Answer:
[tex]C=17x^2+\frac{11662}{x}[/tex]
Given that:
Volume of the storage shed = 833 cubic feet
Cost of the concrete for the base per square foot = $8
Cost of concrete for the root per square foot = $9
Cost of the material for the sides per square foot = $3.50
Let x be the length of the side of the square and h be height of the shed.
The formula to calculate the volume is
V = Bh
where B is the base area.
Since the base is a square with side 'x',
[tex]B=x^2[/tex]
Substitute the given values into the formula of V.
[tex]\begin{gathered} 833=x^2\cdot h \\ =x^2h \\ \Rightarrow h=\frac{833}{x^2} \end{gathered}[/tex]
The base will have the same area with the roof.
Area of the roof = Base area
[tex]=x^2[/tex]
Cost to construct base
[tex]=8x^2[/tex]
Cost to construct the roof
[tex]=9x^2[/tex]
Area of one side = xh
Cost to construct one side = 3.5xh
Cost to construct 4 sides of the box
[tex]\begin{gathered} =4(3.5xh) \\ =14xh \\ =14x\cdot\frac{833}{x^2} \\ =\frac{11662}{x} \end{gathered}[/tex]
The total cost is the sum of these three costs. So, the objective function is
[tex]\begin{gathered} C=8x^2+9x^2+\frac{11662}{x} \\ =17x^2+\frac{11662}{x} \end{gathered}[/tex]
The dimension for the most economical cost will occur when dC/dx = 0. Then
[tex]\begin{gathered} 34x-\frac{11662}{x^2}=0 \\ x^3=\frac{11662}{34} \\ =343 \\ x=7\text{ ft} \end{gathered}[/tex]
The length of side of the base is 7 feet.
Substitute the value of x into the equation of h.
[tex]\begin{gathered} h=\frac{833}{7^2} \\ =17\text{ ft} \end{gathered}[/tex]
The height of the storage shed is 17 feet.
