Question

Question 24 ptsTeresa is saving up to buy a car. She currently has $5,000 in her savingsaccount and the car she wants costs $6,500. The bank only gives her 2%interest, compounded weekly.What is the minimum number of years (whole number) that it will take forher to have enough money to buy the car?How much money would she have left in her account after the purchase?HTML EditoresxBIU A - A - IKEV c ✓FERTTTT 12ptParagraph

Answer 1

The rule of the compound interest is

`A=P(1+(r)/(n))^(nt)`

Where:

A is the future value

P is the initial value

r is the rate in decimal

n is the period

t is the time

let us find them from the question

A = 6500

P = 5000

r = 2% = 2/100 = 0.02

n = 52 weeks in year

t = ?

Let us substitute them in the rule to find t

`6500=5000(1+(0.02)/(52))^(52t)`

To find t divide both sides by 5000 first

`(6500)/(5000)=(1+(0.02)/(52))^(52t)`
`1.3=(1.000384615)^(52t)`

To find t insert log for both sides

`\text{log}(1.3)=\log (1.000384615)^(52t)`

Take the exponent in front of log because log a^b = b log a

`\text{log}(1.3)=52t\lbrack\log (1.000384615)`

Divide both sides by log(1.000384615)

`\begin{gathered} 682.2789435=52t \\ t=(682.2789435)/(52)=13.12074891 \end{gathered}`

So the number of years is 14 years to give her enough money to buy the car