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3 years ago

8

A group of people are available for 7 1/2 hours to paint rooms in a house. It takes them 2 1/4 hours to paint each room. About h

ow many rooms can the group paint?

1 answer:

Schach [20]3 years ago

7 0

The people can paint about 3 rooms.

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Use the given transformation x=4u, y=3v to evaluate the integral. ∬r4x2 da, where r is the region bounded by the ellipse x216 y2

exis [7]

The Jacobian for this transformation is

$J = \begin{bmatrix} x_u & x_v \\ y_u & y_v \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 0 & 3 \end{bmatrix}$

with determinant $|J| = 12$ , hence the area element becomes

$dA = dx\,dy = 12 \, du\,dv$

Then the integral becomes

$\displaystyle \iint_{R'} 4x^2 \, dA = 768 \iint_R u^2 \, du \, dv$

where $R'$ is the unit circle,

$\dfrac{x^2}{16} + \dfrac{y^2}9 = \dfrac{(4u^2)}{16} + \dfrac{(3v)^2}9 = u^2 + v^2 = 1$

so that

$\displaystyle 768 \iint_R u^2 \, du \, dv = 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2 \, du \, dv$

Now you could evaluate the integral as-is, but it's really much easier to do if we convert to polar coordinates.

$\begin{cases} u = r\cos(\theta) \\ v = r\sin(\theta) \\ u^2+v^2 = r^2\\ du\,dv = r\,dr\,d\theta\end{cases}$

Then

$\displaystyle 768 \int_{-1}^1 \int_{-\sqrt{1-v^2}}^{\sqrt{1-v^2}} u^2\,du\,dv = 768 \int_0^{2\pi} \int_0^1 (r\cos(\theta))^2 r\,dr\,d\theta \\\\ ~~~~~~~~~~~~ = 768 \left(\int_0^{2\pi} \cos^2(\theta)\,d\theta\right) \left(\int_0^1 r^3\,dr\right) = \boxed{192\pi}$

3 0

2 years ago

Please answer me please

barxatty [35]

Step-by-step explanation:

please mark me as brainlest

4 0

2 years ago

Read 2 more answers

Marissa is painting one of her bedroom walls. She marks off 1/4 of the wall. Then she divides the marked section into 3 equal pa

Sergio039 [100]

Answer:

1/12

Step-by-step explanation:

The question can be written as (1/4) / 3. This is equivalent to (1/4)*(1/3), which is 1/12.

4 0

3 years ago

4x + 11y = -3 and -6x = 18y - 6 (substitution method)

dolphi86 [110]

Answer:

-3 + -6= -9

because there is answer given

6 0

3 years ago

Solve the following equation -4y+8=4[2y-2]-

aksik [14]

The answer is y=4/3. Hope this helps.

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3 years ago