A telephone line hangs between two poles at 12m apart in the shape of a catenary y=50cos(x/50)-45, where x and y are measured in meters. Find the slope of this curve where it meets the right pole.

Question

A telephone line hangs between two poles at 12m apart in the shape of a catenary y=50cos(x/50)-45, where x and y are measured in meters.  Find the slope of this curve where it meets the right pole.

dumbcow · Answer

In other words find the slope(dy/dx) when x=12
Differentiate the function y
derivative of constant is zero, so the 45 just goes away
for first term use chain rule:
$$d/dx \cos u = -\sin u$$
$$d/dx \frac{x}{50} = \frac{1}{50}$$
$$\rightarrow d/dx 50\cos(x/50) = 50*\frac{1}{50}*-\sin(x/50) = -\sin(x/50)$$
Now substitute in x=12
$$dy/dx = -\sin(12/50)= -0.2377$$