Answer:
[tex]f(x)=72x^{2} +\frac{2000}{x}[/tex]
Step-by-step explanation:
1) First of all, we are going to see what is the area of each element of the box.
The area of the base is x² (because x is the length of the side of the square base).
The area of the top is x² as well.
Now, the area of each side will be x·h where h is the height of the box. The box has 4 sides so the total area of the sides will be 4x·h
However, we can express h in terms of x because we have the total volume of the box:
V = (base area) · height = 50 ft³
50 = x²h
50/x² = h
Therefore, the area of the 4 sides will be: [tex]4(x)(h) = 4x(\frac{50}{x^{2} } )=\frac{200}{x}[/tex]
2) Now we are going to find the function giving the cost of constructing the box:
To find the function, we are going to use the prices we are given.
The price of the base, top and sides will be (for each of them): (price per ft²)(area in ft²)
Therefore the function to find the price (in cents) would be:
[tex]f(x)=42x^{2} +30x^{2} +(10)(\frac{200}{x} )\\f(x)=72x^{2} +\frac{2000}{x}[/tex]
