Question Statement

The number of compact discs $N$ (in millions) purchased each year is increasing exponentially and is given by the formula: $N(t)=7.5(6{)}^{0.5t}$ where $t$ is the number of years after 2024 (e.g., $t=0$ corresponds to 2024, $t=1$ corresponds to 2025).

a. After what amount of time will one billion compact discs be sold in a year? b. What is the doubling time on the sale of compact discs?

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Background and Explanation

This problem requires working with exponential growth functions. To solve for the time variable $t$ located in the exponent, we use logarithms to "bring down" the exponent, allowing us to isolate and solve for the unknown value.

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Solution

Part (a)

To find when one billion discs will be sold, we must first convert one billion into the units used in the formula (millions). Since 1 billion = 1,000 million, we set $N=1000$.

$\begin{array}{rl}1000& =7.5(6{)}^{0.5t}\\ \frac{1000}{7.5}& =6^{0.5t}\\ 133.33& =6^{0.5t}\end{array}$

Now, we take the logarithm of both sides to solve for $t$: $\begin{array}{rl}\log (133.33)& =\log (6^{0.5t})\\ \log (133.33)& =0.5t\log (6)\\ 0.5t& =\frac{\log (133.33)}{\log (6)}\\ t& =\frac{\log (133.33)}{0.5\times \log (6)}\\ t& \approx 5.46\text{ years}\end{array}$

Part (b)

The doubling time is the time required for the initial amount to double. The initial amount (at $t=0$) is $7.5$ million. Therefore, we want to find $t$ when $N=15$ million.

$\begin{array}{rl}15& =7.5(6{)}^{0.5t}\\ \frac{15}{7.5}& =6^{0.5t}\\ 2& =6^{0.5t}\end{array}$

Taking the logarithm of both sides: $\begin{array}{rl}\log (2)& =\log (6^{0.5t})\\ \log (2)& =0.5t\cdot \log (6)\\ 0.5t& =\frac{\log (2)}{\log (6)}\\ t& =\frac{\log (2)}{0.5\times \log (6)}\\ t& \approx 0.774\text{ years}\end{array}$

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Key Formulas or Methods Used

• Exponential Growth Equation: $N(t)=A{b}^{kt}$
• Logarithm Power Rule: $\log (x^y)=y\log (x)$
• Unit Conversion: Recognizing that 1 billion = 1,000 million to match the units of $N$.

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Summary of Steps

1. Identify the target value: For part (a), set $N=1000$; for part (b), set $N$ to double the initial value ($7.5\times 2=15$).
2. Isolate the exponential term: Divide both sides by the coefficient (7.5).
3. Apply Logarithms: Use $\log$ on both sides to move the variable $t$ out of the exponent.
4. Solve for $t$: Divide the resulting values to find the approximate time in years.