Question

A box with rectangular sides has width twice the length of the base. The volume is 24 cubic inches and the total surface area of all six sides is 52 square inches. Write down a (cubic) equation whose solution is the length of the box.

Answer 1

Answer: the cubic equation is

L^3 - 52L +144 =0

Lenght= 4inches

Width = 2inches

Height = 3 inches

Step-by-step explanation:

Let the lenght= L

Let the width = W

Let the height = H

Volume of the box = 24 cubic inch

Total surface area of the box = 52 sq. Inches

W = L/2

W, = 0.5L

Volume = L * B * H

Insert W = 0.5L

L * 0.5L * H = 24

0.5L^2 * H = 24

H = 24/0.5L^2

= 48/L^2

Surface area equation

2(L*W) + 2(L+H) + 2(W+H)= 52

Divide through by 2

(L*W) + (L+H) + (W*H) = 26

Insert W= 0.5L

(L*0.5L) + (L*H) + (0.5L*H) = 26

0.5L^2 + LH + 0.5LH = 26

0.5L^2 + 1.5LH = 26

Insert H = 48/L^2

0.5L^2 + 1.5*48/L = 26

0.5L^2 + 72/L = 26

Multiply through by L to get rid of the denominator

0.5L^3 + 72 - 26L = 0

0.5L^3 - 26L + 72 = 0

Multiply through by 2, we have

L^3 - 52L +144 =0

Try L= 4

4*3 -52(4)+ 144 =0

0= 0

(L - 4) is a factor of cubic equation

Therefore Length (L) = 4 inch

Recall that W= L/2

W= 4/2

W= 2 inches

H= 48/L*2

= 48/4*2

= 48/16

= 3 inches