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 fun 10 feet. On each subseqeunt 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 his plan for very long, but D' Andre is determined to make it work for at least two weeks.
How far will D' Andre have to run on the 14th day? In neartest tenth of a mile. (1 mile= 5280 feet)
If he sticks to the plan how far will he have ran in those two weeks? to the nearest mile.
Is there an equation for that one?^ or do i just have to add them all up?
for the first part I have 10(2)^14-1 and on the 14th day he'll run about 15.5 miles, but what about the second part
Given that f(x) = x² + 4x, evaluate f(-2).
To find out how far D'Andre will have to run on the 14th day, we need to calculate the distance he will run on each day and sum them up.
D'Andre runs 10 feet on the first day and doubles the distance each subsequent day. So we can calculate the distance using the formula:
distance = 10 x (2 ^ (n - 1)), where n is the day number.
To find the 14th day's distance, we substitute n = 14 into the formula:
distance = 10 x (2 ^ (14 - 1))
distance = 10 x (2 ^ 13)
distance = 10 x 8,192
distance = 81,920 feet
Now, let's convert this distance to the nearest tenth of a mile. We know that 1 mile is equal to 5,280 feet. So we can divide the total distance by 5,280 to get the distance in miles:
distance in miles = 81,920 / 5,280
distance in miles ≈ 15.52 miles (rounded to the nearest tenth)
Therefore, D'Andre will have to run approximately 15.52 miles on the 14th day.
To find out how far he will have run in those two weeks, we need to add up the distances for each day. Since there are 14 days, we need to calculate the sum:
total distance = 10 + 20 + 40 + ... + distance on the 14th day
While there is a formula to calculate the sum of a geometric sequence, with this specific sequence, it's easier to use a shortcut.
The sum of a geometric sequence is given by the formula:
sum = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms.
In this case, a = 10, r = 2 (since the distance doubles each day), and n = 14.
Substituting these values into the formula, we get:
sum = 10 * (1 - 2^14) / (1 - 2)
sum = 10 * (1 - 16,384) / (1 - 2)
sum = 10 * (-16,383) / (-1)
sum = 10 * 16,383
sum = 163,830 feet
Now, let's convert this distance to the nearest mile. We divide the total distance by 5,280 to get the distance in miles:
total distance in miles = 163,830 / 5,280
total distance in miles ≈ 31 miles (rounded to the nearest mile)
Therefore, if D'Andre sticks to the plan for two weeks, he will have run approximately 31 miles.