Answer:
- f(t) = 1.15(152/115)^(t/4)
- f(25) ≈ 6.57 million
Step-by-step 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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<em>Alternate expressions of the exponential function</em>
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))
