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Answer :
To solve the exponential equation [tex]\( y = 172(1 + 0.3)^t \)[/tex] and find the value of [tex]\( y \)[/tex] after a certain time [tex]\( t \)[/tex], we will follow these steps:
1. Identify the components of the equation:
- The initial value [tex]\( a \)[/tex] is 172.
- The growth factor is [tex]\( 1 + 0.3 = 1.3 \)[/tex].
- [tex]\( t \)[/tex] represents the time period and is given as an example with a specific value (e.g., [tex]\( t = 5 \)[/tex]).
2. Use the given values to calculate [tex]\( y \)[/tex]:
- We have:
- Initial amount, [tex]\( a = 172 \)[/tex].
- Growth rate factor, [tex]\( r = 1.3 \)[/tex].
- Time period, [tex]\( t = 5 \)[/tex] (as per the example given).
3. Apply the values to the equation:
- Substitute the values into the equation:
[tex]\[
y = 172 \times (1.3)^t
\][/tex]
- With [tex]\( t = 5 \)[/tex]:
[tex]\[
y = 172 \times (1.3)^5
\][/tex]
4. Calculate the result:
- First, compute [tex]\( (1.3)^5 \)[/tex], which involves multiplying 1.3 by itself five times.
- Next, multiply the result by 172 to get the final value of [tex]\( y \)[/tex].
5. Final Result:
- After calculating the expression, the value of [tex]\( y \)[/tex] when [tex]\( t = 5 \)[/tex] is approximately:
[tex]\[
y \approx 638.62
\][/tex]
This represents the value after 5 time periods of exponential growth given the initial amount and growth rate provided.
1. Identify the components of the equation:
- The initial value [tex]\( a \)[/tex] is 172.
- The growth factor is [tex]\( 1 + 0.3 = 1.3 \)[/tex].
- [tex]\( t \)[/tex] represents the time period and is given as an example with a specific value (e.g., [tex]\( t = 5 \)[/tex]).
2. Use the given values to calculate [tex]\( y \)[/tex]:
- We have:
- Initial amount, [tex]\( a = 172 \)[/tex].
- Growth rate factor, [tex]\( r = 1.3 \)[/tex].
- Time period, [tex]\( t = 5 \)[/tex] (as per the example given).
3. Apply the values to the equation:
- Substitute the values into the equation:
[tex]\[
y = 172 \times (1.3)^t
\][/tex]
- With [tex]\( t = 5 \)[/tex]:
[tex]\[
y = 172 \times (1.3)^5
\][/tex]
4. Calculate the result:
- First, compute [tex]\( (1.3)^5 \)[/tex], which involves multiplying 1.3 by itself five times.
- Next, multiply the result by 172 to get the final value of [tex]\( y \)[/tex].
5. Final Result:
- After calculating the expression, the value of [tex]\( y \)[/tex] when [tex]\( t = 5 \)[/tex] is approximately:
[tex]\[
y \approx 638.62
\][/tex]
This represents the value after 5 time periods of exponential growth given the initial amount and growth rate provided.
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