A Magical Circle
The radius of a circle is initially 2 mm. But it’s a magical circle, and it suddenly begins growing before your eyes, its radius growing longer at a constant rate of 3 mm/s. At what rate is the area of the circle increasing at the instant the radius reaches 7 mm?
Magical Rectangle
The length and width of a magical rectangle each suddenly begin growing. The length is growing at a constant rate of 3 ft/s and the width is growing at a constant rate of 2 ft/s. There is an instant in time that the length and width are exactly 10 feet and 7 feet respectively. At what rate is the area of the rectangle increasing?
The Third Baseman’s Perspective
A baseball diamond is a 90 ft square. There are runners on first and second. The batter hits an infield ground ball which is thrown to third base for a force out. Meanwhile the batter is running to first base at 26 ft/s. Consider the instant at which the third baseman looks to throw to first and sees the runner exactly 48 ft down the first base line.
- How far is the runner from the third baseman at this instant?
- At what rate is the straight-line distance between the runner and the third baseman increasing at this instant?
- From the third baseman’s perspective, at what rate, in radians-per-second, is the angle between the runner and home plate changing at this same instant?
Jimmy Running Home
The sun has set. Little Jimmy, so-named because he’s only 3′4″ tall, is out well past curfew and is running home. He runs past a 13-foot tall streetlamp at 11.9 ft/s, his shadow growing on the sidewalk in front of him as he runs. How quickly is the tip of Jimmy’s shadow moving away from the streetlamp the moment Jimmy is 25 ft past that streetlamp?