Question

a quantity with an initial value of 110 grows continuously at a rate of 0.5% per hour. what is the value of the quantity after 6 hours, to the nearest hundredth?

Explanation:

Step1: Recall the continuous growth formula

The formula for continuous growth is \( A = Pe^{rt} \), where \( P \) is the initial amount, \( r \) is the growth rate (in decimal), \( t \) is the time, and \( e \) is the base of the natural logarithm.
Here, \( P = 110 \), \( r = 0.5\% = 0.005 \) (converted to decimal), and \( t = 6 \) hours.

Step2: Substitute the values into the formula

Substitute \( P = 110 \), \( r = 0.005 \), and \( t = 6 \) into \( A = Pe^{rt} \):
\( A = 110 \times e^{(0.005 \times 6)} \)

Step3: Calculate the exponent

First, calculate the exponent: \( 0.005 \times 6 = 0.03 \)

Step4: Calculate the value of \( e^{0.03} \)

Using a calculator, \( e^{0.03} \approx 1.030454534 \)

Step5: Calculate the final amount

Multiply by the initial amount: \( A = 110 \times 1.030454534 \approx 113.3500 \)

Answer:

\( 113.35 \)