Question

question 16
a quantity with an initial value of 8200 grows continuously at a rate of 0.55% per decade. what is the value of the quantity after 97 years to the nearest hundredth?
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Explanation:

Step1: Recall continuous growth formula

The formula for continuous growth is \( A = Pe^{rt} \), where \( P \) is the initial quantity, \( r \) is the growth rate (in decimal), and \( t \) is the time.

Step2: Identify values

Given \( P = 8200 \), \( r = 0.55\% = 0.0055 \) per decade. First, find \( t \) in decades. Since 97 years, \( t=\frac{97}{10}=9.7 \) decades.

Step3: Substitute into formula

Substitute \( P = 8200 \), \( r = 0.0055 \), \( t = 9.7 \) into \( A = Pe^{rt} \). So \( A = 8200 \times e^{0.0055\times9.7} \).

Step4: Calculate exponent

Calculate \( 0.0055\times9.7 = 0.05335 \).

Step5: Calculate \( e^{0.05335} \)

\( e^{0.05335}\approx1.0550 \) (using calculator for \( e^x \)).

Step6: Calculate \( A \)

\( A = 8200\times1.0550\approx8200\times1.055 = 8200 + 8200\times0.055 = 8200 + 451 = 8651 \) (more accurately, \( 8200\times e^{0.05335}\approx8200\times1.05503\approx8651.25 \)).

Answer:

\( \approx 8651 \) (or more precisely \( 8651.25 \) when calculated with more precise \( e^{0.05335} \))