A rectangular box is to have a square base and a volume of 50 ft3. The material for the base costs 44¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 26¢/ft2. Letting x denote the length of one side of the base, find a function in the variable x giving the cost (in dollars) of constructing the box.
To find the cost of constructing the box, we need to determine the dimensions of the box that would yield a volume of 50 ft³ and then calculate the cost based on those dimensions.
Let's start by determining the dimensions of the box. The base of the box is square, so the length and width of the base are equal, and we denote this side as x. The height of the box can be determined by dividing the volume by the area of the base.
The volume of the box is 50 ft³, and the area of the base is x * x = x². Therefore, the height of the box is given by:
height = volume / area of the base = 50 ft³ / (x² ft²) = 50 / x² ft
Now that we have the dimensions of the box in terms of x, we can calculate the cost of constructing the box.
The cost of the base is given as 44¢/ft², and the area of the base is x² ft², so the cost of the base is:
cost of the base = cost per ft² * area of the base = 44¢ * x² ft²
The cost of the sides is given as 10¢/ft², and there are four sides, each with an area of x * height ft². Therefore, the cost of the sides is:
cost of the sides = cost per ft² * (4 * x * height) ft² = 10¢ * (4 * x * (50 / x²)) ft² = 2000 / x ¢
The cost of the top is given as 26¢/ft², and the area of the top is x * x ft², so the cost of the top is:
cost of the top = cost per ft² * area of the top = 26¢ * x² ft²
Now, we can calculate the total cost of constructing the box by summing the costs of the base, sides, and top:
total cost = cost of the base + cost of the sides + cost of the top
= 44¢ * x² + 2000 / x ¢ + 26¢ * x²
= (70 * x² + 2000 / x) ¢
Finally, we need to convert the cost from cents to dollars by dividing by 100:
cost (in dollars) = total cost / 100
= (70 * x² + 2000 / x) / 100
= 0.7 * x² + 20 / x
Therefore, the function in the variable x giving the cost (in dollars) of constructing the box is:
C(x) = 0.7x² + 20/x