A sample of bacteria is growing at a continuously compounding rate. The sample triples in 10 days. Find the formula for the daily rate. Type your answer as a fraction.

The number of bacteria  after  days is given by

where  is the initial number of bacteria.  

Step-by-step explanation:

The number of bacteria  in the sample triples every 10 days, this means after the first 10th day, the number of bacteria is

where  is the initial number of bacteria in the sample.

After the 2nd 10th days, the number of bacteria is

after the 3rd day,

and so on.

Thus, the formula we get for the number of bacteria after the nth 10-days is

where  is is the nth 10-days.

Since,  is 10 days, we have

or

Explanation:

Here's one way:

At vertex A, draw the vertical angle to the given angle 4. That is, extend AC, rather than AB to show the exterior angle at that vertex.

Now, collapse triangle to one with zero area, so that vertices A, B, C lie on top of each other. You can do this using a dilation scale factor f such that f→0.

The result is that the exterior angles divide the circular space around point ABC into three parts that sum to 360°. Essentially, we have copied each of the exterior angles to the same vertex.

The sum of the measures of the exterior angles is 360°.

Additional comment

This identical approach can be used to show the sum of exterior angles around any convex polygon is always 360°.