Question

A right triangle has legs of 18 inches and 24 inches whose sides are

Answer 1

Given:

A right triangle has legs of 18 inches and 24 inches.

The short leg is increasing by 4 in/sec and the long leg is  shrinking at 9 in/sec.

To find:

The rate of change of the hypotenuse.

Solution:

Let x be the shorter leg (Base), y be the larger leg (Perpendicular) and z be the hypotenuse.

We have,

$\dfrac{dx}{dt}=4\text{ in/sec}$

$\dfrac{dy}{dt}=-9\text{ in/sec}$

$x=18$

$y=24$

According to the Pythagoras theorem,

$Hypotenuse^2=Base^2+Perpendicular^2$

$z^2=x^2+y^2$           ...(i)

$z^2=(18)^2+(24)^2$

$z^2=324+576$

$z^2=900$

Taking square root on both sides.

$z=\pm \sqrt{900}$

Side cannot be negative. So,

$z=30$

Differentiating (i) with respect to time t, we get

$\dfrac{d}{dt}z^2=\dfrac{d}{dt}(x^2+y^2)$

$2z\dfrac{dz}{dt}=2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt}$

$2(30)\dfrac{dz}{dt}=2(18)(4)+2(24)(-9)$

$60\dfrac{dz}{dt}=144-432$

$60\dfrac{dz}{dt}=-288$

Divide both sides by 60.

$\dfrac{dz}{dt}=\dfrac{-288}{60}$

$\dfrac{dz}{dt}=-4.8$

Here, negative sign means hypotenuse is decreasing.

Therefore, the hypotenuse is shrinking at 4.8 in/sec.