Math 5 Problem set on Related Rates: (Ele-Glu-Gra-Pho)TBS: Nov. 3 or 4, class timeInstructions: This should be done individually. People will be randomly selected to discuss their answers to the problems during class time. We might have a quiz where one of these problems will be given.Use the following guide to answer the questions:

A. Draw a figure and represent quantities that change over time with variablesB. What rates/details are given? What rate has to be computed?C. What is the relationship of the variables?D. Solve for the rate and make the necessary conclusion.

1. A particle is moving along the curve y=√1+x3. As it reaches the point (2,3), the y-coordinate is

increasing at a rate of 4cm/s. a. How fast is the x-coordinate of the point changing at that instant?b. How fast is the distance of the particle from the origin changing as it reaches the point (2,3)?

2. A snowball melts in such a way that the surface area decreases at a rate of 1cm2/minute. What is the rate at which the DIAMETER decreases when the diameter is 10cm?

3. Two resistors with resistances R1 and R2 are connected in PARALLEL. If their resistances are 100Ω and 80 and they are changing at a rate of 0.3 /s and 0.2 /s respectively, what is the rate of changeΩ Ω Ω of the total resistance?

4. A two-mole ideal gas is placed in a 2L flask that is expanding at 0.1L/min. If the temperature is 300K, and it is increasing at a rate of 1K/min, how fast is the pressure changing?

5. A lighthouse is located on an island 3km away from the nearest point O on a straight shoreline (OP). The light on the lighthouse makes four revolutions per minute (that is 8 rad/min). How fast is theπ beam of light moving when it is at point P which is 4km from O?

6. A 15-foot ladder is leaning against a building. The bottom of the ladder is moving away from the building at a rate of ½ft/s. Find the rate of change of the area of the triangle formed by the building, ground and ladder when the bottom of the ladder is 9ft from the building.

7. Two carts, A and B, are connected by a rope 39 feet long, that passed over a pulley P (see figure below). The point Q is on the floor directly below the pulley. Cart A is being pulled away from P at 2 ft/s.a. If Z is the length of rope from cart A to P, how fast is Z getting longer when cart A is 5 ft from Q?b. How fast is cart B moving toward Q?

8. Two sides of a triangle have lengths 12ch and 15cm. the angle between them is increasing at a rate of π90

rad/s. Use the cosine law to determine the rate at which the third side is increasing when the

angle between the fixed sides is 60 °?

Bonus: (You are not required to answer these questions, but these questions are worth answering for additional review for the upcoming exam.)

B1: A runner sprints at 7m/s around a circular track with radius 100m. The runner’s friend is standing at a distance of 200m from the center of the track. How fast is the distance between the friends changing when the distance between them is 200m?

B2: The minute hand on a watch is 8mm long, and the hour hand is 4mm long. How fast is the distance between the tips of the hands of the clock changing at 1o’clock?

B3: As the last car of a train passes under a bridge (imagine a straight bridge with a train track under it), a car crosses the bridge on the roadway perpendicular to the track and 30ft above it. The train is traveling at a rate of 80ft/s and the car is traveling at 40m/s. how fast are the train and car separating after 2 seconds?

B4: A sidewalk runs parallel to a 20-foot wall and is 5 feet away from the wall. There is a light on the top of the wall directly above a faucet on the foot of the wall. A girl, 4 feet tall, walks on the sidewalk at 5ft/s so that she’s moving closer to the light. At what rate is her shadow changing when her feet are 13 ft from the faucet?

B5: A painting 3-feet high is placed on a wall, and the lower end of the painting is 4 feet from the floor. A tall man starts walking towards the painting at a rate of 2 ft/s (suppose the eye level of the person is 6 feet high). How fast is the angle subtended by the painting to the person’s eye increasing when the man was 5ft from the wall? (Angle subtended by the painting is the angle the entire painting makes to the eyes of the person. Use the arctangent function to find the angle subtended by the painting. The derivative of the arctangent function was already discussed in class.)