Problema Solution
A piece of sheet metal is 2.5 times as long as it is wide. It is to be made into a box with an open top by cutting 3-in. squares from each corner and folding up the sides. Let x represent the width of the original piece.
(a) Represent the length of the original piece of sheet metal in terms of x.
(b) What are the restrictions on x?
(c) Determine a function that represents the volume of the box in terms of x.
(d) For what values of x will the volume of the box be between 600 and 800 inches^3? Give values to the nearest tenth of an inch.
Answer provided by our tutors
let 'x' represent the width, then '2.5x' represents the length
a) the length of the original sheet is 2.5x
b) 3 in. corners are being cut, so x must be greater than 3*2 = 6
c) volume = length * width * height
the height will be 3 inches, the length of the box will be '2.5x-3' and the width of the box will be 'x-3'
so, volume = 3 * (x-3)* (2.5x-3)
d)we want a volume no less than 600 and no more than 800 cubic inches:
for a volume of 600, we have:
600 = 3 * (x-3)* (2.5x-3)
solving for 'x' gives a positive value of x=11.089
for a volume of 800, we have:
800 = 3 * (x-3)* (2.5x-3)
solving for 'x' gives a positive value of x=12.467
in order to stay within a volume range of [600, 800], we round up on the smaller value of 'x' and we round down on the larger value of 'x', giving (to the nearest tenth of an inch):
11.1 <= x <= 12.4