Question

Please help 100 points!! A box in the shape of a rectangular prism has a volume of 72 cubic feet. The box has a length of x feet, a width of (x − 1) feet, and a height of (x + 9) feet. Find the dimensions of the box.

Answer 1

Answer:

Dimensions:

Length = 3 feet

Width = 2 feet

Height = 12 feet

Step-by-step explanation:

72 = length × width × height

72 = x(x-1)(x+9)

72 = x(x²-x+9x-9)

72 = x³ + 8x² - 9x

x³ + 8x² - 9x - 72 = 0

x = 3, -3, -8

Can be obtained using factor theorem. Just try factors of 72 for 0.

Since x is the length, can't be negative.

So x = 3

x - 1 = 2

x + 9 = 12

Dimensions:

Length = 3 feet

Width = 2 feet

Height = 12 feet

Answer 2

Step-by-step explanation:

The volume of a rectangular prism is given by the formula

V = lwh

where V = volume, l = length, w = width, and h = height.

$\therefore \: v = lwh \\ \therefore \: 72= x \times (x - 1) \times (x + 9) \\ let \: us \: use \: trial \: and \: error \: method: \\ let \: us \: plug \: x = 2 \\ \therefore \:lhs = x \times (x - 1) \times (x + 9) \\ = 2 \times (2 - 1)(2 + 9) \\ = 2 \times 1 \times 11 \\ = 22 \neq \: 72 \\ \therefore \:lhs\neq \: rhs \\ hence \: x = 2 \: is \: not \: a \: solution. \\ next \: let \: us \: plug \: x = 3 \\ lhs = x \times (x - 1) \times (x + 9) \\ = 3 \times (3 - 1)(3+ 9) \\ = 3 \times 2\times 12 \\ = 72 \\ = rhs \\ hence \: x = 3 \: is \: a \: solution \: of \: the \: \\ given \: equation. \\ \\ \implies \\ length = x = 3 \: feet \\ width = (x - 1) = (3 - 1) = 2 \: feet \\ height = (x + 9) = (3 + 9) = 12 \: feet$

Therefore, dimensions of the box are 3 ft, 2 ft and 12 ft.