Question

Emmanuel invests $3600 and Kelsey invests $2400. Both investments eam 3.8%annual interest. How much longer will it take Kelsey's investment to reach $10,000than Emmanuel's investment?

Answer 1

Step-by-step explanation

We can build the following equations in order to represent the Emmanuel earnings:

`y_1=3600(1+0.038)^x`
`y_2=10000`

Matching both expressions:

`10000=3,600(1+0.038)^x`
`\mathrm{Switch\: sides}`
`3600\mleft(1+0.038\mright)^x=10000`
`\mathrm{Divide\: both\: sides\: by\: }3600`
`(3600\left(1+0.038\right)^x)/(3600)=(10000)/(3600)`
`\mathrm{Simplify}`
`\mleft(1+0.038\mright)^x=(25)/(9)`
`\mathrm{If\: }f\mleft(x\mright)=g\mleft(x\mright)\mathrm{,\: then\: }\ln \mleft(f\mleft(x\mright)\mright)=\ln \mleft(g\mleft(x\mright)\mright)`
`\ln \mleft(\mleft(1+0.038\mright)^x\mright)=\ln \mleft((25)/(9)\mright)`
`\ln \mleft(\mleft(1+0.038\mright)^x\mright)=x\ln \mleft(1+0.038\mright)`
`x\ln \mleft(1+0.038\mright)=\ln \mleft((25)/(9)\mright)`
`\mathrm{Divide\: both\: sides\: by\: }\ln \mleft(1.038\mright)`
`(x\ln\left(1+0.038\right))/(\ln\left(1.038\right))=(\ln\left((25)/(9)\right))/(\ln\left(1.038\right))`

Simplify:

`x=(\ln\left((25)/(9)\right))/(\ln\left(1.038\right))`

In decimal form, this is equivalent to 27.39 years.

Now, applying the same reasoning to the Kelsey investment:

`y_1=2400(1+0.038)^x`
`y_2=10000`

Matching both expressions:

`10000=2400(1+0.038)^x`
`Switch\: sides`
`2400\mleft(1+0.038\mright)^x=10000`
`\mathrm{Divide\: both\: sides\: by\: }2400`
`(2400\left(1+0.038\right)^x)/(2400)=(10000)/(2400)`
`\mathrm{Simplify}`
`\mleft(1+0.038\mright)^x=(25)/(6)`

Applying the exponent rule:

`\mathrm{If\: }f\mleft(x\mright)=g\mleft(x\mright)\mathrm{,\: then\: }\ln \mleft(f\mleft(x\mright)\mright)=\ln \mleft(g\mleft(x\mright)\mright)`
`\ln \mleft(\mleft(1+0.038\mright)^x\mright)=\ln \mleft((25)/(6)\mright)`

Apply log rule:

`\ln \mleft(\mleft(1+0.038\mright)^x\mright)=x\ln \mleft(1+0.038\mright)`
`x\ln \mleft(1+0.038\mright)=\ln \mleft((25)/(6)\mright)`
`\mathrm{Divide\: both\: sides\: by\: }\ln \mleft(1.038\mright)`
`(x\ln\left(1+0.038\right))/(\ln\left(1.038\right))=(\ln\left((25)/(6)\right))/(\ln\left(1.038\right))`

Simplify:

`x=(\ln\left((25)/(6)\right))/(\ln\left(1.038\right))`

Expressing in decimal form:

`x=38.26`

This is equivalent to 38.26 years to reach 10000 to the Kesley investment.

Comparing both investments:

Kesley Investment - Emmanuel Investment =

= 38.26 - 27.39 = 10.87

In conclusion, It will take 10.87 more years for Kelsey to reach $10,000