Can you help me? Thank you so much.

Question

11) The Algebros start a side business selling t‐shirts.  The first year they make $500 (mostly from family and friends).  The next several years the revenue increased by about 6.8% each year.  a) How much revenue did they make in their fifth year? b) How much total revenue did they make after five years? 

Answer 1

Revenue in year n is 500*(1.068)ⁿ. You can do the arithmetic when n=5.

The sum is a geometric series with a₁=500 and r=1.068.

The sum of a geometric series is (a₁rⁿ⁺¹-a₁)/r-1)

Answer 2

Given: 1st Year Revenue = $500

Revenue increases 6.8% each year

Find:

How much revenue did they make in year #5?

How much total revenue did they make after 5 years?

SOLUTION:

Rn = Revenue for nth year

Rn = 500(1 + .068)^(n-1)

R5 = Revenue for 5th year;

R5 = 500(1 + .068)^(n-1)

R5 = 500(1 + .068)^4

R5 = 650.511

Revenue during 5th year = $650.51

Total Revenue after 5 years:

R = R1 + R2 + R3 + R4 + R5

R = 500 + 500(1.068)^1 + 500(1.068)^2 + 500(1.068)^3 + 500(1.068)^4

R = 500*[ 1 + 1.068 + 1.068^2 + 1.068^3 + 1.068^4 ]

R = 500[ 1 + 1.068 + 1.141 + 1.218 + 1.301 ]

R = 500[ 5.728 ]

R = 2864

Total Revenue after 5 years = $2864

Answer 3

Hi Lan!

This looks like an exponential growth problem.

So I would use the the formula y= a( 1+ r) ^t

a will be the initial investment and r will be 6.8% as decimal. t will be the year.

So first you will need to find a, but the good news is that when you let r be .068 and t=1 you know that y=500 bc that is how much of a profit they had after one year.

Plugging in those values, it will look like this

y= a(1+r) ^t

500 = a( 1+ .068) ¹

500 = a( 1.068) ¹

500 = a(1.068)

500/1.068 = a

468.14 = a [not exact I rounded to the nearest hundredth]

now that you know a, you can now solve for y bc that will tell you how much they made in year 5.

a = 468.16

r= 6.8% or .068

t= 5

y= a(1+ r) ^t

y= 468.16( 1 + .068) ⁵

y = 468.16(1.068) ⁵

y= 650.5048

y= 650.50

By the fifth year they would have made $650.50 in revenue.

B) let t = ,2,3,4 and 5 then simplify:

468.16(1.068) + 468.16(1.068)² + 468.16(1.068)³+ 468.16(1.068)⁴ + 468.16(1.068)⁵

500 + 534+ 570.30 + 609 + 650.50 = $ 2863.80

I hope this helps. Have a great day!

Mrs. S