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A certain box has a width that is 2 inches more than its length and a height that is 5 inches less than its length. If each of the two smallest faces of the box has an area of 36 square inches, what is the volume if the box?

Sagot :

Answer:

V = 396 cubic inches

Step-by-step explanation:

width = w; length = l; height = h; volume = V; area of smallest face = a

base units are inches

w = 2 + l

h = l - 5

Height is smallest and length is second smallest (h = l -, l = l, w = l +), so a is for h and l.

a = h × l = (l - 5) × l

36 = l^2 - 5l ==> l^2 - 5l - 36 = 0

Factor ==> (l - 9) × (l + 4) = 0

l = 9 and l = -4

Since length cannot be negative, 9 is the only Real answer.

l = 9

h = l - 5 = 9 - 5 = 4

w = 2 + l = 2 + 9 = 11

Volume of rectangular prism/box is length times width times height.

V = l × w × h = 9 × 11 × 4 = 396

The volume of the box with the given dimensions is;

Volume = 396 in³

Let us denote the properties of the box as follows;

Length of box = l

Width of box = w

Height of box = h

Area of the smallest face of box = a  

We are told that the width is 2 inches more than the length. Thus;

w = 2 + l

We are told that the height is 5 inches less than its length. Thus;

h = l - 5

Since length is smaller than width but bigger than height, the height and length are the 2 smallest faces

Thus,

a = h × l

plugging in the relevant values gives;

a = (l - 5) × l

a = l² - 5l

We are told that the area of the two smallest faces is 36 in². Thus;

l² - 5l = 36

l² - 5l - 36 = 0

Using online quadratic equation solver, we have; l = 9 inches

Plugging in 9 for h in; h = l - 5  

h = 9 - 5

h = 4

Also, plugging in 9 for l into; w = 2 + l, we have;

w = 2 + 9

w = 11

Volume of box is given by;

Volume = length × width × height

Volume = 9 × 11 × 4

Volume = 396 in³

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