Thread: Average Value of a Function

1. Average Value of a Function

I'm working on a school assignment and need someone to check it.
Thanks!

The growth rate for a 24-hour period of a bacteria colony is measured to be r(t)=96e^(.24t), where r(t) is measured in bacteria per hour. At t=0, there are 400 bacteria in the colony. Find the average groth rate for the time period 0 equal to or less than t equal to or less than 24.
How many bacteria would be in the colony at the end of the 24-hour period?Include units.
Solution:
integral with bounds [24,0] 96e^(.24t)
integral ==> 400e^(.24t)
400e^(.24(24)) - 400e^(.24(0)) = 126539 bacteria per hour.
&
126539 * 24=3036944 bacteria per day.

Is this the correct solution? My other solution was to solve for
r(0)+integral with bounds [24,0] 96e^(.24t). I added the r(0) because it's the initial bacteria count.

Sorry it's a little hard to read.

2. Originally Posted by iyppxstahh
I'm working on a school assignment and need someone to check it.
Thanks!

The growth rate for a 24-hour period of a bacteria colony is measured to be r(t)=96e^(.24t), where r(t) is measured in bacteria per hour. At t=0, there are 400 bacteria in the colony. Find the average groth rate for the time period 0 equal to or less than t equal to or less than 24.
How many bacteria would be in the colony at the end of the 24-hour period?Include units.
Solution:
integral with bounds [24,0] 96e^(.24t)
integral ==> 400e^(.24t)
400e^(.24(24)) - 400e^(.24(0)) = 126539 bacteria per hour.
&
126539 * 24=3036944 bacteria per day.

Is this the correct solution? My other solution was to solve for
r(0)+integral with bounds [24,0] 96e^(.24t). I added the r(0) because it's the initial bacteria count.

Sorry it's a little hard to read.
I get the same antiderivative as you, and the same value for the integral, but I think you set up the problem slightly wrong. Taking the integral gives you total growth, so you have 126539 bacteria, not bacteria per hour. To get average growth rate, you must divide by the interval length, in this case 24 hours, so I get

average growth rate = 126539 / 24 = 5272.45833333... bacteria / hour

Let x(0) = 400 = number of bacteria at time t = 0. It is bad to say r(0) = 400, because r(t) is already defined as rate.

We can use average growth rate to find the number of bacteria by taking x(24) = x(0) + (24 * average).

Or we can simply use the intermediate result from before and say that x(24) = x(0) + 126539 = 400 + 126539 = 126939 bacteria after 24 hours.

As I said in another post to a similar question of yours just now, I'm a bit rusty, so I wouldn't mind if someone else confirmed my results.

3. Right, thank you! I understand the problem as well as my mistake now.