Question

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A famous fast food chain opened two branches in different parts of a city. Branch A made a profit of $50,000 in the first year, and the profit increased by 4% every year. Branch B made a profit of $35,000 in the first year, and the profit increased by 5.5% every year.

Which function can the fast food chain use to determine its total profit, P(x), after x years, and how much money will the chain have made in profit after 4 years?

A)P(x) = 5,000(10(1.055)x + 7(1.04)x); $97,341.65
B)P(x) = 5,000(10(1.04)x + 7(1.055)x); $73,017.93
C)P(x) = 5,000(10(1.055)x + 7(1.04)x); $106,576.25
D)P(x) = 5,000(10(1.04)x + 7(1.055)x); $101,851.79

Answer 1

Answer: D) P(x) = 5,000(10(1.04)x + 7(1.055)x); $101,851.79

Explanation:

The exponential growth equation is given by :-

`y=I(1+r)^t` , where x is time , I is the initial value and and r is the rate of growth.

For Branch A:

I= $50,000

r= 4%=0.04

Exponential growth equation for Branch A :
`y_1=50000(1+0.04)^x`

`y_1=50000(1.04)^x` (1)

For Branch B:

I= $35,000

r= 5.5%=0.055

Exponential growth equation for Branch A :
`y_2=35000(1+0.055)^x`

`y_2=35000(1.055)^x` (2)

Add (1) and (2)

`P(x)=y_2+y_2\\\\ P(x)=50000(1.04)^x+35000(1.055)^x\\\\ P(x)= 5,000(10(1.055)^x + 7(1.04)^x)`

For x= 4, we have

`P(x)=5000(10(1.055)^4 + 7(1.04)^4)\\\\=101851.790772\approx101851.79`

Hence, the function can the fast food chain use to determine its total profit, P(x), after x years, :

`P(x)= 5,000(10(1.055)^x + 7(1.04)^x)`

Amount of money will the chain have made in profit after 4 years = $101851.79

Answer 2

The function can the fast food chain use to determine its total profit, P(x), after x years :
`P(x)=5000[10(1.04)^x+7(1.055)^x`

Total profit after 4 years = $101,851.79

Explanation:

Branch A:

First year profit = $50,000

The profit increased by 4% every year

Formula :
`A =P(1+r)^t`

So, Total profit after x years =
`P(x)=50,000(1+0.04)^x`

=
`P(x)=50,000(1.04)^x`

Branch B:

First year profit = $35000

The profit increased by 5.5% every year

Formula :
`A =P(1+r)^t`

So, Total profit after x years =
`P(x)=35,000(1+0.055)^x`

=
`P(x)=35,000(1.055)^x`

Thus the function of total profit of fast food chain :

`P(x)=50,000(1.04)^x+35,000(1.055)^x`

`P(x)=5000[10(1.04)^x+7(1.055)^x`

Now to find total profit after four years .

Substitute x = 4

`P(4)=5000[10(1.04)^4+7(1.055)^4]`

`P(4)=101851.790772`

Hence Option D is correct.

The function can the fast food chain use to determine its total profit, P(x), after x years :
`P(x)=5000[10(1.04)^x+7(1.055)^x`

Total profit after 4 years = $101,851.79