We need to make an open topped box with a square base and fixed volume of 8 cubic feet.
The material for the base costs $ 6 per square foot and the material for the side costs $4 per square foot. Write a function for the cost as a function of the length and side of the base
each side of the base ---- x ft
height of box ------------ y ft
x^2 y = 8
y = 8/x^2
cost = 6x^2 + 4xy
= 6x^2 + 4x(8/x^2)
cost = 6x^2 + 32/x
To write a function for the cost as a function of the length and side of the base, we can break down the open topped box into its components: the base and the four sides.
First, let's find the dimensions of the box's base. Since the box has a square base, both the length and the width of the base will be the same. Let's denote this common side length as s.
The volume of the box is given as 8 cubic feet. Since the base has a square shape, the area of the base (A_base) can be calculated by squaring the side length (s). So, we have:
A_base = s^2
The height of the box can be calculated by dividing the volume (V) by the area of the base:
height = V / A_base
height = 8 / (s^2)
Now, let's calculate the cost of the base and the four sides separately:
Cost of the base:
The cost of the base is given as $6 per square foot, and the area of the base is A_base = s^2. Therefore, the cost of the base (C_base) can be calculated as:
C_base = 6 * A_base
C_base = 6 * s^2
Cost of the sides:
The cost of the sides is given as $4 per square foot. Since there are four sides, the total area of the sides (A_sides) can be calculated as:
A_sides = 4 * (length * height)
A_sides = 4 * (length * (8 / (s^2))) substituting height = 8 / (s^2)
The cost of the sides (C_sides) can be calculated as:
C_sides = 4 * $4 * A_sides
C_sides = 16 * A_sides
C_sides = 16 * (length * (8 / (s^2)))
Finally, the total cost (C_total) can be calculated by summing the cost of the base and the cost of the sides:
C_total = C_base + C_sides
C_total = 6 * s^2 + 16 * (length * (8 / (s^2)))
Hence, the cost function is:
C(s, length) = 6s^2 + 16 * (length * (8 / s^2))