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 dy/dx when x = 5 feet and y = 8 feet.
dy/dx=____

Answer 1

Check the picture below.

the assumption being that "y" is encapsulating a function in x-terms, whilst "x" is just a simple variable.

`S(x,y)=(x)(x)+(x)(x)+6(x)(y)\implies S(x,y)=2x^2+6xy \\\\[-0.35em] ~\dotfill\\\\ 210=2x^2+6xy\implies 0=4x+6\stackrel{ \textit{product rule} }{\left(1\cdot y+x\cdot \cfrac{dy}{dx} \right)} \\\\\\ 0=4x+6y+6x\cfrac{dy}{dx}\implies -4x-6y=6x\cfrac{dy}{dx}\implies \cfrac{-4x-6y}{6x}=\cfrac{dy}{dx} \\\\\\ \left. \cfrac{dy}{dx} \right|_(x=5,y=8)\implies \cfrac{-4(5)-6(8)}{6(5)}\implies \cfrac{-68}{30}\implies \cfrac{-34}{15}`