Mike’s mom made a batch of soup that was 8 4/5 cups. Each serving is 1 1/2 cups. How many servings of soup did Mike’s mom make?

Mathematics

1 answer:

kipiarov [429]2 years ago

6 0

Answer:

B) 5 and 13/15 servings

Step-by-step explanation:

Divide 8 4/5 by 1 1/2 to get 5 and 13/15

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$x=y=0\implies 2z=2\implies z=1\implies(0,0,1)$

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$x=z=0\implies4y=8\impliesy=2\implies(0,2,0)$

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Both planes share the same x- and y-intercepts, but the second plane's z-intercept is higher, so Plane 2 acts as the roof of the bounded region.

Meanwhile, in the (x, y)-plane where z = 0, we see the bounded region projects down to the triangle in the first quadrant with legs x = 0, y = 0, and x + y = 2, or y = 2 - x.

So the volume of the region is

$\displaystyle\int_0^2\int_0^{2-x}\int_{\frac{2-x-y}2}^{8-4x-4y}\mathrm dz\,\mathrm dy\,\mathrm dx=\displaystyle\int_0^2\int_0^{2-x}\left(8-4x-4y-\frac{2-x-y}2\right)\,\mathrm dy\,\mathrm dx$

$=\displaystyle\int_0^2\int_0^{2-x}\left(7-\frac72(x+y)\right)\,\mathrm dy\,\mathrm dx=\int_0^2\left(7(2-x)-\frac72x(2-x)-\frac74(2-x)^2\right)\,\mathrm dx$

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