Jack and jill each inherit $1200, they do the following with this money:

Question

Jack and Jill each inherit $1200.

 Jack invests $1000 of this money and spends the rest. He anticipates that the value of

this investment will grow by approximately 8% per year.

 Jill purchases a motor boat for her $1200. The boat will lose approximately 5% of its

value per year.

a) Write exponential equations for these situations where y is the amount in dollars over x

years. (3 pts.)

Jack’s investment value equation: Jill’s motor boat value equation:

b) How long will it take for Jack’s investment to grow to $1200? (2 pts.)

c) Determine when the two equations from part (a) intersect and explain the meaning of

this intersection point in terms of Jack and Jill’s investments. (3 pts.)

Answer 1

(a) So this is a common compound interest problem. I'm not sure if you are looking at this from a finance perspective or a math perspective so I shall go through both.

The regular formula is:

Future Value=Investment*(1+rate-of-return)^the the power of the number of years.

For Jack it's:

Future Value=1000*(1+8%)^X

Or

Y=1000*(1.08)^x

For Jill it's:

Future Value=1000*(1-5%)^Number of years

Or

Y=1200*(0.95)^^x

(b) To find the number of years for Jack's investment to grow to 1200 we substitute 1200 for Y

1200=1000*(1.08)^x

Then we divided both sides by 1000

1.2=1.08^x

And then we find the natural log of both sides

ln(1.2)=ln(1.08^x)

And then we divide the natural log of 1.2 by the natural log of 1.08

ln(1.2)/ln(1.08)=X=2.369

(c)

To find the intersection point we have to set the equations equal to each other.

1000*1.08^x=1200*0.95^x

Again, we divide both sides by 1000 (we could divide both sides by 1200 but it makes things simpler.

1.08^x=1.2*0.95^x

Then we divide both these sides by 0.95 (again we could use 1.08 but it makes things simpler)

1.08^x/0.95^x=1.2

(1.08/0.95)^x=1.2

(1.13684)^x=1.2

Again, you take the natural log of both sides.

ln(1.13684^x)=ln(1.2)

And again you divide both sides

ln(1.2)/ln(1.13684)=X=1.42156

This equates to about 17 months.

Bear in mind, I rounded for simplicity.