Question

you use a rectangular piece of cardboard 20 in by 30 to construct a box. you square the sides x in. from each corner of the piece of cardboard and then fold up the sides to form the bottom.

A. write a function A to represent the area of the base of the box in terms of x.

B. What is a reasonable domain for the function A?

C. Write an equation if the area of the base must be 416in^2

D. Solve the equation in part C for the values of x in the reasonable domain
What are the dimensions of the base of the box.

Answer 1

length of base = 30-2x
width = 20-2x

Area of base = (30-2x)(20-2x)

clearly 20-x and 30-2x have to be positve, or
0 < x < 10


(30-2x)(20-2x) = 416
600 - 100x + 2x^2 = 416
2x^2 - 100x + 184
x^2 - 50x + 92 = 0

I will complete the square
x^2 - 50x + 625 = -92 + 625
(x-25)^2 = 533
x-25 = ± √533

x = 25 ± √533

so staying within the domain x = 25+√533
or appr. 1.91

Answer 2

7+7