Finding the profit from a demand equation

Question

The demand equation for a company's product is $3p + x = 400$, where $x(t)$ units can be sold at a price of $p$ each. If the demand increases at a rate of 3 units per year when the demand reaches 50 units, the fixed costs are 200, and the variable cost per unit is 60. What is the rate of change of the profit per year?

I can't understand this statement. I will put what I have understood:

Profit: $U$

Total Cost: $C$

Total revenue: $R$

$U=R-C$

$R=px$

$C=200+60x$

$U=R-C$

$U=px-(200+60x)$

What I can't understand is the phrase "If the demand increases at a rate of 3 units per year when the demand reaches 50 units"

This means that $dx/dt=3$ then $\int_{50}^{x(t)} dx=2\int_{t=0\text{ or }1 \text{?}}^{t}dt$?

Thanks.

Answer 1

The wording of the question could be improved. Assuming the question is asking for the rate of change of profit per year when demand is 50 units and has a rate of change of 3 units per year, you can apply the chain rule where you left off:

$$\begin{align} U&=x\left({400-x \over 3}\right)-(200+60x)\\ \implies {\partial U \over \partial t}&={\partial U \over \partial x}{\partial x \over \partial t}\\ &=\left({400 \over 3}-{2\over 3}x-60\right)(3). \end{align}$$

When $x=50$, we thus get $$\partial U/ \partial t ={\\\$} 120/\text{year}$$.