Question
Lily star right 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 1352 copies. Several of the songs 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 monthly number of Al bums sold, y, x months after release.
Use whole NUmbers, decimals or simplified fractions for the values of an and b.
After how many months will the number of copies sold be greater than 5,000?
Write an exponential equation in the form y =a(b)^x that can model the monthly number of Al bums sold, y, x months after release.
Use whole NUmbers, decimals or simplified fractions for the values of an and b.
After how many months will the number of copies sold be greater than 5,000?
Answers
GPT-4o mini
To create an exponential equation in the form \( y = a(b)^x \) that models the monthly sales of Lily Star's album, we can start by determining the values of \( a \) and \( b \).
1. **Identify the initial sales**:
- At the beginning of March (month 0), \( y = 0 \) copies sold.
- By the end of March (month 1), \( y = 1,040 \) copies sold.
- In April (month 2), \( y = 1,352 \) copies sold.
2. **Setting up the equation**:
- We know that \( a \) is the initial value of sales after the first month. Hence, we will take \( a = 1,040 \) and \( x = 1 \).
3. **Determine the growth factor \( b \)**:
- The sales from month 1 to month 2 can be expressed as:
\[
1,352 = 1,040(b)^1
\]
\[
b = \frac{1,352}{1,040}
\]
\[
b \approx 1.29615
\]
Thus, the exponential equation modeling the album sales becomes:
\[
y = 1040(1.29615)^x
\]
4. **Finding the month when sales exceed 5,000**:
- We want to find \( x \) when \( y > 5000 \):
\[
5000 < 1040(1.29615)^x
\]
Dividing both sides by 1040:
\[
\frac{5000}{1040} < (1.29615)^x
\]
\[
4.80769 < (1.29615)^x
\]
Taking the logarithm of both sides:
\[
\log(4.80769) < x \cdot \log(1.29615)
\]
Solving for \( x \):
\[
x > \frac{\log(4.80769)}{\log(1.29615)}
\]
Calculating the values:
- \( \log(4.80769) \approx 0.682 \)
- \( \log(1.29615) \approx 0.113 \)
Thus:
\[
x > \frac{0.682}{0.113} \approx 6.04
\]
Since \( x \) must be a whole number, Lily's album sales will exceed 5,000 copies after approximately **7 months** (in month 7 since the album's release).
1. **Identify the initial sales**:
- At the beginning of March (month 0), \( y = 0 \) copies sold.
- By the end of March (month 1), \( y = 1,040 \) copies sold.
- In April (month 2), \( y = 1,352 \) copies sold.
2. **Setting up the equation**:
- We know that \( a \) is the initial value of sales after the first month. Hence, we will take \( a = 1,040 \) and \( x = 1 \).
3. **Determine the growth factor \( b \)**:
- The sales from month 1 to month 2 can be expressed as:
\[
1,352 = 1,040(b)^1
\]
\[
b = \frac{1,352}{1,040}
\]
\[
b \approx 1.29615
\]
Thus, the exponential equation modeling the album sales becomes:
\[
y = 1040(1.29615)^x
\]
4. **Finding the month when sales exceed 5,000**:
- We want to find \( x \) when \( y > 5000 \):
\[
5000 < 1040(1.29615)^x
\]
Dividing both sides by 1040:
\[
\frac{5000}{1040} < (1.29615)^x
\]
\[
4.80769 < (1.29615)^x
\]
Taking the logarithm of both sides:
\[
\log(4.80769) < x \cdot \log(1.29615)
\]
Solving for \( x \):
\[
x > \frac{\log(4.80769)}{\log(1.29615)}
\]
Calculating the values:
- \( \log(4.80769) \approx 0.682 \)
- \( \log(1.29615) \approx 0.113 \)
Thus:
\[
x > \frac{0.682}{0.113} \approx 6.04
\]
Since \( x \) must be a whole number, Lily's album sales will exceed 5,000 copies after approximately **7 months** (in month 7 since the album's release).