The total cost of producing $x$ toys is given by $C(x) = -2350 + 80x - 0.04x^2$. The price function is given by $P(x) = 2x - 35$. We need to find the marginal profit function.

1. Problem Description

The total cost of producing $x$ toys is given by $C(x) = -2350 + 80x - 0.04x^2$. The price function is given by $P(x) = 2x - 35$. We need to find the marginal profit function.

2. Solution Steps

First, we need to find the revenue function, $R(x)$. The revenue is the price per toy times the number of toys sold, so $R(x) = x \cdot P(x)$.

$R(x) = x(2x - 35) = 2x^2 - 35x$

Next, we need to find the profit function, $Pr(x)$. Profit is revenue minus cost, so $Pr(x) = R(x) - C(x)$.

$Pr(x) = (2x^2 - 35x) - (-2350 + 80x - 0.04x^2) = 2x^2 - 35x + 2350 - 80x + 0.04x^2$

$Pr(x) = (2 + 0.04)x^2 + (-35 - 80)x + 2350 = 2.04x^2 - 115x + 2350$

Finally, we need to find the marginal profit function, which is the derivative of the profit function, $Pr'(x)$.

$Pr'(x) = \frac{d}{dx}(2.04x^2 - 115x + 2350) = 2(2.04)x - 115 + 0 = 4.08x - 115$

3. Final Answer

The marginal profit function is $Pr'(x) = 4.08x - 115$.