Answer:
The cost function of manufacturing this box is:
[tex]C_{T} = 2.15\cdot x^{2}+30\cdot x[/tex], where [tex]x[/tex] is the length of the box, measured in US dollars.
Step-by-step explanation:
From the statement we know that:
1) The box has a square base ([tex]w = l = x[/tex]).
2) The box has a depth of 10 inches ([tex]h = 10\,in[/tex]).
3) The cost of manufacturing the top of the box is $ 1.20 per square inch.
4) The cost of manufacturing the bottom of the box is $ 0.95 per square inch.
5) The cost of manufacturing each side of the box is $ 0.75 per square inch.
The total cost of manufacturing the box is:
[tex]C_{T} =C_{bottom} + C_{top} + 4\cdot C_{side}[/tex]
Where:
[tex]C_{bottom}[/tex] - Cost of the bottom of the box, measured in US dollars.
[tex]C_{top}[/tex] - Cost of the top of the box, measured in US dollars.
[tex]C_{side}[/tex] - Cost of the side of the box, measured in US dollars.
Dimensionally, cost equals unit cost, measured in US dollars by square inch, multiplied by surface area, measured in square inches. Now, the expression is expanded:
[tex]C_{T} = c_{bottom}\cdot x^{2}+c_{top}\cdot x^{2}+4\cdot c_{side}\cdot h\cdot x[/tex]
[tex]C_{T} =(c_{bottom}+c_{top})\cdot x^{2}+4\cdot c_{side}\cdot h \cdot x[/tex]
If we know that [tex]c_{bottom} = \$\,0.95\,in^{-2}[/tex], [tex]c_{top} = \$\,1.20\,in^{-2}[/tex], [tex]c_{side} = \$\,0.75\,in^{-2}[/tex] and [tex]h = 10\,in[/tex], the cost function of manufacturing this box is:
[tex]C_{T} = 2.15\cdot x^{2}+30\cdot x[/tex], where [tex]x[/tex] is the length of the box, measured in US dollars.
