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Chapter 4 | Applications of Derivatives
4.1 EXERCISES For the following exercises, find the quantities for the given equation.
7. Two airplanes are flying in the air at the same height: airplane A is flying east at 250 mi/h and airplane B is flying north at 300mi/h. If they are both heading to the same airport, located 30 miles east of airplane A and 40 miles north of airplane B , at what rate is the distance between the airplanes changing?
dy dt
at x =1 and y = x 2 +3 if dx
dt =4.
1. Find
dy dt =−1.
2. Find dx dt
at x =−2 and y =2 x 2 +1 if
3. Find dz dt
at ( x , y ) = (1, 3) and z 2 = x 2 + y 2 if
dy dt =3.
dx dt =4
and
For the following exercises, sketch the situation if necessary and used related rates to solve for the quantities. 4. [T] If two electrical resistors are connected in parallel, the total resistance (measured in ohms, denoted by the Greek capital letter omega, Ω) is given by the equation 1 R = 1 R 1 + 1 R 2 . If R 1 is increasing at a rate of 0.5Ω/min and R 2 decreases at a rate of 1.1Ω/min, at what rate does the total resistance change when R 1 =20Ω and R 2 =50Ω ? 5. A 10-ft ladder is leaning against a wall. If the top of the ladder slides down the wall at a rate of 2 ft/sec, how fast is the bottom moving along the ground when the bottom of the ladder is 5 ft from the wall?
8. You and a friend are riding your bikes to a restaurant that you think is east; your friend thinks the restaurant is north. You both leave from the same point, with you riding at 16 mph east and your friend riding 12mph north. After you traveled 4mi, at what rate is the distance between you changing? 9. Two buses are driving along parallel freeways that are 5mi apart, one heading east and the other heading west. Assuming that each bus drives a constant 55mph, find the rate at which the distance between the buses is changing when they are 13mi apart, heading toward each other. 10. A 6-ft-tall person walks away from a 10-ft lamppost at a constant rate of 3ft/sec. What is the rate that the tip of the shadow moves away from the pole when the person is 10ft away from the pole?
6. A 25-ft ladder is leaning against a wall. If we push the ladder toward the wall at a rate of 1 ft/sec, and the bottom of the ladder is initially 20ft away from the wall, how fast does the ladder move up the wall 5sec after we start pushing?
11. Using the previous problem, what is the rate at which the tip of the shadow moves away from the person when the person is 10 ft from the pole?
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