Q65. - Calculus

Question

Suppose a right triangle has legs of lengths a and b and a

hypotenuse of length h and that this triangle is changing

size, so that the length of its hypotenuse does not change.

Prove that the ratio of the rates of change $\frac{da}{dt}$and $\frac{db}{dt}$is equal to $-\frac{b}{a}$.

Step-by-Step Solution

Answer

$\frac{{\displaystyle \frac{da}{dt}}}{{\displaystyle \frac{db}{dt}}}=-\frac{b}{a}$

Step 1. Given Information

The length of the two sides of a right-angle triangle is $"a" \& "b"$

and the length of its hypotenuse is $"h".$

Step 2. Calculation

By Pythagora's theorem

$h^2=a^2+b^2$

Differentiating both sides with respect to$"t"$we get

$2a\frac{da}{dt}+2b\frac{db}{dt}=0$, since hypotenuse remains constant.

or, $a\frac{da}{dt}=-b\frac{db}{dt}$

Therefore, $\frac{{\displaystyle \frac{da}{dt}}}{{\displaystyle \frac{db}{dt}}}=-\frac{b}{a}$