Question

score on last try: 5 of 10 pts. see details for more. at least one scored part is incorrect. jump to first changeable incorrect part. you can retry this question below. 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)=2x^{2}+4xy b. if the surface area of the rectangular box is 210 square feet, find \\(\frac{dz}{dy}\\) when x = 5 feet and y = 8 feet. question help: video jump to answer submit question question 15

Explanation:

Step1: Write surface - area formula

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

Step2: Set $S = 210$

We know that $S = 210$, so $2x^{2}+4xy=210$.

Step3: Solve for $y$ in terms of $x$

\[

$$\begin{align*} 4xy&=210 - 2x^{2}\\ y&=\frac{210 - 2x^{2}}{4x}=\frac{105}{2x}-\frac{x}{2} \end{align*}$$

\]

Step4: Differentiate $y$ with respect to $x$

Using the power rule, if $y=\frac{105}{2x}-\frac{x}{2}=\frac{105}{2}x^{-1}-\frac{1}{2}x$, then $\frac{dy}{dx}=-\frac{105}{2x^{2}}-\frac{1}{2}$.

Step5: Evaluate at $x = 5$

\[

$$\begin{align*} \frac{dy}{dx}\big|_{x = 5}&=-\frac{105}{2\times5^{2}}-\frac{1}{2}\\ &=-\frac{105}{50}-\frac{1}{2}\\ &=-\frac{21}{10}-\frac{5}{10}\\ &=-\frac{21 + 5}{10}=-\frac{26}{10}=-\frac{13}{5} \end{align*}$$

\]

Answer:

$-\frac{13}{5}$