To model Lily Starbright's album sales using an exponential equation of the form \( y = a(b)^x \), we can determine the values of \( a \) and \( b \) based on the sales data provided.
Step 1: Determine initial values
Let \( y \) be the number of albums sold and \( x \) be the number of months after release.
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At the beginning of March (which we'll call \( x = 0 \)), the initial sales are: \[ y(0) = 1040 \quad \text{(initial sales at release)} \]
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At the end of March (which would be \( x = 1 \)), she sold: \[ y(1) = 1040 \quad \text{(sales by the end of March)} \]
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In April (which is \( x = 2 \)), she sold: \[ y(2) = 1352 \]
Step 2: Set up the equations
Using the sales at \( x = 0 \):
\[ y(0) = a(b)^0 = a = 1040 \]
Now we have: \[ y = 1040(b)^x \]
Using the sales in April (\( x = 2 \)): \[ y(2) = 1040(b)^2 = 1352 \]
Step 3: Solve for \( b \)
Now we will solve for \( b \): \[ 1040(b)^2 = 1352 \] \[ (b)^2 = \frac{1352}{1040} \] \[ (b)^2 = 1.298076923 \] \[ b = \sqrt{1.298076923} \approx 1.141 \]
Step 4: Write the exponential model
Now we can write the exponential equation:
\[ y = 1040(1.141)^x \]
Step 5: Find when y exceeds 5000
To find out when sales will exceed 5,000 copies, we set up the inequality:
\[ 1040(1.141)^x > 5000 \]
Dividing both sides by 1040:
\[ (1.141)^x > \frac{5000}{1040} \approx 4.807692308 \]
Next, take the logarithm of both sides:
\[ x \cdot \log(1.141) > \log(4.807692308) \]
Solve for \( x \):
\[ x > \frac{\log(4.807692308)}{\log(1.141)} \]
Calculating the logarithms:
\[ \log(4.807692308) \approx 0.682 \] \[ \log(1.141) \approx 0.0607 \]
Now calculate \( x \):
\[ x > \frac{0.682}{0.0607} \approx 11.24 \]
Conclusion
Since \( x \) represents months after the release of the album, Lily can expect to sell more than 5,000 copies after approximately 12 months (since we round up 11.24 to the nearest whole number).