Question

Alwx is singing a songwriter released her first album in the beginning of March by the end of March she had sold 1040 copies in April. She sold 1352 copies Several songs on the album are now trending so expect her sales to continue to increase

write a exponential equation in the form of y=a(b)^x

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

Answer 1

To create an exponential equation to model the sales of Alwx's album, we can use the data provided.

Let:
- \( y \) = number of copies sold,
- \( x \) = number of months since the album's release,
- \( a \) = initial number of copies sold at \( x = 0 \) (March),
- \( b \) = growth factor representing the rate of increase in sales per month.

From the problem:
- In March (\( x = 0 \)), she sold 1040 copies.
- In April (\( x = 1 \)), she sold 1352 copies.

Now we can calculate the growth factor \( b \):

1. From March to April, the increase in sales is:
\[
\text{Increase} = 1352 - 1040 = 312
\]

2. The growth factor \( b \) can be calculated using the formula:
\[
b = \frac{y_1}{y_0}
\]
where \( y_0 = 1040 \) (copies sold in March) and \( y_1 = 1352 \) (copies sold in April):
\[
b = \frac{1352}{1040} \approx 1.3
\]

Now, we can write the exponential equation:
\[
y = 1040(1.3)^x
\]

Next, we need to find out after how many months the number of copies sold monthly will be greater than 5000. We can set up the inequality:
\[
1040(1.3)^x > 5000
\]

To solve for \( x \), divide both sides by 1040:
\[
(1.3)^x > \frac{5000}{1040} \approx 4.8077
\]

Next, take the logarithm of both sides:
\[
\log((1.3)^x) > \log(4.8077)
\]

Using the power property of logarithms:
\[
x \cdot \log(1.3) > \log(4.8077)
\]

Now solving for \( x \):
\[
x > \frac{\log(4.8077)}{\log(1.3)}
\]

Calculating the logarithms:
\[
\log(4.8077) \approx 0.6826
\]
\[
\log(1.3) \approx 0.1139
\]

Now, substitute these values into the equation:
\[
x > \frac{0.6826}{0.1139} \approx 5.99
\]

Rounding up, since \( x \) must be a whole number, we find:
\[
x \geq 6
\]

Thus, after **6 months**, the number of copies sold monthly will be greater than 5,000.