Suppose a company manufactures two types of wood-burning stoves: a freestanding model and a fireplace-insert model. The cost function for producing x freestanding and y Fireplace-insert stoves is $C = 32 \sqrt { x y } + 175 x + 205 y + 1050 . \text { Find the marginal costs } \frac { \partial C } { \partial x } \text { when }$ $x = 70$ and $y = 80$. Round your answer to the nearest integer.
A) 192
B) 234
C) 210
D) 402
E) 403

Question

Quizplus · Accepted Answer

The marginal cost with respect to $x$, $\frac{\partial C}{\partial x}$, is found by differentiating the cost function $C = 32 \sqrt{xy} + 175x + 205y + 1050$ with respect to $x$. Differentiating gives $\frac{\partial C}{\partial x} = 32 \cdot \frac{1}{2} \cdot (xy)^{-1/2} \cdot y + 175$. Substituting $x = 70$ and $y = 80$ into this expression gives $\frac{\partial C}{\partial x} = 32 \cdot \frac{1}{2} \cdot \frac{80}{\sqrt{70 \cdot 80}} + 175$, which simplifies to approximately 192 when rounded to the nearest integer.

Suppose a Company Manufactures Two Types of Wood-Burning Stoves: a Freestanding