A 10 ft ladder leans against a wall at an angle θ with the
horizontal, as shown in the accompanying figure. The top
of the ladder is x feet above the ground. If the bottom of
the ladder is pushed toward the wall, find the rate at which
x changes with respect to θ when θ = 60 ◦ . Express the
answer in units of feet/degree.
In all of my earlier questions I had added process how I solved. But, in this question. I am unable to understand the first process. The is saying to solve $dx=d\theta$, isn't it? I have length of ladder which is 10 ft. But, I think that's not $x$. $x$x is base that's what I think. So, the question is saying to find base changes with respect to angle. Earlier in all of my question I had an equation like this $x=2a^2$(just a sample) But, this time I don't have any equation given. I was thinking to solve it using $\tan$. But, I don't know what to write in first line.
$$dx/d\theta = ?$$
Best Answer
You are required to find the rate of change of x with respect to theta(𝜃). You have an equation in hand, i.e, using $$ sin𝜃=\frac{x}{10}$$ Differentiating it with respect to 𝜃 we get $$ cos𝜃=\frac{\frac{dx}{d𝜃}}{10}$$ $$ \frac{dx}{d𝜃}=10cos𝜃$$
We want to find the rate at 𝜃=60 deg,so substituting it,we get,
$$\frac{dx}{d𝜃}=5 feet/deg$$
Hope it helps!!