a square - based, box - shaped shipping crate is designed to have a volume of 125 ft³. the material used to…

Explanation:

Step1: Define variables

Let the side - length of the square base be $x$ (in feet) and the height be $h$ (in feet). The volume $V=x^{2}h = 125$, so $h=\frac{125}{x^{2}}$. Let the cost per square - foot of the side material be $c$. Then the cost per square - foot of the base material is $\frac{8}{5}c$ and the cost per square - foot of the top material is $\frac{2}{5}c$.

Step2: Write the cost function

The surface area of the base is $x^{2}$, the surface area of the top is $x^{2}$, and the surface area of the four sides is $4xh$. The cost function $C$ is $C=\frac{8}{5}cx^{2}+\frac{2}{5}cx^{2}+4cxh$. Combine like terms: $C = 2cx^{2}+4cxh$.

Step3: Substitute $h$ into the cost function

Substitute $h=\frac{125}{x^{2}}$ into $C$: $C(x)=2cx^{2}+4cx\cdot\frac{125}{x^{2}}=2cx^{2}+\frac{500c}{x}$, where $x\gt0$.

Step4: Find the derivative of the cost function

Differentiate $C(x)$ with respect to $x$. Using the power rule, $C^\prime(x)=4cx-\frac{500c}{x^{2}}$.

Step5: Set the derivative equal to zero

Set $C^\prime(x) = 0$: $4cx-\frac{500c}{x^{2}}=0$. Since $c\neq0$ (because there is a non - zero cost for the material), we can divide both sides by $c$ to get $4x-\frac{500}{x^{2}}=0$. Multiply through by $x^{2}$: $4x^{3}-500 = 0$. Then $x^{3}=125$, so $x = 5$.

Step6: Find the value of $h$

Substitute $x = 5$ into the volume formula $h=\frac{125}{x^{2}}$. We get $h=\frac{125}{25}=5$.

Answer:

The length of one side of the crate's base is $5$ ft. The height of the crate is $5$ ft.