Question

suppose that a ramp makes an angle of α degrees with the ground. if the ramp has slope m, then we can relate the slope of the ramp and the angle the ramp makes with the ground according to (\frac{alpha}{180}=arctan(m)). if the angle of the ramp is increasing at a rate of 5.2 degrees per minute, what is the rate of change of the slope of the ramp at the instant when the slope is 5? rate of change of slope =

Explanation:

Step1: Recall the relationship between angle and slope

We know that $\frac{\pi\alpha}{180}=\arctan(m)$. Differentiate both sides with respect to time $t$. The derivative of the left - hand side is $\frac{\pi}{180}\frac{d\alpha}{dt}$, and the derivative of the right - hand side is $\frac{1}{1 + m^{2}}\frac{dm}{dt}$ using the chain rule.

Step2: Substitute given values

We are given that $\frac{d\alpha}{dt}=5.2$ degrees per minute. First, convert degrees to radians. Since $1$ degree $=\frac{\pi}{180}$ radians, $\frac{d\alpha}{dt}=5.2\times\frac{\pi}{180}$ radians per minute. We want to find $\frac{dm}{dt}$ when $m = 5$.
From $\frac{\pi}{180}\frac{d\alpha}{dt}=\frac{1}{1 + m^{2}}\frac{dm}{dt}$, we can solve for $\frac{dm}{dt}$.
$\frac{dm}{dt}=(1 + m^{2})\frac{\pi}{180}\frac{d\alpha}{dt}$.
Substitute $m = 5$ and $\frac{d\alpha}{dt}=5.2$ (in degrees per minute, convert to radians: $\frac{d\alpha}{dt}=5.2\times\frac{\pi}{180}$ radians per minute) into the formula.
$\frac{dm}{dt}=(1 + 5^{2})\times\frac{\pi}{180}\times5.2$.
$\frac{dm}{dt}=(1 + 25)\times\frac{\pi}{180}\times5.2$.
$\frac{dm}{dt}=26\times\frac{\pi}{180}\times5.2$.
$\frac{dm}{dt}=\frac{26\times5.2\pi}{180}\approx\frac{135.2\pi}{180}\approx2.36$.

Answer:

$2.36$