Respuesta :
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,
[tex]\dfrac{dx}{dt}=4\text{ in/sec}[/tex]
[tex]\dfrac{dy}{dt}=-9\text{ in/sec}[/tex]
[tex]x=18[/tex]
[tex]y=24[/tex]
According to the Pythagoras theorem,
[tex]Hypotenuse^2=Base^2+Perpendicular^2[/tex]
[tex]z^2=x^2+y^2[/tex] ...(i)
[tex]z^2=(18)^2+(24)^2[/tex]
[tex]z^2=324+576[/tex]
[tex]z^2=900[/tex]
Taking square root on both sides.
[tex]z=\pm \sqrt{900}[/tex]
Side cannot be negative. So,
[tex]z=30[/tex]
Differentiating (i) with respect to time t, we get
[tex]\dfrac{d}{dt}z^2=\dfrac{d}{dt}(x^2+y^2)[/tex]
[tex]2z\dfrac{dz}{dt}=2x\dfrac{dx}{dt}+2y\dfrac{dy}{dt}[/tex]
[tex]2(30)\dfrac{dz}{dt}=2(18)(4)+2(24)(-9)[/tex]
[tex]60\dfrac{dz}{dt}=144-432[/tex]
[tex]60\dfrac{dz}{dt}=-288[/tex]
Divide both sides by 60.
[tex]\dfrac{dz}{dt}=\dfrac{-288}{60}[/tex]
[tex]\dfrac{dz}{dt}=-4.8[/tex]
Here, negative sign means hypotenuse is decreasing.
Therefore, the hypotenuse is shrinking at 4.8 in/sec.
