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Suppose the food company decides it would like to grow from the 5 stores it has now so that it will have at least 600 stores, but no more than 800 stores 5 years from now.Come up with a plan for the company to achieve this where it adds the same number of stores each year.Come up with a plan for the company to achieve this where the number of stores multiplies by the same factor each year. (might need to round the outcome to the nearest whole store.)

Sagot :

From the statement we know that:

• a food company decides it would like to grow from the 5 stores,

,

• to at least 600 stores, but no more than 800 stores in 5 years from now.

1) A plan for the company to achieve this where it adds the same number of stores each year.

One plan consists in adding 120 stores per year, so the company we will have:

[tex]5+120\cdot5=5+600=605[/tex]

stores after 5 years. This plan satisfies the requirement of the company because 600 < 605 < 800, as required in the statement.

2) A plan for the company to achieve this where the number of stores multiplies by the same factor each year (might need to round the outcome to the nearest whole store).

If the company will grow the same factor each year, the number of stores after the 5 years will be:

[tex]N=5\cdot x^5.[/tex]

Where x is the growth factor. If set our goal to be N stores, we can find the value of x taking the logarithm on both sides:

[tex]\begin{gathered} \frac{N}{5}=x^5, \\ \ln (\frac{N}{5})=\ln (x^5), \\ \ln (\frac{N}{5})=5\cdot\ln x, \\ \ln x=\frac{1}{5}\cdot\ln (\frac{N}{5}), \\ x=e^{\frac{1}{5}\cdot\ln (\frac{N}{5})}. \end{gathered}[/tex]

If we set the number of stores to be approximately N = 650, we find that the growth factor must be:

[tex]x=e^{\frac{1}{5}\cdot\ln (\frac{650}{5})}\cong2.65.[/tex]

With this growth factor, the company will have:

[tex]N=5\cdot2.65^5\cong653[/tex]

after 5 years.

Answers

1. The plan consists in adding 120 stores per year. After 5 years the company will have 605 stores.

2. The plan consists in multiplying the number of stores by 2.65 each year. After 5 years the company will have 653 stores.

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