Question

6. A rectangular box with an open top is being constructed so that its base is twice as long as it is wide. In addition, the base of the box cost $2 per square foot while the sides cost $1.50 per square foot. We only want to spend $16 on materials for the box. a. Draw a labeled diagram that represents the situation using relevant variables. b. Write a formula for the box's surface area, A, in terms of only two variables. Briefly explain your thinking. C. Find an expression for the cost of the box, C, in terms of the same variables from (b). Briefly explain your thinking. Then use the fact that cost is constrained to solve for one variable in terms of another.

Answer 1

a)

h w I=2w

Let the width of the rectangular box is 'w'

The length of the rectangular box be 'l'=2w (Given)

and The height of Rectangular box be 'h'

b)

Remember the top end is open

Surface area(A) = [area of base] + [2x area of side with base w]+ [2x area of side with base 2w]

Area of base =

1 xb=2w?

Area of base with side w =

Area of base with side 2w =

S=2w2 + 2(wh) + 2(2wh)

$S=2w^2+6(wh)$

This is the expression for surface area of rectangular box without top end

c)

Given that , Base of box costs per sqr. ft

Sides of the box costs per sqr. ft

Expence for base = cost for base of box per sqr. ft x Base area =$2\times 2w^2$

Expence for side = cost for side of box per sqr. ft x side area =$1.5\times 6wh$

$Total.spend=4w^2+9wh$

Maximum amount we can spend for the box is

i,e. $16=4w^2+9wh$

This is the required relation