a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 125 cubic…

a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 125 cubic feet. the concrete for the base costs $4 per square foot, the material for the roof costs $9 per square foot, and the material for the sides costs $6.50 per square foot. find the dimensions of the most economical shed. let x represent the length of one of the sides of the base and let c represent the cost of the shed. what is the objective function? c = (type an expression.)

a storage shed is to be built in the shape of a box with a square base. it is to have a volume of 125 cubic feet. the concrete for the base costs $4 per square foot, the material for the roof costs $9 per square foot, and the material for the sides costs $6.50 per square foot. find the dimensions of the most economical shed. let x represent the length of one of the sides of the base and let c represent the cost of the shed. what is the objective function? c = (type an expression.)

Answer

Explanation:

Step1: Find base and roof area

The base and roof are squares with side - length $x$. Area of base = Area of roof=$x^{2}$.

Step2: Find side - wall area

The volume of the box $V = x^{2}h=125$, so $h=\frac{125}{x^{2}}$. Each side - wall has area $xh$, and there are 4 side - walls. So the total side - wall area is $4xh = 4x\cdot\frac{125}{x^{2}}=\frac{500}{x}$.

Step3: Calculate cost function

Cost of base is $4x^{2}$ (since base area is $x^{2}$ and cost per square foot is $4$). Cost of roof is $9x^{2}$ (since roof area is $x^{2}$ and cost per square foot is $9$). Cost of sides is $6.5\times\frac{500}{x}$. The objective function $C$ (total cost) is the sum of the cost of the base, roof, and sides. $C = 4x^{2}+9x^{2}+6.5\times\frac{500}{x}$. $C=13x^{2}+\frac{3250}{x}$.

Answer:

$13x^{2}+\frac{3250}{x}$