Suppose that a side a is increasing at a rate of 2 meter/second and c is fixed at 10 meters. find the rate...?

Question

Suppose that a side a is increasing at a rate of 2 meter/second and c is fixed at 10 meters. Find the rate of change of the length of side b when b=5 meters.

Rate of change =

Answer 1

There isn't enough information to solve the problem. It sounds like you are referring to a triangle, but you aren't telling us what kind of triangle. Is it a right triangle? The answer would be different for each angle between sides b and c.

If it is a right triangle, then we know:

a^2 = b^2 + c^2 so b^2 = a^2 - c^2 or
b = sqrt(a^2 - c^2)

we are interested in the formula for db/dt, so we need to use the chain rule; if H(t) = F(G(t)) then:

dH/dt = (dF/du)(du/dt) where

u = G(t) so du/dt = dG/dt

In this case:
F(u) = sqrt(u)
G(t) = a(t)^2 - c^2

So we compute the formula for H'(t) - the rate at which the b changes. But we need to evaluate H'(t) when b = 5. But we know that at that time a = sqrt(b^2 + c^2) so we have both a and a' and so can evaluate H' (aka db/dt)