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 consider the different components of the box and their respective costs.
Let's break down the components of the box:
1. Base: The base is a square with side length x ft. The cost of the base is given as 44¢/ft².
2. Sides: The box has four sides, each with a length of x ft and a height of h ft. The cost of the sides is given as 10¢/ft².
3. Top: The top of the box is also a square with a side length of x ft. The cost of the top is given as 26¢/ft².
To find a function that gives the cost of constructing the box in dollars, we need to calculate the area of each component and multiply it by the respective cost per square foot.
1. Area of the base (B):
Since the base is a square, its area is given by the formula A = x × x = x² ft².
2. Area of the sides (S):
Each side has a length of x ft and a height of h ft. The area of each side is given by A = x × h ft². Since there are four sides in total, the total area of all sides is 4 × x × h ft².
3. Area of the top (T):
The area of the top is the same as the base, given by A = x² ft².
So, the total cost (C) of constructing the box can be expressed as:
C = cost of the base + cost of the sides + cost of the top
C = (Area of base × cost/ft² for the base) + (Area of sides × cost/ft² for the sides) + (Area of top × cost/ft² for the top)
C = (x² × 44¢/ft²) + (4 × x × h × 10¢/ft²) + (x² × 26¢/ft²)
Converting the costs from cents to dollars (1 dollar = 100 cents), the equation becomes:
C = (x² × 0.44 $/ft²) + (4 × x × h × 0.10 $/ft²) + (x² × 0.26 $/ft²)
Finally, the function giving the cost (in dollars) of constructing the box in terms of the variable x is:
C(x) = 0.44x² + 0.4xh + 0.26x²
Note: The height (h) of the box is not given in the problem statement. If the height is provided, you can substitute its value into the equation to get a specific function in terms of x.