Question

The number of fish (F) in a fish tank increases when the number of plants (P) increases. Write the correct equation for this scenario, and solve for the number of fish when the number of plants is 3.

Answer 1

Assuming the number of fish depends only on the number of plants, the first sentence translates to saying there is some \(k>0\) such that
\[\frac{\mathrm dF}{\mathrm dP}=kP\]In other words, \(F(P)\) is an increasing function because the derivative is positive.

You can solve for \(F\) directly to obtain a general solution, but the solution will be in terms of the unknown \(k\) and the constant of integration. So just knowing what \(F\) is for only one given \(P\) is not enough information to find the exact value of \(F(3)\). Are you given any more conditions?

Answer 2

The equation of a line that passes through two given points, (x1, y1) and (x2, y2) is

y−y1=y2−y1x2−x1(x−x1)

This equation is called the two-point form of the equation of a line.

You have two points. Call one point (x1, y1), and the other point (x2, y2), and find the equation of the line using the equation above.
After you have the equation, replace x with P and y with F.

Then calculate F for P = 3.