Suppose a corporation manufactures candles at two locations. The cost of producing $x _ { 1 } \text { units at location } 1 \text { is } C _ { 1 } = 0.06 x _ { 1 } ^ { 2 } + 4 x _ { 1 } + 500 \text { and the cost of producing } x _ { 2 } \text { units at location } 2 \text { is }$ $C _ { 2 } = 0.05 x _ { 2 } ^ { 2 } + 4 x _ { 2 } + 400$. The candles sell for $\$ 43$ per unit. Find the quantity that should be produced at each location to maximize the profit $P = 43 \left( x _ { 1 } + x _ { 2 } \right) - C _ { 1 } - C _ { 2 }$.
A) $x _ { 1 } = 345 ; x _ { 2 } = 380$
B) $x _ { 1 } = 325 ; x _ { 2 } = 390$
C) $x _ { 1 } = 315 ; x _ { 2 } = 408$
D) $x _ { 1 } = 390 ; x _ { 2 } = 325$
E) $x _ { 1 } = 408 ; x _ { 2 } = 315$

Question

Quizplus · Accepted Answer

To maximize profit, we need to find the derivative of the profit function with respect to $x_1$ and $x_2$, set them to zero, and solve for $x_1$ and $x_2$. The profit function is $P = 43(x_1 + x_2) - (0.06x_1^2 + 4x_1 + 500) - (0.05x_2^2 + 4x_2 + 400)$. Taking the derivative of $P$ with respect to $x_1$ and $x_2$ and setting them to zero gives us the optimal production levels to maximize profit. Without performing the full calculation here, the correct answer is based on the calculation of these derivatives and solving the resulting equations, which leads to $x_1 = 325$ and $x_2 = 390$.

Suppose a Corporation Manufactures Candles at Two Locations C2=0.05x22+4x2+400C _ { 2 } = 0.05 x _ { 2 } ^ { 2 } + 4 x _ { 2 } + 400C2​=0.05x22​+4x2​+400

Suppose a Corporation Manufactures Candles at Two Locations $C _ { 2 } = 0.05 x _ { 2 } ^ { 2 } + 4 x _ { 2 } + 400$