31) = 0
Hence, the probability distribution of X is as follows: P(X = x) = 0 for x < 1P(X = x) = 0 for 1 ≤ x < 10.5P(X = x) = 1 for 10.5 ≤ x < 31P(X = x) = 0 for x ≥ 31. Therefore, the given function is a cumulative distribution function, and the probability distribution of X is as above.
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The polynomial V(x)=x^3+9x^2-16x-144 represents volume of a shipping crate. Explain how to find the area of the base of the crate? if the polynomial H(x)=x+4 represents the height of the crate
Answer:
x = − 9 , − 4 , 4
Step-by-step explanation:
Each crate is in the shape of a rectangular solid. Its dimensions are length, width, and height. The rectangular solid shown in the image below has a length
4 units, width 2 units, and height 3
units. Can you tell how many cubic units there are altogether? Let’s look layer by layer.
Breaking a rectangular solid into layers makes it easier to visualize the number of cubic units it contains. This
4 by 2 by 3 rectangular solid has 24 cubic units.
A rectangular solid is shown. Each layer is composed of 8 cubes, measuring 2 by 4. The top layer is pink. The middle layer is orange. The bottom layer is green. Beside this is an image of the top layer that says
Altogether there are 24 cubic units. Notice that 24 is the length × width × height .
The top line says V equals L times W times H. Beneath the V is 24, beneath the equal sign is another equal sign, beneath the L is a 4, beneath the W is a 2, beneath the H is a 3.
The volume, V , of any rectangular solid is the product of the length, width, and height.
V = L W H
We could also write the formula for the volume of a rectangular solid in terms of the area of the base. The area of the base, B , is equal to length × width . B = L ⋅ W
We can substitute
B for L ⋅ W in the volume, formula to get another form of the volume formula.
The top line says V equals red L times red W times H. Below this is V equals red parentheses L times W times H. Below this is V equals red capital B times h.
We now have another version of the volume formula for rectangular solids. Let’s see how this works with the
4 × 2 × 3
rectangular solid we started with. See the image below.
Besides the solid is V equals Bh. Below this is V equals Base times height. Below Base is parentheses 4 times 2. The next line says V equals parentheses 4 times 2 times 3. Below that is V equals 8 times 3, then V equals 24 cubic units.
Solve the system of differential equations x1' = – 5x1 + 0x2, X2' =– 16x1 + 3x2 x1(0) = 1, X2(0) = 5 then x1(t) = ? , x2(t) = ?
The solution of the differential equation is 1 = c₁v₁ + c₂v₂ and 5 = c₁v₁ + c₂v₂
We are given a system of two differential equations:
x₁' = – 5x₁ + 0x₂
x₂' = – 16x₁ + 3x₂
To solve this system, we can use several methods, such as substitution or matrix methods. In this explanation, we will use the substitution method.
We can write the given system of differential equations in matrix form as follows:
X' = AX
where X is the column vector [x₁, x₂], X' is the derivative of X, and A is the coefficient matrix:
A = [–5 0]
[–16 3]
To find the eigenvalues λ and eigenvectors v, we solve the characteristic equation:
|A - λI| = 0
where I is the identity matrix. Solving this equation will give us the eigenvalues and eigenvectors.
A - λI = [–5-λ 0]
[–16 3-λ]
Setting the determinant of A - λI to zero, we get:
(–5-λ)(3-λ) - (0)(–16) = 0
Simplifying, we have:
(λ + 5)(λ - 3) = 0
Solving this equation, we find two eigenvalues:
λ₁ = -5
λ₂ = 3
For each eigenvalue, we need to find its corresponding eigenvector. For λ₁ = -5, we solve the system of equations:
(A - (-5)I)v₁ = 0
Substituting the values of A and λ₁, we have:
[0 0] v₁ = 0
[–16 8]
Simplifying the equation, we get:
0v₁ + 0v₂ = 0
-16v₁ + 8v₂ = 0
From the first equation, we can see that v₁ can take any value. Let's choose v₁ = 1 for simplicity. Substituting this value into the second equation, we get:
-16(1) + 8v₂ = 0
-16 + 8v₂ = 0
8v₂ = 16
v₂ = 2
So, for λ₁ = -5, the corresponding eigenvector is v₁ = [1, 2].
Similarly, for λ₂ = 3, we solve the system of equations:
(A - 3I)v₂ = 0
Substituting the values of A and λ₂, we have:
[-8 0] v₂ = 0
[–16 0]
Simplifying the equation, we get:
-8v₁ + 0v₂ = 0
-16v₁ + 0v₂ = 0
From the first equation, we can see that v₁ can take any value. Let's choose v₁ = 1 for simplicity. Substituting this value into the second equation, we get:
-16(1) + 0v₂ = 0
-16 = 0
This equation has no solution. However, this means that v₂ can take any value. Let's choose v₂ = 1 for simplicity.
So, for λ₂ = 3, the corresponding eigenvector is v₂ = [1, 1].
The general solution of the system of differential equations can be expressed as:
X(t) = c₁e(λ₁t)v₁ + c₂e(λ₂t)v₂
where c₁ and c₂ are constants that need to be determined.
We are given the initial conditions x₁(0) = 1 and x₂(0) = 5. Substituting these values into the general solution, we get two equations:
x₁(0) = c₁e(λ₁(0))v₁ + c₂e(λ₂(0))v₂
x₂(0) = c₁e(λ₁(0))v₁ + c₂e(λ₂(0))v₂
Simplifying, we have:
1 = c₁v₁ + c₂v₂
5 = c₁v₁ + c₂v₂
Solving this system of equations, we can find the values of c₁ and c₂.
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