Lily Starbright, a singer-songwriter, released her first album at the beginning of March. By the end of March, she had sold 1,040 copies. In April, she sold 1,352 copies. Several of the songs on the album are now trending, so Lily expects her sales to continue to increase.

Write an exponential equation in the form y=a(b)x that can model the expected monthly number of albums sold, y, x months after release.

After how many months will the number of copies sold monthly be greater than 5,000?

1 answer

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.

  • At the beginning of March (which we'll call \( x = 0 \)), the initial sales are: \[ y(0) = 1040 \quad \text{(initial sales at release)} \]

  • At the end of March (which would be \( x = 1 \)), she sold: \[ y(1) = 1040 \quad \text{(sales by the end of March)} \]

  • 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).