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3 years ago

Find the volume of the shape

Mathematics

1 answer:

Jet001 [13]3 years ago

3 0

Answer:

42.875 cm^3

Step-by-step explanation:

Since cubes have the same length, width, and heigh measurements all around we can infer that every side is equal to 3.5 cm.

V = base × height × width

V = 3.5 cm × 3.5 cm × 3.5 cm

V = 42.875 cm^3

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Homer is giving some cookies to each of

Stells [14]

Answer:

7 cookies

Step-by-step explanation:

We have 3 brothers

First , Second, Third

Let the total number cookies be represented by A

1 cookie = 1

Working backwards, we start from the third brother

If we work backwards

It means he gave everything away

We start from the youngest

He was given 1/2 of what was left and 1/2 a cookie

This means,

1/2 + 1/2 = 1

The second brother

He got half of what is left and 1/2 a cookies

Half of what is left from brother

= What the youngest brother got + 1/2 + 1/2 a cookie

= 1 + 1/2 + 1/2

= 1.5 + 1/2

= 2 cookies

For the first brother

He got 1/2 of the cookies + 1/2 cookie

= 1.5 × 2 + 1/2 + 1/2 cookies

= 3 1/2 + 1/2

= 4 cookies

The first brother got 4 cookies

The second brother got 2 cookies

The third broth got 1 cookies

3 0

3 years ago

Evaluate the spherical coordinate integral

expeople1 [14]

Rewrite the equations of the given boundary lines:

y = -x + 1 ==> x + y = 1

y = -x + 4 ==> x + y = 4

y = 2x + 2 ==> -2x + y = 2

y = 2x + 5 ==> -2x + y = 5

This tells us the parallelogram in the x-y plane corresponds to the rectangle in the u-v plane with 1 ≤ u ≤ 4 and 2 ≤ v ≤ 5.

Compute the Jacobian determinant for this change of coordinates:

$J=\begin{bmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{bmatrix}=\begin{bmatrix}1&1\\-2&1\end{bmatrix}\implies|\det J|=3$

Rewrite the integrand:

$-3x+4y=-3\cdot\dfrac{u-v}3+4\cdot\dfrac{2u+v}3=\dfrac{5u+7v}3$

The integral is then

$\displaystyle\iint_R(-3x+4y)\,\mathrm dx\,\mathrm dy=3\iint_{R'}\frac{5u+7v}3\,\mathrm du\,\mathrm dv=\int_2^5\int_1^45u+7v\,\mathrm du\,\mathrm dv=\boxed{333}$