chain rule - math

Question

The velocity of an object is given by  dh/dt= 19.21sin(1.7t + 0.3) − 16.32cos(1.7t + 0.3) cm/min

Where h is the height of the object in centimeters and t is time in minutes. The initial height, at t = 0, of this object is 25 centimeters.

1. Find the formula for the height of this object. If you use decimals in your formulas you must be accurate to two decimal digits.

h(t) = _____________

2.Find the velocity of this object at the first instant t > 0 it reaches a height of 45 cm. Be accurate to one decimal digit.

___________

Answer 1

If dh/dt= 19.21sin(1.7t + 0.3) − 16.32cos(1.7t + 0.3) then we can multiply both sides of the equation by dt and then integrate both sides to get:

The antiderivative of "dh" is just "h"

The antiderivative of "19.21sin(1.7t + 0.3) − 16.32cos(1.7t + 0.3)" is "-19.21/1.7cos(1.7t+0.3) - 16.32/1.7sin(1.7t + 0.3)" and then, of course, we add the constant C so:

h = -19.21/1.7cos(1.7t+0.3) - 16.32/1.7sin(1.7t + 0.3) + C

h = -11.3cos(1.7t + 0.3) - 9.6sin(1.7t + 0.3) + C

Using the fact that at t=0, h=25 we can solve for C:

25 = -11.3cos(1.7(0) + 0.3) - 9.6sin(1.7(0) + 0.3) + C

25 = -11.3cos(0.3) - 9.6sin(0.3) + C

25 = -11.3(0.95534) - 9.6sin(0.29552) + C

25 = -10.7953 - 2.8370 + C

C = 25 + 10.7953 + 2.8370 = 38.6323

So: h(t) = -11.30cos(1.70t + 0.30) - 9.60sin(1.70t + 0.30) + 38.63

To find the velocity when h = 45, first find "t" when h = 45:

The easiest way is to graph it. Doing so, you find that t = 1.423

Then plug in t = 1.423 into the velocity equation v(t) = dh/dt= 19.21sin(1.7t + 0.3) − 16.32cos(1.7t + 0.3)

Make sure your calculator is in radians.