Question

There were about 1.86 million Hispanic-owned businesses in 1999 and 2.83 million in 2004. a) Find an exponential model for this data in which t=0 corresponds to 1999 and the number of businesses is measured in millions. f(t)= b) Assuming the model remains accurate, how many million Hispanic-owned businesses will there be in 2031? million businesses

Answer 1

The exponential model for the given data can be represented by the function f(t) = 1.86 * 1.072^t. In 2031, there will be approximately 4.31 million Hispanic-owned businesses.

Step-by-step explanation:

To find an exponential model for the data, we need to use the general form of an exponential function: f(t) = ab^t, where t is time and f(t) is the number of businesses.

Given that t=0 corresponds to 1999, we can substitute the given values:

1999: f(0) = 1.86

2004: f(5) = 2.83

Using these values, we can set up a system of equations:

1.86 = ab^0

2.83 = ab^5

Since b^0 is equal to 1, the first equation becomes:

1.86 = a

Substituting this value of a in the second equation, we have:

2.83 = 1.86 * b^5

Dividing both sides of the equation by 1.86, we get:

1.52 = b^5

Taking the fifth root of both sides, we have:

b = 1.52^(1/5) ≈ 1.072

Therefore, the exponential model for this data is:

f(t) = 1.86 * 1.072^t

To find the number of businesses in 2031, we need to substitute t = 32 (2031 - 1999) into the exponential model:

f(32) = 1.86 * 1.072^32 ≈ 4.31 million businesses

Answer 2

a) The exponential model for the number of Hispanic-owned businesses, with t representing years since 1999 and f(t) in millions, is:

f(t) = 0.89 * (1 + 52.15%)^t

b) Assuming the model remains accurate, there will be approximately 6055.38 million Hispanic-owned businesses in 2031.

Step-by-step explanation:

Calculating the growth rate:

Growth rate = (2.83 million - 1.86 million) / 1.86 million ≈ 52.15%

Initial value and formula:

The initial value (a) is 1.86 million, as it represents the number of businesses in 1999 (t = 0).

The exponential model is f(t) = a * (1 + growth rate)^t.

Plugging in values:

f(t) = 1.86 million * (1 + 52.15%)^t

Simplifying, we get: f(t) = 0.89 * (1 + 52.15%)^t

Number of businesses in 2031:

For 2031, t = 2031 - 1999 = 32.

Substituting t = 32 into the model: f(32) ≈ 6055.38 million

Therefore, the exponential model provides a good fit for the data and predicts approximately 6055.38 million Hispanic-owned businesses in 2031. Remember, this is based on an assumption of the model's continued accuracy and should be interpreted cautiously.