Currently, BC is helping $R=5,000$ refugees. The number of refugees that BC must help is rising at a rate of $\frac{dR}{dt}=1,000$ refugees per year. Currently, the number of staff members is $N=100$ and is increasing at a rate of $\frac{dN}{dt}=10$ staff members per year. BC would like to know how quickly the number $H$ of refugees helped by each staff member is changing, and it wants you to predict how many additional refugees each staff member will be responsible for helping in three years’ time.
Instructions:
1: The total number of refugees helped is given by the formula $R=HN$.
Determine the value of $\frac{dH}{dt}$ using the information given above.
2: Use differentials and your result from part (a) to estimate how many additional refugees per year each staff member will have to help three years from now.
I dont really know how to approch this. i tried this but it doesnt really make sense.
The relationship between the number of refugees helped ($R$), the number of staff members ($N$), and the number of refugees helped by each staff member ($H$) is given by the formula $R = HN$.
Taking the derivative of both sides with respect to time ($t$): $$\frac{dR}{dt} = \frac{d}{dt} (HN)$$
Using the product rule on the right-hand side: $$\frac{dR}{dt} = H \frac{dN}{dt} + N \frac{dH}{dt}$$
Given that $\frac{dR}{dt} = 1000$ (rate of increase in refugees) and $\frac{dN}{dt} = 10$ (rate of increase in staff members), we can substitute these values into the equation and solve for $\frac{dH}{dt}$: $$1000 = H \cdot 10 + N \cdot \frac{dH}{dt}$$ $$1000 = 100H + 100 \cdot \frac{dH}{dt}$$
Now, solve for $\frac{dH}{dt}$ by rearranging the equation:
$$\frac{dH}{dt} = \frac{1000 - 100H}{100}$$
Shouldnt my asnwer be in function of t and not in function of $H$? also i tried subsituting $H$ by $R/N$ and then replascing the numbers and it gave me -40 which makes less sense. how do I approach this question. also im lost for the second part.