A rectangular box is to have a square base and a volume of 50 ft3. The material for the base costs 42¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 24¢/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.
Since it is rectangular rather than a cube, we have no indication how the other side or the height relate to x.
To find a function in the variable x giving the cost of constructing the box, we need to determine the dimensions of the box.
Let's denote the length of one side of the base as x. Since the base is square, the other side of the base will also have a length of x.
Now, let's consider the dimensions of the box. We know that the volume of the rectangular box is 50 ft3. The volume of a rectangular box is calculated by multiplying the length, width, and height of the box. In this case, the height of the box will be x as well.
Therefore, we have the equation:
Volume = length * width * height
50 = x * x * x
50 = x^3
Now, let's find the cost of constructing the box. The cost depends on the area of the different surfaces of the box and the cost per square foot for each surface.
The base has an area of x * x = x^2, and its cost is 42¢/ft2. So, the cost of the base is 42¢ * x^2.
The four sides of the box (excluding the base) have an area of 4 * x * x = 4x^2, and their cost is 10¢/ft2. So, the cost of the sides is 10¢ * 4x^2 = 40¢ * x^2.
The top of the box also has an area of x * x = x^2, and its cost is 24¢/ft2. So, the cost of the top is 24¢ * x^2.
Now, to find the total cost, we add up the cost of the base, sides, and top:
Cost = Cost of base + Cost of sides + Cost of top
Cost = 42¢ * x^2 + 40¢ * x^2 + 24¢ * x^2
To convert the cost from cents to dollars, we divide by 100:
Cost (in dollars) = (42¢ * x^2 + 40¢ * x^2 + 24¢ * x^2)/100
Cost (in dollars) = (106¢ * x^2)/100
Cost (in dollars) = 1.06x^2
Therefore, the function in the variable x giving the cost of constructing the box is:
Cost(x) = 1.06x^2, where x is the length of one side of the base in feet.