Answer to Problem of the Month for 4/2020

After viewing graphs of the number of confirmed cases of the coronavirus in China and the U.S. ( Click here to see the graphs), I wondered if the cases in Colorado followed a similar pattern.

Here is the data for the first eighteen days from March 5, 2020 to March 22, 2020 in Colorado:

I gathered the data in the following table so that it can be used in this month's problem:

Day Cases

0 2

1 8

2 8

3 8

4 12

5 17

6 34

7 49

8 77

9 101

10 131

11 160

12 183

13 216

14 277

15 363

16 475

17 591

Since it appears to be an exponential function, use the data for day #11 (160 cases) and day #15 (363 cases) to solve for A and B in the general exponential function

y = A e^{B x}.

Now take this equation that you just found and let x = 9, then solve for y to see how close the model is to the actual number of cases.

You must show your algebra to get credit.

Solution to the Problem:
The answers are y = 16.81538 e^{0.204807 x} and y = 106.22.

For x = 11: 160 = A e^{11 B}
For x = 15: 363 = A e^{15 B}

Solving for A in the second equation, we get: A = 363 / (e^15B)

Now subsitute in the first equation to get: 160 = (363 / (e^15B)) e^11B

This simplifies to: 160 = 363 e^{- 4B} or 160 e^4B = 363

Dividing by 160: e^4B = 363 / 160

Now, take the natural log of each side: 4B = ln (363 / 160)

So, B = 0.204807

Now substitute back to get: A = 16.81538

Now take this equation, y = 16.81538 e^{0.204807 x} and replace x by 9 to get:

y = 16.81538 e^{((0.204807)(9))} and therefore y = 106.22.

Correctly solved by:

1. Anna T. Vice	Landon School, Bethesda, Maryland
2. Kelly Stubblefield	Mobile, Alabama
3. James Alarie	Flint, Michigan
4. Brijesh Dave	Mumbai City, Maharashtra, India