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Fun with related rates (calculus problems)

We started this blog less than two weeks ago and we're already neglecting it. What bad bloggers we are. I don't know what Neal's excuse is, but mine is that my calculus class has taken over my life this week -- our first exam is tomorrow.

So, in lieu of more interesting content, here's a taste of what's been occupying me:

"A plane flying horizontally at an altitude of 1/2 km and a speed of 200 km/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 km away from the station."

"A street torch is mounted atop a 12 ft pole. A 6 ft woman strolls away from the pole with the speed of 1 ft/s along a straight path. How fast is the tip of her shadow moving when she is 7 ft away from the pole?"

"A spotlight on the ground shines on the wall 8 m away. If a child 1 m tall walks from the spotlight toward the wall at the speed of 0.9 m/s, how fast is the length of her shadow on the wall decreasing when she is at the halfway point?"

"Sand is being dumped from a conveyor belt and forms a conical pile. Assuming that the height of this cone is always exactly 3 times the size of the radius of its base, and that the sand is added at the rate of 10 m^3/min, how fast is the height increasing when the pile is 15 m high?"

"The length of a rectangle is increasing at a rate of 8 cm/s and its width is increasing at a rate of 3 cm/s. When the length is 20 cm and the width is 10 cm, how fast is the area of the rectangle increasing?"

"If a snowball melts so that its surface area decreases at a rate of 1 cm^2/min, find the rate at which the diameter decreases when the diameter is 10 cm."

"At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at 4:00 PM?"

"Two cars start moving from the same point. One travels south at 60 mi/h and the other travels west at 25 mi/h. At what rate is the distance between the cars increasing two hours later?"

"A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s, how fast is the boat approaching the dock when it is 8 m from the dock?"

The challenge of these related rates problems is thinking carefully about all the functions and variables involved. It's not like the relatively straightforward numerical problems, which can always just be plugged into Wolfram Alpha to check answers or see the steps to solve the problem when stuck. (Hat tip to my classmate Peter for showing me how to use Wolfram Alpha -- it's a great resource for math students!)

Enjoy working your way through those problems. I'll report back next week about how I do on my exam.

About Jackie

MBA student, Girl Friday for engineers & scientists, and science fiction & fantasy geek.