Question

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The number of people that drink Root Beer instead of Coca-Cola in Outback, Kentuck has been declining at the rate of 6.5% per year. As of Sept. 1983, the number of people who still drink Root Beer was 5480. (NOT MULTIPLE CHOICE)

a. Write an exponential equation that will represent the number of Root Beer loyalists in Outback for any year.

b. How many Root Beer loyalists should there be as of September 2023? Show or explain how you know.

Answer 1

Part (a)

`y=5480(0.935)^x`

where:

• x is the number of years after 1983.
• y is the total number of Root Beer loyalists.

Part (b)

373

Step-by-step explanation:

`\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function}\\\\$y=ab^x$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value ($y$-intercept). \\ \phantom{ww}$\bullet$ $b$ is the base (growth/decay factor) in decimal form.\\\end{minipage}}`

The initial number of people who drink Root Beer (in September 1983):

• a = 5480

If the number of people that drink Root Beer instead of Coca-Cola has been declining at the rate of 6.5% per year, then:

• b = 100% - 6.5% = 93.5% = 0.935

Therefore, an exponential equation that represents the number of Root Beer loyalists in Outback for any year is:

`y=5480(0.935)^x`

where:

• x is the number of years after 1983.
• y is the total number of Root Beer loyalists.

The number of years between September 1983 and September 2023 is:

• 2023 - 1984 = 40 years

To calculate how many Root Beer loyalists should there be as of September 2023, substitute x = 40 into the equation found in part (a):

`\implies y=5480(0.935)^(40)`

`\implies y=5480(0.0679930362...)`

`\implies y=372.601838...`

`\implies y=373`

Therefore, there should be approximately 373 Root Beer loyalists as of September 2023.

Answer 2

Answers in bold:

• Part A) y = 5480*0.935^x
• Part B) 373 people

=========================================================

Step-by-step explanation:

Part A

x = number of years after 1983

Example: x = 2 is the year 1985 since 1985-1983 = 2.

The number of people who drink this beverage declines by 6.5% each year. The multiplier is 1+r = 1+(-0.065) = 0.935

Meaning that if 6.5% of the people leave, then 93.5% remain. The two percentages add to 100%.

This value 0.935 is the base of the exponent portion. We have 0.935^x as the exponent portion.

Out front is the value 5480 which is the starting amount of people.

This leads to the exponential equation y = 5480*0.935^x

It is of the form y = ab^x

a = 5480 = starting population

b = 0.935 = decay factor

---------------------------------------

Part B

The timespan from 1983 to 2023 is 40 years (because 2023-1983 = 40).

Which means we plug x = 40 into the equation we found in part A.

y = 5480*0.935^x

y = 5480*0.935^40

y = 5480*0.06799303620922

y = 372.601838426526

y = 373

Be sure to round to the nearest whole number.

We expect or predict there will be approximately 373 people who remain loyalists to this root beer as of September 2023.