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Answer :
To find out how the function [tex]\( f(t) = 1.3^t \)[/tex] changes over the interval from [tex]\( t = 2 \)[/tex] to [tex]\( t = 4 \)[/tex], we need to calculate the percentage change in the function's value over this interval.
1. Calculate the Value at [tex]\( t = 2 \)[/tex]:
- We evaluate the function at [tex]\( t = 2 \)[/tex]:
[tex]\[
f(2) = 1.3^2 = 1.69
\][/tex]
2. Calculate the Value at [tex]\( t = 4 \)[/tex]:
- We then evaluate the function at [tex]\( t = 4 \)[/tex]:
[tex]\[
f(4) = 1.3^4 = 2.8561
\][/tex]
3. Determine the Change in Value:
- The difference in the function's values is:
[tex]\[
\Delta f = f(4) - f(2) = 2.8561 - 1.69 = 1.1661
\][/tex]
4. Calculate the Percentage Change:
- To find the percentage change from [tex]\( t = 2 \)[/tex] to [tex]\( t = 4 \)[/tex], use the formula for percentage change:
[tex]\[
\text{Percentage Change} = \left( \frac{\Delta f}{f(2)} \right) \times 100
\][/tex]
- So the calculation is:
[tex]\[
\text{Percentage Change} = \left( \frac{1.1661}{1.69} \right) \times 100 \approx 69\%
\][/tex]
Therefore, the function [tex]\( f(t) = 1.3^t \)[/tex] increases by approximately 69% over the interval from [tex]\( t = 2 \)[/tex] to [tex]\( t = 4 \)[/tex].
1. Calculate the Value at [tex]\( t = 2 \)[/tex]:
- We evaluate the function at [tex]\( t = 2 \)[/tex]:
[tex]\[
f(2) = 1.3^2 = 1.69
\][/tex]
2. Calculate the Value at [tex]\( t = 4 \)[/tex]:
- We then evaluate the function at [tex]\( t = 4 \)[/tex]:
[tex]\[
f(4) = 1.3^4 = 2.8561
\][/tex]
3. Determine the Change in Value:
- The difference in the function's values is:
[tex]\[
\Delta f = f(4) - f(2) = 2.8561 - 1.69 = 1.1661
\][/tex]
4. Calculate the Percentage Change:
- To find the percentage change from [tex]\( t = 2 \)[/tex] to [tex]\( t = 4 \)[/tex], use the formula for percentage change:
[tex]\[
\text{Percentage Change} = \left( \frac{\Delta f}{f(2)} \right) \times 100
\][/tex]
- So the calculation is:
[tex]\[
\text{Percentage Change} = \left( \frac{1.1661}{1.69} \right) \times 100 \approx 69\%
\][/tex]
Therefore, the function [tex]\( f(t) = 1.3^t \)[/tex] increases by approximately 69% over the interval from [tex]\( t = 2 \)[/tex] to [tex]\( t = 4 \)[/tex].
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