If the base \(b\) of a triangle is decreasing at a rate of \(10\) inches per minute while its height is increasing at a rate of \(10\) inches per minute, which of the following must be true about the area \(A\) of the triangle?

Summary Submit

$$ A=\frac{1}{2}bh $$

Use the product rule to differentiate the area equation.

$$ \frac{dA}{dt} = \frac{1}{2}\left(b\frac{dh}{dt}+h\frac{db}{dt}\right) $$

Substitute known values and simplify:

$$ \frac{dA}{dt} = \frac{1}{2}(10b-10h) $$ $$ \frac{dA}{dt} = 5(b-h) $$

\(\dfrac{dA}{dt}\) is negative if \(b \lt h\).