Reference no: EM13914725
1. Four students disagree as to the value of the integral 0∫π sin8x dx. Josh says that it is equal to π. Jordan says that it is equal to 35π/128. Chad claims it is equal to 3π/90 - 1 and Lindsay thinks it is equal to π/2. One of them is correct. Which one is it? Do not evaluate the integral. Your answer must depend on eliminating the three wrong answers. In order to do this, consider what 0∫π sin8x dx represents geometrically.
2. A rubber ball rebounds to two-thirds the height from which it falls. If it is dropped from a height of 4 feet and is allowed to continue bouncing indefinitely, what is the total distance it travels?
3. A hare runs a two-mile race with a tortoise. The tortoise has a one-mile head start. Even though the hare runs ten times faster than the tortoise, it seems unable to catch up for the following reason: When the hare gets to the spot where the tortoise started, the tortoise is one-tenth of a mile farther on; when the hare gets to this one-and-one-tenth-mile point, the tortoise has moved on from it and is still ahead; and so on. Explain this seeming paradox.
4. Construct a sequence of figures T0, T1, T2,...., Tn, ... (see below) in the following way. T0 is an equilateral triangle of side length one; T1 is obtained from T0 by replacing the middle third of each edge of T0 by an outward equilateral triangle (of side length 1/3); T2 is obtained from T1 by replacing the middle third of each edge of T1 by an outward equilateral triangle (of side length 1/9); and so on. (The area of an equilateral triangle with side length s is √(3)s2/4.)
a) Find Sn, the number of edges in the perimeter of Tn.
b) Find Ln, the length of each edge in the perimeter of Tn.
c) Find Pn, the perimeter of Tn.
d) Find limn→∞ Pn.
e) How much area is added in going from Tn to Tn+1?
f) Find An, the area of Tn.
g) Find limn→∞An.
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