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.
Use whole numbers, decimals, or simplified fractions for the values of a and b.

y =

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

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:

msm555 msm555 · Answer 2 · 2024-04-17T00:07:15-04:00

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:

[tex]\bold \sf y[/tex] as the number of albums sold in a given month [tex]\bold \sf x[/tex] months after the release.
[tex]\bold \sf a[/tex] as the initial number of albums sold at the beginning (March), which is 1,040.
[tex]\bold \sf b[/tex] as the growth factor or multiplier.

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.