Question

A small bank vault is being designed in the shape of a rectangular prism. The vault’s sides and top should all be 3 feet thick. The outer length of the vault should be twice the outer width. The outer height should be the same as the outer width. What should the outer dimensions of the vault be if is to have 972 cubic feet of space?
(^ incase the words got cut off in the pic) ​

Answer 1

Answer:   24 ft by 12 ft by 12 ft

This represents the exterior length, width and height in that order.

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Work Shown:

The steps you have written so far are 100% correct.

From there, you multiply the interior width, length and height to get the interior volume 972. Solve for x

So,

(length)*(width)*(height) = volume

(2x-6)(x-6)(x-3) = 972

(2x-6)(x^2-9x+18) = 972

2x(x^2-9x+18)-6(x^2-9x+18) = 972

2x^3-18x^2+36x-6x^2+54x-108 = 972

2x^3-24x^2+90x-108 = 972

2x^3-24x^2+90x-108-972 = 0

2x^3-24x^2+90x-1080 = 0

2(x^3-12x^2+45x-540) = 0

x^3-12x^2+45x-540 = 0

From here, we can use the factor by grouping method

x^3-12x^2+45x-540 = 0

(x^3-12x^2)+(45x-540) = 0

x^2(x-12)+45(x-12) = 0

(x^2+45)(x-12) = 0

Setting each factor equal to 0 leads to....

• x^2+45 = 0 doesn't have any real valued solutions for x, since sqrt(-45) is a complex number. We'll ignore this case and move on.
• x-12 = 0 solves to x = 12

So x = 12 is the only solution here.

If x = 12, then we have these exterior dimensions:

• length = 2x = 2*12 = 24 ft
• width = x = 12 ft
• height = x = 12 ft

The outer dimensions are 24 ft by 12 ft by 12 ft

From that, the inner dimensions would be:

L = 24-6 = 18 ft

W = 12-6 = 6 ft

H = 12-3 = 9 ft.

Then note how L*W*H = 18*6*9 = 972 ft^3 to help confirm the answer.