D’Andre has decided that he needs to get more exercise, so he is building a running regimen for himself. He decides that, beginning on Sunday, he’ll run 10 feet. On each subsequent day, he’ll double the distance he ran the previous day. So he’ll run 20 feet on Monday, 40 feet on Tuesday, and so on. His Track coach (who also happens to be his math teacher) told D’Andre that he won’t be able to keep to his plan for very long, but D’Andre is determined to make it work for at least two weeks.
a. According to his plan, how far will D’Andre have to run on the 14th day? Give your answer to the nearest tenth of a mile (1 mile = 5,280 feet).
b. If D’Andre sticks to his plan, how far will he have run in those 2 weeks? Give your answer to the nearest tenth of a mile.
a. On the 14th day, D'Andre will have to run 10 feet * 2^(14-1) = 10 feet * 2^13 = 10 feet * 8192 = 81,920 feet. Converting this to miles, we have 81,920 feet / 5,280 feet/mile = 15.52 miles.
b. To calculate how far D'Andre will have run in 2 weeks, we need to sum up the distances he ran each day:
10 feet + 20 feet + 40 feet + ... + 81,920 feet.
This is a geometric series with a first term of 10 feet and a common ratio of 2. The sum of a geometric series is given by the formula: S = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, n is the number of terms.
Plugging in the values:
S = 10 * (1 - 2^14) / (1 - 2) = 10 * (1 - 16384) / -1 = 10 * -16383 / -1 = 163,830 feet.
Converting this to miles, we have 163,830 feet / 5,280 feet/mile = 31.07 miles.
Therefore, if D'Andre sticks to his plan, he will have run approximately 31.1 miles in 2 weeks.