To train for a triathlon Cartman swims 125m daily for one week.Each subsequent week he doubles the daily amount until he swims 4km each day for one week. The total distance he has swam is?
What is the equation for this,since he doubles every week
1 km = 1000 m
7(125 + 250 + 500 + 1000 + 2000 + 4000) = ?
I hope this helps.
Gurpreet Ginder
LINT BALLS
To find the total distance Cartman has swum, we can start by finding the number of weeks he has trained and the distance he swims each week.
Let's assume the number of weeks Cartman has trained is "n" and the distance he swims each day in the first week is "d" meters. We can then determine the distance he swims in each subsequent week as he doubles the daily amount.
In the first week, the distance is d meters per day. In the second week, it becomes 2 * d meters per day. In the third week, it becomes 2 * (2 * d) meters per day. And so on, until the nth week when it becomes 2 * 2 * ... * d meters per day, which is equal to 2^n * d meters per day.
Since we know that Cartman swims 125m daily for the first week and eventually reaches 4km (4000m) daily in the final week, we can set up an equation to find the value of n:
2^n * d = 4000
To find n, we need to solve for it. We can do this by dividing both sides of the equation by d and taking the logarithm base 2 of both sides:
log2(2^n * d) = log2(4000)
log2(2^n) + log2(d) = log2(4000)
n + log2(d) = log2(4000)
Now, we know that d is equal to 125m. So let's substitute that value in:
n + log2(125) = log2(4000)
Next, we need to isolate n by subtracting log2(125) from both sides:
n = log2(4000) - log2(125)
Using the properties of logarithms, we can simplify this further:
n = log2(4000/125)
n = log2(32)
Finally, we need to evaluate log2(32):
n = 5
Therefore, Cartman has trained for 5 weeks.
To find the total distance he has swum, we need to sum up the distances for each week. Since the distances double each week, we can use a geometric progression formula to find the sum:
total distance = d * (1 - 2^n) / (1 - 2)
Plugging in the values:
total distance = 125 * (1 - 2^5) / (1 - 2)
total distance = 125 * (1 - 32) / (1 - 2)
total distance = 125 * (-31) / (-1)
total distance = 3,875
Therefore, Cartman has swum a total distance of 3,875 meters.