Get expert insights and reliable answers to your questions on IDNLearn.com. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
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.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.