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loris [4]
3 years ago
10

A rectangular box has a base that is 4 times as long as it is wide. The sum of the height and the girth of the box is 200 feet.

(a) Express the volume V of the box as a function of its width w. Determine the domain of V (w).
Mathematics
1 answer:
enyata [817]3 years ago
6 0

Answer: V(W) = (1/3)*(*W^2*800ft - 8W^3) and the domain is 0 < W < 100ft.

Step-by-step explanation:

The dimensions of the box are:

L = length

W = width

H = heigth.

We know that:

L = 4*W

And the girth of a box is equal to: G = 2*W + 2*H

then we have:

2*W + 2*H + H = 200ft

2W + 3*H = 200ft

Then we have two equations:

L = 4*W

2W + 3*H = 200ft

We want to find the volume of the box, which is V = W*L*H

and we want in on terms of W.

Then, first we can replace L by 4*W (for the first equation)

and:

2*W + 3*H = 200ft

3*H = 200ft - 2*W

H = (200ft - 2*W)/3.

then the volume is:

V(W) = W*(4*W)*(200ft - 2*W)/3

V(W) = (1/3)*(*W^2*800ft - 8W^3)

The domain of this is the set of W such that the volume is positive, then we must have that:

W^2*800ft > 8W^3

To find the maximum W we can see the equality (the minimum extreme is 0 < W, because the width can only be a positive number)

W^2*800ft = 8W^3

800ft = 8*W

100ft = W.

This means that if W is equal or larger than 100ft, the equation gives a negative volume.

Then the domain is 0 < W < 100ft.

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\bigstar\:{\underline{\sf{In\:right\:angled\:triangle\:ABC\::}}}\\\\

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⠀⠀⠀

\bf{\dag}\:{\underline{\frak{By\:using\:Pythagoras\: Theorem,}}}\\\\

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⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

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⠀⠀⠀

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⠀⠀⠀

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⠀⠀⠀

  • tan A = Perpendicular/Base = {\sf{\pink{\dfrac{4}{7}}}}

⠀⠀⠀

\therefore\:{\underline{\sf{Hence,\: {\pmb{Option\:A)}}\:{\sf{is\:correct}}.}}}

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