Question

There were about 1.22 million Hispanic-owned businesses in 2012 and 1.73 million in 2017.

Find an exponential model for this data in which t = 0 corresponds to 2012 and the number of businesses is measured in millions. Round computed values to at least 4 decimal places

Answer 1

The exponential model for the data is N = 1.22 * e^(0.06852t), where N is the number of businesses and t is the number of years from 2012.

Step-by-step explanation:

To find an exponential model for the data, let's use the formula N = N0 * e^(kt), where N is the number of businesses, N0 is the initial number of businesses, t is the number of years from the initial time, k is the growth rate, and e is the base of the natural logarithm.

From the given data, we can set up two equations:

1.22 = N0 * e^(k * 0) --> N0 = 1.22

1.73 = 1.22 * e^(k * 5) --> 1.41803 = e^(5k)

Taking the natural logarithm of both sides, we get:

ln(1.41803) = ln(e^(5k)) --> ln(1.41803) = 5k

Solving for k, we divide both sides by 5:

k = ln(1.41803) / 5 ≈ 0.06852

Therefore, the exponential model is N = 1.22 * e^(0.06852t). This model predicts the number of Hispanic-owned businesses in millions based on the year, with t = 0 corresponding to 2012.