Answer: The rate of change of the distance between the bottom of the rod and the base of the wall when the height is 35 inches = 1.02675 inches per second.
Step-by-step explanation:
Let h = height, x= distance between the bottom of the rod and the base of the wall.
Given: [tex]\dfrac{dh}{dt}[/tex] = 5 inches per second
Consider the diagram (given in the attachment below):
By Pythagoras theorem, we have
[tex](174)^2=h^2+x^2\ \ (i)\\\\$30276=35^2+x^2 \ \ \ [h=35 \ in.]\\\\ 30276=35^2+x^2 \\\\ 30276=1225+x^2\\\\ x^2=30276-1225\\\\ x^2 = 29051 \\\\ x=\sqrt{29051}\approx170.447 in.[/tex]
Differentiate both sides of (i) w.r.t t, we get
[tex]0=2x\dfrac{dx}{dt}+2h\dfrac{dh}{dt}\\\\\dfrac{dx}{dt}=\dfrac{-h}{x}\dfrac{dh}{dt}[/tex]
[tex]\dfrac{dx}{dt}=\dfrac{-35}{170.44}(-5)=1.02675\text{ inches per second }[/tex]
Hence, the rate of change of the distance between the bottom of the rod and the base of the wall when the height is 35 inches = 1.02675 inches per second.
