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.

1 answer

To write an exponential equation in the form \( y = a(b)^x \) that models the expected monthly number of albums sold by Lily Starbright, we begin by identifying the initial parameters:

  1. Initial Value (a): This is the number of albums sold in the first month (March). Lily sold 1,040 copies by the end of March, so we take: \[ a = 1040 \]

  2. Growth Factor (b): To determine the growth factor, we need to compare the sales from March to April. The sales in April were 1,352 copies. The growth can be calculated as follows: \[ \text{Growth Factor} = \frac{\text{Sales in April}}{\text{Sales in March}} = \frac{1352}{1040} \] Let's calculate this value: \[ \frac{1352}{1040} \approx 1.298076923 \] We can round this value to 1.30 for simplicity, which indicates a percentage increase of about 30%.

Now, we can formulate the equation. Let \( x \) be the number of months after the release:

  1. For March (the first month), \( x = 0 \), so: \[ y = 1040(1.30)^0 = 1040 \]

  2. For April (the second month), \( x = 1 \): \[ y = 1040(1.30)^1 \approx 1352 \]

Putting this all together, the exponential model that represents the expected monthly sales can be written as: \[ \boxed{y = 1040(1.30)^x} \]

This equation will model the expected number of albums sold \( y \) \( x \) months after the album's release.