Question

Help me solve this problem!!

Answer 1

• f(t) = 1.15(152/115)^(t/4)
• f(25) ≈ 6.57 million

Explanation:

(a) The ratio in 4 years is 1.52/1.15 = 152/115. So, the exponential function can be written using the form ...

f(t) = (initial value) × (ratio in period)^(t/(length of period))

Here, the initial value is 1.15 million, the ratio in the period of 4 years is 152/115, so the exponential function is ...

f(t) = 1.15(152/115)^(t/4)

Since no calculations were done, no rounding is necessary.

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(b) After 25 years, this formula predicts the number of Hispanic owned businesses to be ...

f(25) = 1.15(152/115)^(25/4) ≈ 6.57 . . . . . million

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Alternate expressions of the exponential function

We can divide out the ratio we used and write the function using that:

f(t) = 1.15·1.321739^(t/4)

or we can take the 4th root of it to give ...

f(t) = 1.15·1.0722263^t

or we can take the log of it to get ...

f(t) = 1.15e^(0.069737t)

and we can even fold the initial constant into the exponent:

f(t) = e^(0.069737t +0.139762)

or rearrange to a slightly different form:

f(t) = e^(0.069737(t +2.004126))

Answer 2

Answer: Approximately 6573481.358 Hispanic businesses

Explanation:

In 2009, the number of Hispanic owned businesses was 1.15 million

In 2013, the number of Hispanic owned businesses was 1.52 million

Since the growth is exponential, we would apply

Tt = ar^t-1

Where Tt is the number of Hispanic owned businesses after t years

a is the number of Hispanic owned businesses in 2009

r is the common ratio or exponential growth rate of the number of Hispanic owned business

at 2013 , t = 5(t= 0 in 2009 to t= 5 in 2013)

1520000 = 1150000 × r^5-1

1520000 / 1150000 = r^4

152/115 = r^4

1.32174 = r^4

Taking 4th root of both sides,

r = 1.07222

We want to determine the number of Hispanic owned business that would be there in 2034. It becomes

T2034 = 1150000 × 1.07222^25

T2034 = 1150000 × 5.7160

T2034 = 6573481.3579901

Approximately 6573481.358 Hispanic businesses