Question

1. a rectangular shelter is to be made using a rectangular sheet that is 5m wide by 2m long. what value of x will maximize the volume of the shelter?

Explanation:

Step1: Express the dimensions of the shelter

The shelter is likely formed by folding the sheet. Let's assume the height of the shelter is $x$. The base - length of the shelter is $2$m, and the base - width is $5 - 2x$ (since we are folding up $x$ from each side of the 5 - m width). The volume $V$ of a rectangular - prism (the shelter) is given by $V(x)=2\times(5 - 2x)\times x=10x-4x^{2}$.

Step2: Find the derivative of the volume function

Differentiate $V(x)$ with respect to $x$. Using the power rule $\frac{d}{dx}(ax^{n})=nax^{n - 1}$, we have $V^\prime(x)=\frac{d}{dx}(10x - 4x^{2})=10-8x$.

Step3: Set the derivative equal to zero to find critical points

Set $V^\prime(x)=0$. So, $10 - 8x = 0$. Solving for $x$ gives $8x=10$, and $x=\frac{10}{8}=\frac{5}{4}=1.25$m.

Step4: Check the second - derivative to confirm it's a maximum

Differentiate $V^\prime(x)$ to get the second - derivative $V^{\prime\prime}(x)=\frac{d}{dx}(10 - 8x)=-8$. Since $V^{\prime\prime}(x)=-8<0$, when $x = 1.25$m, the volume function $V(x)$ has a maximum.

Answer:

$x = 1.25$m