Find the surface area of that part of the plane 4x 5y z=74x 5y z=7 that lies inside the elliptic cylinder x225 y2100=1
Answers
Solve for \(z=f(x,y)\) in the plane equation.
\(4x + 5y + z = 7 \implies f(x,y) = z = 7 - 4x - 5y\)
Let \(E\) be the set of points in the plane \(z=0\) bounded by the ellipse \(\frac{x^2}{25}+\frac{y^2}{100}=1\), i.e.
\(E = \left\{(x,y) ~:~ \dfrac{x^2}{25} + \dfrac{y^2}{100} \le 1\right\}\)
Then the area of the plane bounded by the elliptic cylinder is
\(\displaystyle \iint_E dA = \iint_E \sqrt{1 + \left(\dfrac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2} \, dx \, dy \\\\ ~~~~~~~~ = \iint_E \sqrt{1 + (-4)^2 + (-5)^2} \, dx\, dy \\\\ ~~~~~~~~ = \sqrt{42} \iint_E dx \, dy\)
which is simply √42 times the area of the ellipse in the plane \(z=0\). This ellipse has a minor axis of length 5 and a major axis of length 10, so its area is π•5•10 = 50π, and so the area of the plane in question is 50√42 π.
To confirm this result: In polar coordinates, with
\(\begin{cases}x(r,\theta) = 5r\cos(\theta) \\ y(r,\theta) = 10r\sin(\theta) \\ \frac{x^2}{25}+\frac{y^2}{100}=r^2\end{cases}\)
the area element is
\(dA = \begin{vmatrix}\frac{\partial x}{\partial r}&\frac{\partial x}{\partial\theta} \\ \frac{\partial y}{\partial r} & \frac{\partial x}{\partial\theta}\end{vmatrix} \, dr \, d\theta \\\\ ~~~~~~~~ = \begin{vmatrix}5\cos(\theta)&-5r\sin(\theta)\\10\sin(\theta)&10r\cos(\theta)\end{vmatrix} \, dr \, d\theta \\\\ ~~~~~~~~ = \left(50r\cos^2(\theta)+50r\sin^2(\theta)\right)\,dr\,d\theta \\\\ ~~~~~~~~ = 50r \, dr \, d\theta\)
The ellipse can be parameterized by
\(E = \left\{(r,\theta) ~:~ 0\le\theta\le2\pi \text{ and } 0\le r\le1\right\}\)
so that the integral for the area of the ellipse in the plane \(z=0\) is
\(\displaystyle \iint_E dA = \int_0^{2\pi} \int_0^1 50r \, dr \, d\theta \\\\ ~~~~~~~~ = 100\pi \int_0^1 r \, dr \\\\ ~~~~~~~~ = 100\pi\left(\frac12-0\right) = 50\pi\)
Related Questions
2 right triangles have identical angle measures but different side lengths. The first triangle has side lengths of 6, 10, 8 and the second triangle has side lengths of 3, 5, 4.
Which transformation maps the large triangle onto the small triangle?
dilation
reflection
rotation
translation
Answers
Answer:
dilation is the answer
(The Capital Gate in Abu Dhabi) While you're at the top of the tower, you see an ant walking along the edge of the building. If the ant were to walk straight down the side of the tower until it reached the ground, how far would the ant travel? Which trigonometric ratio would you use to find this distance? Use the ratio to find the measurement. (4 points: 1 point for the method, 2 points for shown work, 1 point for the answer) (from the top of the tower to the base, it's 51.84 meters).
Answers
Keys would land approximately 142.66 meters from the base. Ant would travel approximately 158.69 meters using the Pythagorean theorem.
To determine how far from the base of the Capital Gate Tower the keys would land, we can use trigonometry. Given that the tower is 150 meters tall and makes a 72° angle with the ground, we can calculate the horizontal distance from the base.
Let's consider the right triangle formed by the height of the tower, the distance from the base to where the keys land, and the vertical distance from the top of the tower to where the keys land.
Using the sine function, we can relate the angle and the side lengths of the triangle:
sin(72°) = opposite/hypotenuse
sin(72°) = x/150
Rearranging the equation, we get:
x = 150 * sin(72°)
x ≈ 150 * 0.9511
x ≈ 142.66
Therefore, the keys would land approximately 142.66 meters from the base of the tower.
Next, let's determine the distance the ant would travel if it walked straight down the side of the tower until it reached the ground. We know that from the top of the tower to the base, it's 51.84 meters.
The distance the ant would travel is equal to the hypotenuse of a right triangle formed by the height of the tower and the distance it travels.
Using the Pythagorean theorem, we can calculate the distance:
Distance = \(\sqrt{(51.84^2 + 150^2)}\)
Distance ≈\(\sqrt{ (2685.4656 + 22500)}\)
Distance ≈ \(\sqrt{25185.4656}\)
Distance ≈ 158.69
Therefore, the ant would travel approximately 158.69 meters from the top of the tower to the base. The trigonometric ratio used to find this distance is the Pythagorean theorem, which relates the sides of a right triangle.
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Suppose you have 28 coins that are all dimes and quarters. The total value of the coins is $5.35. Write a system of equations that models this situation and solve the system to find the number of quarters and dimes.
Answers
Answer:
1) The system of equations that models this situation is given as follows
0.1·x + 0.25·y = 5.35...(1)
x + y = 28...(2)
2) The number of dimes = 11 dimes
The number of quarters = 17 quarters
Step-by-step explanation:
1) The given parameters are;
The numbers of quarters and dimes = 28
The total value of the coins = $5.25
Let x represent the number of dimes and y represent the number of quarters, and given that a dime = $0.1 and a quarter = $0.25, we have;
The system of equations that models this situation is given as follows
$0.1·x + $0.25·y = $5.35...(1)
x + y = 28...(2)
2) Making y the subject of both equations gives;
From equation (1), we have;
0.1·x + 0.25·y = 5.35
0.25·y = 5.35 - 0.1·x
y = 5.35/0.25 - 0.1/0.25·x = 21.4 - 0.4·x
y = 21.4 - 0.4·x
From equation (2), we have;
x + y = 28
y = 28 - x
Equating both equations to find the common solution, gives;
21.4 - 0.4·x = 28 - x
x - 0.4·x = 28 - 21.4
0.6·x = 6.6
x = 6.6/0.6 = 11
x = 11
The number of dimes = x = 11
y = 28 - x = 28 - 11 = 17
y = 17
The number of quarters = y = 17.
Find the distance between the points (-16, 20) and (-16, -14).2007(-16, 20)161284-20-16-12-8-4048121620-4-812(-16, -14)-16-20units
Answers
Point A
(-16, 20)
Point B
(-16, -14)
The Distance Formula itself is actually derived from the Pythagorean Theorem which is a
\(\begin{gathered} d=\sqrt[]{(y2-y1)^2+(x2-x1)^2} \\ d=\sqrt[]{(-14-20)^2+(-16-(-16}))^2 \\ d=\sqrt[]{(-34)^2} \\ d=34 \end{gathered}\)The distance would be 34 units
add 6 to the quotient of j and k
Answers
Answer:
j/k + 6
Step-by-step explanation:
Add 6 to the quotient of j and k comes out to:
j/k + 6
There are 4 red marbles and 3 blue marbles in a bag. Once a marble is drawn it is not replaced. Find the probability of drawing two red marbles in a row.
Answers
Answer:
2/7
Step-by-step explanation:
4 red marbles and 3 blue marbles in a bag = 7 marbles
P(red) = red/total = 4/7
No replacement
3 red marbles and 3 blue marbles in a bag = 6 marbles
P(red) = red/total = 3/6 = 1/2
P(red, no replacement, red) = 4/7 * 1/2 = 2/7
Which is an equation of the line through the origin and (-5,6)?
A. y=-5/6x
B. y=-6x
C. y=-5x
D. y=- 6/5x
Answers
Answer:
y=-6/5x
Step-by-step explanation:
Simplify 3(x+2=5).Explain how you found your answer
Answers
Answer:
3(x+2=5)
=3(x=5-2)
=3(x=3)
=3(3)
=3×3
=9
If a = -8, b = -7, c = 6, then verify that (a+b) + c = a+ (b+c)
tysm for de help ✨
Answers
The given information is,
→ a = -8
→ b = -7
→ c = 6
Let's verify the problem,
→ (a+b) + c = a+ (b+c)
→ (-8 - 7) + 6 = -8 + (-7 + 6)
→ -15 + 6 = -8 - 1
→ -9 = -9
→ [ LHS = RHS ]
Hence, it is equal and verified.
What is the equation of a vertical line passing through (-5, -2)?
O y = -2
O x = -5
O y = -3
O x = -7
Answers
Answer:
x=-5
Step-by-step explanation:
if point the line is passing through is -5, -2 and the line is vertical than x would have to stay the same no matter where the y is because it's vertical.
Prove that the points P(-1,1) ,Q(2,3) and R(2,1) are the vertices of a salence triangle. Plz ans fast.
Answers
Step-by-step explanation:
P(-1,1), Q(2,3), R(2,1)
Let P(-1,1)be x1and y1
Q(2,3)be x2 and y2
therefore, by distance formula
d(PQ)= root (x2-x1)^2+(y2-y1)^2
= root[ 2-(-1)] + (3-1)
=root (2+1)^2+2^2
= root (3square +2square )
= root 9+4
d( PQ) = root 13.........(1)
Now find distance QR and PR then prove that the three sides are not equal and therefore it is a scalene triangle
2nd also do the same.
James owes His friend Sam $65. He pays Him back $38. How much does James still owe?
Answers
Answer:
$27
Step-by-step explanation:
An artist makes a batch of orange paint by mixing 3/4 cup of yellow paint whith 3/4 of red paint if mixes 15 batches how many cups of orange paint will have
Answers
Answer:
he would have 22 and 1/2 cups
Step-by-step explanation:
6/4 x 15/1 = 45/2 which also equals 22 1/2
Answer:
22 and 1/2 cups :)
Step-by-step explanation:
It is known that x1 and x2 are roots of the equation 3x^2 2x k=0, where 2x1=−3x2. find k.
Answers
Value of the k is -2 for the equation \(3x^{2} +2x + k = 0\).
Given equation is,
\(3x^{2} + 2x + k = 0\)
From the quadratic equation we get,
a = 3, b = 2, c = k
Let us consider roots of the equation are x₁ and x₂
We know that sum of roots is -b/a
x₁ + x₂ = - 2/3
and product of roots is c/a
x₁x₂ = k / 3
It is given that 2x₁ = -3x₂
⇒ x₁ = - 3/2 x₂
Putting value of x₁ in the first equation,
-3/2 x₂ + x₂ = - 2/3
-3x₂ + 2x₂ = -2/3
- x₂ = - 2/3
⇒ x₂ = 2/3
and x₁ = (- 3/2)(2/3)
x₁ = -1
Therefore x₁x₂ = k/3 implies that,
-1(2/3) = k/3
- 2/3 = k/3
k = -2.
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6(10) A pair of fair dice is rolled. Let X denote the product of the number of dots on the top faces. Find the probability mass fimction of X
7.(10) Let X be a discrete random variable with probability mass function p given by:
A -4 -1 0 3 5
p(a) 1/4 5/36 1/9 1/6 1/3
Determine and graph the probability distribution function of X
Answers
The probability distribution function of X is:
X = | -4 | -1 | 0 | 3 | 5
PDF = | 0 | 1/4 | 7/36 | 5/18 | 11/36
To determine the probability distribution function (PDF) of a discrete random variable X with probability mass function (PMF) p, we need to calculate the cumulative probabilities for each value of X.
The cumulative probability P(X ≤ x) for a given value x is obtained by summing up the probabilities for all values of X less than or equal to x. This gives us the cumulative distribution function (CDF) of X.
For the given PMF p:
X | -4 | -1 | 0 | 3 | 5
p(X) | 1/4 | 5/36 | 1/9 | 1/6 | 1/3
The CDF for X can be calculated as follows:
P(X ≤ -4) = 0
P(X ≤ -1) = P(X = -4) = 1/4
P(X ≤ 0) = P(X = -4) + P(X = -1) = 1/4 + 5/36 = 19/36
P(X ≤ 3) = P(X = -4) + P(X = -1) + P(X = 0) = 1/4 + 5/36 + 1/9 = 13/18
P(X ≤ 5) = P(X = -4) + P(X = -1) + P(X = 0) + P(X = 3) = 1/4 + 5/36 + 1/9 + 1/6 = 35/36
Now we have the cumulative probabilities for each value of X. The PDF of X is obtained by taking the differences between consecutive cumulative probabilities:
PDF(X = -4) = P(X ≤ -4) = 0
PDF(X = -1) = P(X ≤ -1) - P(X ≤ -4) = 1/4 - 0 = 1/4
PDF(X = 0) = P(X ≤ 0) - P(X ≤ -1) = 19/36 - 1/4 = 7/36
PDF(X = 3) = P(X ≤ 3) - P(X ≤ 0) = 13/18 - 19/36 = 5/18
PDF(X = 5) = P(X ≤ 5) - P(X ≤ 3) = 35/36 - 13/18 = 11/36
Thus, the probability distribution function of X is:
X | -4 | -1 | 0 | 3 | 5
PDF | 0 | 1/4 | 7/36 | 5/18 | 11/36
To graph the PDF, you can create a bar graph where the x-axis represents the values of X and the y-axis
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the directions say, determine whether the triangles are similar by AA, SSS, SAS, or not similar if the triangles are similar write a valid similarity stating why (9, 10, 11, 12, 14, 15)
Answers
Answer:
9) ASA
10) SAS
11) SAS
12) SAS
13) not similar
14) AA
15) not similar
Step-by-step explanation:
Shandra saved $21. She wants to buy a book that costs $16 and a bracelet that costs one third of her savings. Does she have enough money to buy the book and the bracelet?
Answers
No, Shandra does not have enough money to buy the book and the bracelet.
To determine this, we first need to calculate the cost of the bracelet. Since it costs one-third of her savings, we can divide her savings into 3:
$21 / 3 = $7
So the bracelet costs $7. Now we can add the cost of the book and the bracelet together to find the total cost:
$16 + $7 = $23
Since Shandra has $21 in savings, she does not have enough money to buy both the book and the bracelet. She would need an additional $2 to make the purchase.
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Tori decided to buy a DVD for $18. The sales tax is 7%. How much would the total cost of the DVD be with tax included?
If Tori had $20 to buy the DVD did she have enough money?Type yes or no.
Answers
Answer:
Yes, Tori had enough money to buy the DVD.
Step-by-step explanation:
The total cost of the DVD with tax included would be $18.13. Yes, Tori had enough money to buy the DVD.
A critical? value, z Subscript alphaz??, denotes the? _______.
a. area to the left of z = ??
b. z-score with an area of ?? to its right.
c. z-score with an area of ?? to its left.
d. area to the right of z=??
Answers
A critical value, z Subscript alphas is (c) z-score with an area of ?? to its left.
A critical value, denoted as z (Subscript α/2), is a point on the standard normal distribution curve, which is used in hypothesis testing. It helps to determine whether to accept or reject the null hypothesis. In this context, the critical value denotes the z-score with an area of ?? to its left, which represents the probability of observing a value more extreme than the critical value in the left tail of the distribution.
The critical value z Subscript α/2 signifies the z-score with an area of ?? to its left on the standard normal distribution curve, which is crucial for hypothesis testing.
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Family spends $240 per week on food. If this is %16 of their income, what is the family's weekly income?
Answers
Answer: The family's weekly income is $1500
-------------------------------------------
Work Shown:
x = weekly income (in dollars)
16% of x = amount spent on food per week
16% of x = 240
(16/100)*x = 240
0.16*x = 240
0.16*x/0.16 = 240/0.16
x = 1500
Pls Mark as brainliest
Answer:
1500
Step-by-step explanation:
16% of x = amount spent on food per week
16% of x = 240
(16/100)*x = 240
0.16*x = 240
0.16*x/0.16 = 240/0.16
x = 1500
Hope this helps :)
Find the slope of the line if it exists.
Answers
The slope of the line is Zero, Hence, no slope exists for the line drawn.
The slope of a line represents the rate of change in y-values per change in the x-values. The slope of a line that exists would never be exactly vertical or horizontal.
Here, the line given is exactly vertical with a slope value of zero.
Therefore, no slope exists for the line given.
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Martin is x years old. Jennifer is 3 years younger than Martin. Connor is twice as old as Martin. Write an expression for Jennifer’s age.
Answers
Answer:
x-3
Step-by-step explanation:
martin = x
jennifer = x-3
connor = 2x
what is jennifer's age?
1. Paul uses a coordinate plane to design
his model town layout.
Paul moves the market 2 units left and 3
units down. He says the ordered pair for
the new location of the market is (0,6).
Explain Paul's mistake and write the
correct ordered pair for the new location of
the market.
PLZ ALSO INCLUDE WHAT HIS MISTAKE WAS!
ANSWER FOR Brainiest!!!
Answers
Simplify the expression. Write your answer as a power.
6^10/6^4
Answers
Step-by-step explanation:
6¹⁰/6⁴ = 6⁶
(basically you subtract 4 from 10 so you have 6)
This is because when dividing exponentes you subtract them from one another and ten minus four is 6
what properties are used to solve x/3+1=7
Answers
Answer:
x=18
Step-by-step explanation:
x/3+1=7
-1=-1
x/3=6
×3=×3
x=18
Error Analysis
A student says that √7 is a rational number because you can write √7 as the ratio or quotient
of √7(fraction) √7 over 1 , is the student correct? Explain your answer.
Answers
Answer:
No
Step-by-step explanation:
Ive had the same equation before.
Which equation of a line passes through the points (-1,3) and (2,-3)?
Answers
Answer:
y=-2x+1
Step-by-step explanation:
m=(y2-y1)/(x2-x1)
m=(-3-3)/(2-(-1))
m=-6/(2+1)
m=-6/3
m=-2
y-y1=m(x-x1)
y-3=-2(x-(-1))
y-3=-2(x+1)
y=-2x-2+3
y=-2x+1
Let f(x) = -5x-1 and g(x)=x2 + 5.
= X
Find (f o g)(-2).
Answers
Answer:
- 46
Step-by-step explanation:
evaluate g(- 2) then substitute the result into f(x) , that is
g(- 2) = (- 2)² + 5 = 4 + 5 = 9 , then
f(9) = - 5(9) - 1 = - 45 - 1 = - 46
Dixon orchards has purchased land on which to plant lime and orange trees. each lime tree requires 100 square feet of area. each orange tree requires 200 square feet of area. dixon orchards determines that they have no more than 3,100 square feet of land on which to plant the lime and orange trees. if x represents the number of lime trees and y represents the number of orange trees, which inequality can be used to describe the situation?
Answers
100x + 200y < 3100 inequality can be used to describe the situation.
According to the question
Dixon orchards has purchased land on which to plant lime and orange trees.
Each lime tree requires 100 square feet of area.
Each orange tree requires 200 square feet of area.
Dixon orchards determines that they have no more than 3,100 square feet of land on which to plant the lime and orange trees.
If x represents the number of lime trees and y represents the number of orange trees.
Inequality:
lime tree requires 100 square feet of area that means100x
orange tree requires 200 square feet of area that means 200y
They have no more than 3,100 square feet of land
Rewrite the inequality
100x + 200y < 3100
Therefore,
100x + 200y < 3100 inequality can be used to describe the situation.
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A bird is sitting at the top of a
tree that is 62 feet tall. it spots an
appetizing piece of bread on
the
ground 52 feet away from the foot
of the tree. if it flies in a straight line
lown to the breadcrumb, how far
oes it fly? round your answer to the
earest foot.
Answers
Answer:
81 feet
Step-by-step explanation:
draw a sketch it forms a right angled triangle use formula of a2+b2=c2 making it 62squared plus 52 squared answer you get do the square root