Express as a function of two variables.
An open box is to contain a volume of 12 cubic meters. Given that the material for the sides of the box costs $2 per square meter and the material for the bottom costs $4 per square meter, express the total cost C of the box as a function of the length l and width w.
let the length be l and the width be w
let the height be h
lwh = 12
h = 12/(lw)
base is lw
sides are 2lh + 2lw
cost = 4lw + 2(2lh + 2hw)
= 4lw + 4lh + 4hw
= 4lw + h(4l+4w)
= 4lw + (12/(lw)(4l + 4w)
=4lw + 48/w + 48/l
To express the total cost C of the box as a function of the length l and width w, we need to determine the surface area of the box and then calculate the cost based on the given material costs.
The volume of the box is given as 12 cubic meters, and since it's an open box, one of the dimensions (height) is not specified. However, we can define the height as h.
The formula for the volume of a rectangular box is V = lwh, where l is the length, w is the width, and h is the height. Since we're given that the volume is 12 cubic meters, we have:
12 = lwh
To determine the surface area of the box, we need to consider the sides and the bottom of the box. The sides consist of four identical rectangular faces, each with dimensions l × h, and the bottom has an area of l × w. Therefore, the total surface area A of the box can be calculated as:
A = 4lh + lw
Now that we have the surface area A, we can express the total cost C as a function of the length l and width w using the given material costs. The cost of the sides is $2 per square meter and the cost of the bottom is $4 per square meter. Therefore, the cost is given by:
C = 2(4lh) + 4(lw)
Simplifying this expression, we get the total cost of the box as a function of the length l and width w:
C = 8lh + 4lw