rate of change, please help with explanation and formula to figure out future ?s

Question

each week an album sells 4.5% fewer copies than it did the previous week. this week it sold 13,400 copies. at this rate in how many week will it sell fewer than 2,000 copies? round up to the nearest week.

Answer 1

Week Number of copies sold

1 13,400

? < 2,000

An equation that would be a good representation of the rate of decrease of the total albums sold would be…

f(w) = (.955)w-1(13,400)

X would represent the current week

(.955) would represent the percent of the total from the previous week.

The exponent, x-1, represents the previous week.

The resulting value would be value of albums sold.

When you plug this equation in you result in having 43 weeks until the albums sold is less than 2000, specifically 1937

2000 = 13400 * (.955)(x-1)

.14925 = (.955)x-1

Based on this, we can use the log of both sides to base .955 to set the value of one side to the exponent of the other…Remember alogab = b

Log(.14925) to base .955 = log(.955) to base .955 = x -1

Based on this we can got to the next step…

42 = x-1

43 = x

It will be approximately 43 until the albums sold is less than 2000.

Answer 2

This is an interesting 'Algebra 1' question because logarithms are certainly the best method to solve this but logs are not usually introduced until Algebra 2 (although I'm sure some advanced curriculums may include it sooner). So if we're using techniques learned in Algebra 1 to solve this, and not simply going off of our knowledge and experience with exponents and logarithms, this would end up being a guessing game once a formula is created.

Nonetheless, we are indeed given a or the starting point : 13,400 and the rate of change from week to week: 4.5%. Since the rate of change is a percentage, this sequence is a geometric sequence because each successive term after the 0th term will be multiplied by this rate. The standard form for writing exponential functions is: a*r^x; where a is the initial value, r is the common ration and x is your time interval, in this case, weeks.

For example:

h(w) = 13,400 - (13,400*0.045)^w

**Factor out 13,400

h(w) = 13,400(1- 0.045)^w

h(w) = 13,400(.955)^w

Here are examples of week 0 (starting value) and week 1 to help gel how the formula works:

h(0) = 13,400(.955)⁰ --> any number raised to the 0th power is always equal to 1; so: 13,400(1) = 13,400

h(1) = 13,400(.955)¹ --> 13,400 * .955 = 12,797

So this would be your formula for calculating each week, but since the question is asking for the week at which the album sales will be less than 2,000, this only acts as a starting point. First thing we have to do is set up the formula with the unknown that we need to find, which is the week, and try and isolate terms one step at a time.

2,000 = 13,400(.955)^w

Divide by 13,400 on each side to isolate the term with an exponent

2,000/13,400 = .955^w

0.14925 = .955^w

Since these terms are equal to one another, you have to ask yourself: ".955 raised to what power is approximate/equal to 0.14925?" The answer to this question isn't very intuitive, but without knowledge of logarithms, you may have to plug numbers in to see what brings you closest to this value.

As others have stated and shown, .955 is raised to ~41 will equal or come very close to about 1.49. Since this value would put us right around 2,000, we would add another week since the question is asking for the week when album sales are below 2,000, so it would be 42 weeks.

We can try at h(41) and h(42):

h(41) = 13,400(.955)⁴¹= 13,400(0.1514) --> 2,028.76

h(42) = 13,400(.955)⁴²= 13,400(0.14459) --> 1937.51

All in all, if you have already been introduced to logarithms then this problem can be manipulated very easily. However, if following a traditional Algebra 1 curriculum, you may not learn about logarithms until further along so you may have to plug in values to see what works.

Answer 3

So there are a few ways to look at this algebraic sequence.

Given

13,400 copies sold as starting point

4.5% less each week which is the same as (100-4.5 = 95.5%) 95.5% the previous week in sales.

we can easily see a pattern

Week 0 =13,400 copies sold

Week 1 = 13,400 x 0.955¹ = 12,797 copies sold

Week 2 = 13,400 x 0.955 x 0.955 = 13, 400 x 0.955² (95.5% of week 1 this is the same as 12,797 x 0.955) = 12,221 copies

Week 3 = 13,400 x 0.955 x 0.955 x 0.955 = 13,400 x 0.955³ = 11,671 copies sold.

From here we can see a pattern that the number of copies sold is equal to the original amount x the percentage of the previous week raised to the number of weeks. we could sit at the calculator and keep doing this until we got to 2,000 copies sold but that would be silly! We can use the pattern and solve for n - 1 the number of weeks needed to get to 2,000 copies.

copies sold = 2,000 = 13,400 x 0.955ⁿ divide both sides by 13,400

2,000/13,400 = 0.955ⁿ

0.14925 = 0.955ⁿ to get rid of the exponent you must take the log of both sides but this is not base 10

log(0.14925) = log (0.955ⁿ)

log (0.14925) to base 0.955 = n log of an exponent is just the base but both sides are now base 0.955 not base 10.so we need to use the change of base formula

ln (0.14925)/ln (0.955) =n notice we took the natural log of the lefthand side of the equation and divide by the natural log (ln) of the base.

Solving for x 41.3 = x -1 → n = 42 weeks

We can plug back in to make sure it works!

2,000 = 13,400 x 0.955^41.3

I hope this explanation helped :)

Answer 4

Hi Melissa,

Without repeating explanations already given, I just want to offer the formulas that apply to this problem which can also be used in many other instances.

1. It helps to start by identifying this problem as an exponential function, so the basic formula that can be used is
2. ƒ(x) = ab^x
3. where a is the initial amount,
4. b is the base (what is often call the growth factor), and
5. x is the number of weeks.
6. Then it's a matter of substituting the values to get 2,000 = 13,400(0.955)^x
7. Note: the value for b (growth factor) is determined by subtracting 100% - 4.5% and converting to decimal form.
8. Then solve the resulting equation using logarithms as described.

1. A similar formula known as exponential growth or decay gives you essentially the same equation but takes into account the calculation for the growth factor.
2. ƒ(n) = Α(1 - r)ⁿ
3. where A is the initial amount,
4. r is the rate of growth (+) or decay (-), and
5. n is the number of recurring periods/events
6. Plugging in the values gives 2,000 = 13,400(1 - 0.045)ⁿ
7. Helpful side note: This is also the basis for the Compound Interest formula: A = P(1 + APR/n)^nt

1. A final formula that can be applied in your problem is the Geometric Sequence formula and think of each week as the next term with week 0 as representing the first term of 13,400 starting out.
2. a_n = a₁rⁿ
3. where n is the number of terms (weeks),
4. a₁ is the first term of the sequence (13,400), and
5. r is the common ratio (in this case, determined as a % of previous terms or 0.955)
6. Substitute the values to get 2,000 = 13,400(0.955)ⁿ
7. Note: All the formulas result in the same equation and need to be solved using logarithms.

I hope this is helpful as you learn and tackle future problems and scenarios!

Blessings.

Answer 5

it's a geometric sequence

an= a1r^(n-1)

a1 = 13400

an = 2000

r = 1-.045=.955

2000 = 13400(.955^(n-1)) solve for n

2000/13400 = .955^(n-1)

.955^(n-1) = 20/134 = 10/67

n-1 = log(10/67) to the base .955

n-1 = ln(10/67)/ln.955

n = 1 +41.31=42.31

round up= 43 weeks

Answer 6

Current albums sold = initial albums sold * ( 100% - 4.5%)^(the number of weeks gone by)

2000 = 13400^(0.955)^w

Divide both sides by 13400:

(2000/13400) = 0.955^x

Take the log base 0.955 of both sides (use the change of base formula* if needed):

x ≈ 41.31070426

13400*(0.955)^(41.31070426) ≈ 2001,

but after 42 weeks, copies sold ≈ 1938.

*change of base formula:

log _b (y) = log (y) / log (b), where log with no subscript is log base 10. ln could also be used.