Answer:
To model the expected monthly number of albums sold, we can use the exponential growth formula:
where:
- `y` is the number of albums sold,
- `a` is the initial amount (the number of albums sold in the first month),
- `b` is the growth factor (the rate at which the sales are increasing each month), and
- `x` is the number of months after the release.
Given that Lily sold 1,040 albums in March (the first month) and 1,352 albums in April (the second month), we can calculate the growth factor `b` as the ratio of the sales in April to the sales in March:
So, the exponential equation that models the expected monthly number of albums sold is:
Now, to find out after how many months the number of copies sold monthly will be greater than 5,000, we can set `y` to 5,000 and solve for `x`:
Solving this equation requires the use of logarithms, which might be beyond the scope of this discussion. However, you can use a calculator or a software tool to find the approximate value of `x`. Please note that the actual sales may vary due to a variety of factors, and this model is a simplification that assumes constant exponential growth.
Step-by-step explanation:
Answer:
[tex]\sf y = 1,040 \cdot (1.3)^x [/tex]
6 months
Step-by-step explanation:
To model the expected monthly number of albums sold, [tex]\bold \sf y[/tex], [tex]\bold \sf x[/tex] months after release using an exponential equation of the form:
[tex]\large\boxed{\boxed{\bold \sf y = a \cdot b^x}}[/tex]
We can use the given sales data for March and April.
Let's denote:
To find [tex]\bold \sf b[/tex], we can use the sales data from April. In April, Lily sold 1,352 copies.
Therefore, we can set up the following equation based on the exponential model:
[tex]\sf 1,352 = a \cdot b^1 [/tex]
Substituting [tex]\bold \sf a = 1,040[/tex] and [tex]\bold \sf x = 1[/tex] (for April):
[tex]\sf 1,352 = 1,040 \cdot b [/tex]
Now, solve for [tex]\bold \sf b[/tex]:
[tex]\sf b = \dfrac{1,352}{1,040} [/tex]
[tex]\sf b \approx 1.3 [/tex]
Now that we have [tex]\bold \sf b \approx 1.3[/tex], we can write the exponential equation to model the monthly number of albums sold ([tex]\bold \sf y[/tex]) as follows:
[tex]\sf y = 1,040 \cdot (1.3)^x [/tex]
This equation predicts the expected monthly album sales [tex]\bold \sf y[/tex] [tex]\bold \sf x[/tex] months after the release.
Nextsf, to determine after how many months the sales will be greater than 5,000 copies ([tex]\bold \sf y > 5,000[/tex]), we set up the inequality:
[tex]\sf 1,040 \cdot (1.3)^x > 5,000 [/tex]
To solve for [tex]\bold \sf x[/tex], we can take the logarithm of both sides to isolate [tex]\bold \sf x[/tex]:
[tex]\sf \log(1,040 \cdot (1.3)^x) > \log(5,000) [/tex]
[tex]\sf \log(1,040) + \log((1.3)^x) > \log(5,000) [/tex]
[tex]\sf \log(1,040) + x \cdot \log(1.3) > \log(5,000) [/tex]
Now, solve for [tex]\bold \sf x[/tex]:
[tex]\sf x \cdot \log(1.3) > \log(5,000) - \log(1,040) [/tex]
[tex]\sf x > \dfrac{\log(5,000) - \log(1,040)}{\log(1.3)} [/tex]
[tex]\sf x > 5.984874512 [/tex]
[tex]\sf x > 6 \textsf{(in nearest whole number)}[/tex]
Therefore, the approximate number of months [tex]\bold \sf x[/tex] needed for [tex]\bold \sf y[/tex] to exceed 5,000 copies is 6 months.