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grigory [225]
3 years ago
8

The width of a box in the shape of a rectangular prism is 4 inches more than the height h . The length is the difference of 9 in

ches and the height. A. Write a polynomial that represents the volume of the box in terms of its height (in inches). B. The volume of the box is 180 cubic inches.
Mathematics
1 answer:
Doss [256]3 years ago
3 0

Answer:

(a) V = -H^3+ 5H^2 + 36H

(b) Length = 5in; Width = 9in; Height = 5in or Length = 3in; Width = 10in; Height = 6in

Step-by-step explanation:

Let:

W=Width; L = Length; H = Height

Given

W = 4 + H

L = 9 - H

Solving (a): The polynomial for the volume in terms of height

Volume (V) is calculated using:

V = L * W * H

V = (9 - H) * (4 + H) * H

Expand

V = (9 - H) * (4H + H^2)

Open brackets

V = 36H + 9H^2 - 4H^2 - H^3

V = 36H + 5H^2 - H^3

Rewrite as:

V = -H^3+ 5H^2 + 36H

Solving (b): The dimensions when Volume = 180

Substitute 180 for V in: V = -H^3+ 5H^2 + 36H

180 = -H^3+ 5H^2 + 36H

Rewrite as:

H^3 - 5H^2 - 36H + 180 = 0

Factorize:

H^2(H - 5) - 36(H - 5) = 0

(H^2 - 36)(H - 5) = 0

Express H^2 - 36 as difference of two squares

(H - 6)(H + 6)(H - 5) =0

This gives:

H - 6 =0==> H = 6

H + 6 =0==> H = -6

H - 5 =0==> H = 5

Since the height can not be negative, then H = 5 or H = 6

Recall that:

W = 4 + H

L = 9 - H

When H = 5

W = 4 + 5 = 9

L = 9 - 5 = 4

When H = 6

W = 4 + 6 = 10

L = 9 - 6 = 3

Hence, the dimensions of the box are:

Length = 5in; Width = 9in; Height = 5in

or

Length = 3in; Width = 10in; Height = 6in

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