optimization\nquestion 1, 4.5.21\nhw score: 10%, 1 of 10 points\npoints: 0 of 2\na storage shed is to be…
optimization\nquestion 1, 4.5.21\nhw score: 10%, 1 of 10 points\npoints: 0 of 2\na storage shed is to be built in the shape of a box with a square base. it is to have a volume of 729 cubic feet. the concrete for the base costs $3 per square foot, the material for the roof costs $5 per square foot, and the material for the sides costs $4.50 per square foot. find the dimensions of the most economical shed.\nthe length of one side of the sheds base is \nthe height of the shed is
Answer
Explanation:
Step1: Define variables
Let the side - length of the square base be $x$ (in feet) and the height of the box be $h$ (in feet). The volume $V=x^{2}h = 729$, so $h=\frac{729}{x^{2}}$.
Step2: Calculate the cost function
The area of the base is $A_{base}=x^{2}$, and its cost is $3x^{2}$. The area of the roof is $A_{roof}=x^{2}$, and its cost is $5x^{2}$. The area of each side is $xh$, and there are 4 sides, so the cost of the sides is $4\times4.5xh = 18xh$. The total cost function $C(x)=3x^{2}+5x^{2}+18xh=8x^{2}+18xh$. Substitute $h = \frac{729}{x^{2}}$ into the cost function: $C(x)=8x^{2}+18x\times\frac{729}{x^{2}}=8x^{2}+\frac{13122}{x}$.
Step3: Find the derivative of the cost function
Differentiate $C(x)$ with respect to $x$. Using the power - rule, if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. $C^\prime(x)=16x-\frac{13122}{x^{2}}$.
Step4: Set the derivative equal to zero
Set $C^\prime(x)=0$ to find the critical points: $16x-\frac{13122}{x^{2}} = 0$. Multiply through by $x^{2}$ to get $16x^{3}-13122 = 0$. Then $x^{3}=\frac{13122}{16}=\frac{6561}{8}$, and $x = \sqrt[3]{\frac{6561}{8}}=\frac{18}{2}=9$.
Step5: Find the height
Substitute $x = 9$ into the volume formula $h=\frac{729}{x^{2}}$. So $h=\frac{729}{81}=9$.
Answer:
The length of one side of the shed's base is 9 ft. The height of the shed is 9 ft.