Question

consider a closed rectangular box with a square base with side x and height y. a. find an equation for the surface area of the rectangular box. s(x,y)= b. if the surface area of the rectangular box is 210 square feet, find \\(\frac{dy}{dx}\\) when x = 5 feet and y = 8 feet.

Explanation:

Step1: Find surface - area formula

A closed rectangular box with a square base of side $x$ and height $y$ has surface area $S(x,y)=2x^{2}+4xy$.

Step2: Differentiate with respect to $y$

Treat $x$ as a constant. Using the sum - rule and power - rule of differentiation, $\frac{dS}{dy}=\frac{d}{dy}(2x^{2})+\frac{d}{dy}(4xy)$. Since $\frac{d}{dy}(2x^{2}) = 0$ (because $x^{2}$ is a constant with respect to $y$) and $\frac{d}{dy}(4xy)=4x$, we have $\frac{dS}{dy}=4x$.

Step3: Evaluate $\frac{dS}{dy}$ at given values

We are given $x = 5$ feet. Substitute $x = 5$ into $\frac{dS}{dy}$. So, when $x = 5$ and $y = 8$, $\frac{dS}{dy}=4\times5=20$.

Answer:

$\frac{dS}{dy}=20$ when $x = 5$ and $y = 8$ and the surface - area formula is $S(x,y)=2x^{2}+4xy$