Need help with an algebra question

Question

A box with an open top has a square base and four sides of equal height. The volume of the box is 288 ft ^3.

The height is 2 ft greater than both the length and the width. If the surface area is 228 ft ²

what are the dimensions of the​ box?

Answer 1

Since there is a SQUARE BASE, the LENGTH is the same as the WIDTH.

Volume = (Length) (Width) (Heighth) = (W)(W)(H)

The HEIGHT is 2 feet greater than the length or width.

Volume = (W)(W)(W + 2) = W^2(W + 2) = W^3 + 2W^2

288 = W^3 + 2W^2

REMEMBER there is NO TOP SIDE.

Surface Area = Bottom + Right Side + Left Side + Front Side + Back Side

228 = WW + W(W + 2) + W(W + 2) + W(W + 2) + W(W + 2)

228 = WW + 4(W(W + 2))

228 = W^2 + 4(W^2 + 2W)

228 = W^2 + 4W^2 + 8W

228 = 5W^2 + 8W

0 = 5W^2 + 8W - 228

5W^2 + 8W – 228 = 0

Use the QUADRATIC FORMULA to solve for W: A = 5, B = 8, and C = -228

W = [ -B (+ or -) SQUARE ROOT(B^2 – 4AC) ] / 2A

= [ -8 (+ or -) SQUARE ROOT(64 – (4)(5)(-228) ] / 10

= [ -8 (+ or -) SQUARE ROOT(64 + 4,560) ] / 10

= [ -8 (+ or -) SQUARE ROOT(4,624) ] / 10

= [ -8 (+ or -) 68 ] / 10

= (- 8 + 68) / 10 and (-8 – 68) / 10

= 60 / 10 and -76 / 10

= 6 and – 7.6

Since the WIDTH is not a negative number, then the WIDTH = 6 feet.

Since the WIDTH and the LENGTH are the same, then the LENGTH = 6 feet.

And, the HEIGHT is 2 more feet than the WIDTH or LENGTH.

HEIGHT = 6 + 2 = 8 feet.

ANSWERS:

LENGTH = 6 feet

WIDTH = 6 feet

HEIGHTH = 8 feet

CHECKING:

V = LWH = (6)(6)(8) = (36)(8) = 288 cubic feet

Surface Area = (6)(6) + (6)(8) + (6)(8) + (6)(8) + (6)(8) = 36 + 4(48) = 36 + 192 = 228 ft^2

Hope this helps you Lisette.