# Finding the profit from a demand equation

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.

\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 $$\partial U/ \partial t ={\\\} 120/\text{year}$$.