A rectangular box is to have a square base and a volume of 20 ft3. The material for the base costs 42¢/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.
the height h is 20/x^2
The cost in cents is thus
c = 42x^2 + 26x^2 + 4*10*x(20/x^2)
= 68x^2 + 800/x
To find the cost of constructing the box, we need to consider the cost of each component. The box has a square base, so the length of one side of the base is x.
The cost of the base is given by the area of the base (x^2) multiplied by the cost per square foot, which is 42¢/ft². So, the cost of the base is 42 * x^2 cents.
The sides of the box have a height of x and a width of (20/x^2) ft³, since the volume of the box is 20 ft³. The cost of each side is given by the area of the side (height times width) multiplied by the cost per square foot, which is 10¢/ft². Since there are four sides, the total cost of the sides is 4 * 10 * (x * (20/x^2)) cents.
The cost of the top is given by the area of the top (x^2) multiplied by the cost per square foot, which is 26¢/ft². So, the cost of the top is 26 * x^2 cents.
To convert the costs to dollars, we divide each cost by 100. Thus, the cost of the base is (42 * x^2) / 100 dollars, the cost of the sides is (4 * 10 * (x * (20/x^2))) / 100 dollars, and the cost of the top is (26 * x^2) / 100 dollars.
To find the total cost of constructing the box, we add up the costs of the base, sides, and top. Thus, the total cost function C(x) is:
C(x) = (42 * x^2) / 100 + (4 * 10 * (x * (20/x^2))) / 100 + (26 * x^2) / 100
Simplifying this equation, we have:
C(x) = (42x^2 + (4 * 10 * (20/x^2)) * x + 26x^2) / 100
C(x) = (42x^2 + (80/x) * x + 26x^2) / 100
C(x) = (42x^2 + 80 + 26x^2) / 100
C(x) = (68x^2 + 80) / 100
C(x) = (17x^2 + 20) / 25
Therefore, the cost of constructing the box, C(x), in dollars is (17x^2 + 20) / 25.
To find a function in the variable x giving the cost of constructing the box, we need to consider the cost for each component of the box: the base, the sides, and the top.
Let's start by finding the dimensions of the box. Since the box is rectangular and has a square base, the length, width, and height of the box will all be equal to x.
The volume of a rectangular box is given by the formula volume = length × width × height. In this case, the volume is given as 20 ft³. Therefore, we have the equation:
x × x × x = 20
Simplifying, we find:
x³ = 20
Now, let's find the cost for each component of the box. The cost of the base is given as 42¢/ft², and since the base is a square with side length x, the area of the base is x² ft². Therefore, the cost of the base is:
Cost of the base = 42¢/ft² × x² ft²
The cost of the sides is 10¢/ft², and there are four sides to the box. Since each side is a rectangle with dimensions x and x, the area of each side is x² ft². Therefore, the total cost of the sides is:
Cost of the sides = 10¢/ft² × 4 × x² ft²
Finally, the cost of the top is 26¢/ft², and since the top is a square with side length x, the area of the top is x² ft². Therefore, the cost of the top is:
Cost of the top = 26¢/ft² × x² ft²
To find the total cost of constructing the box, we add up the costs of the base, sides, and top:
Total cost = Cost of the base + Cost of the sides + Cost of the top
Total cost = 42¢/ft² × x² ft² + 10¢/ft² × 4 × x² ft² + 26¢/ft² × x² ft²
Simplifying, we have:
Total cost = (42 + 40 + 26)¢/ft² × x² ft²
Total cost = 108¢/ft² × x² ft²
To convert the cost to dollars, we divide by 100:
Total cost in dollars = (108/100) × x²
Therefore, the function in the variable x giving the cost of constructing the box is:
Cost(x) = (108/100) × x²