Certainly! Let's solve this step-by-step:
1. Understanding the Variables:
- The initial value of the quantity is [tex]\( 160 \)[/tex].
- The continuous growth rate is [tex]\( 0.65\% \)[/tex] per hour.
- The time period given is 402 minutes.
2. Convert Minutes to Hours:
To use the growth rate effectively, we need to convert the time from minutes to hours.
[tex]\[
\text{Time in hours} = \frac{402}{60} = 6.7 \text{ hours}
\][/tex]
3. Continuous Growth Formula:
The formula to calculate the value of a quantity after a certain period of continuous growth is:
[tex]\[
A = P \times e^{rt}
\][/tex]
where:
[tex]\[
A \text{ is the final amount}
\][/tex]
[tex]\[
P \text{ is the initial amount}
\][/tex]
[tex]\[
r \text{ is the growth rate}
\][/tex]
[tex]\[
t \text{ is the time period}
\][/tex]
4. Substitute the Values into the Formula:
Here, [tex]\( P = 160 \)[/tex], [tex]\( r = 0.0065 \)[/tex] (since [tex]\( 0.65\% = 0.0065 \)[/tex] in decimal form), and [tex]\( t = 6.7 \)[/tex] hours.
To calculate the value after 6.7 hours:
[tex]\[
A = 160 \times e^{0.0065 \times 6.7}
\][/tex]
5. Evaluate the Expression:
Using [tex]\( e \)[/tex] as the base of the natural logarithm, we evaluate the exponent:
[tex]\[
e^{0.0065 \times 6.7} \approx 1.044512218
\][/tex]
Then multiply this result by the initial amount:
[tex]\[
A = 160 \times 1.044512218 \approx 167.12195497877417
\][/tex]
6. Round the Final Value:
To find the final value to the nearest hundredth:
[tex]\[
A \approx 167.12
\][/tex]
Hence, after 402 minutes, the value of the quantity, rounded to the nearest hundredth, is [tex]\( 167.12 \)[/tex].
