Question

for the numbers below, calculate the growth or decay rate

1. 1.17\t2. 1.03
2. 0.87\t4. 1.24
3. 0.33\t6. 0.55
4. 1.51\t8. 0.90
5. 0.75\t10. 1.36

how do exponents work for you?
you deposit money into a savings account. you start by depositing $500. your account earns 2% apr (annual percentage rate). you dont add to the account. you leave your money there for 4 years. how much do you now have?
using the formula outlined above
\\( y = 500(1.02)^4 \\)
when we calculate how the following to our account after 4 years
540.81

Explanation:

Step1: Recall growth/decay rate formula

For a number \( b \) in the form of an exponential base (like in \( y = a(b)^x \)), the growth or decay rate \( r \) is calculated as \( r=b - 1 \). If \( r>0 \), it's a growth rate (expressed as a decimal, can be converted to percentage by multiplying by 100); if \( r < 0 \), it's a decay rate.

Step2: Calculate for 1.17

\( r=1.17 - 1=0.17 \). As a percentage, \( 0.17\times100 = 17\% \) growth rate.

Step3: Calculate for 1.03

\( r = 1.03-1=0.03 \). As a percentage, \( 0.03\times100=3\% \) growth rate.

Step4: Calculate for 0.87

\( r=0.87 - 1=- 0.13 \). As a percentage, \( - 0.13\times100=-13\% \) (13% decay rate).

Step5: Calculate for 1.24

\( r = 1.24-1 = 0.24 \). As a percentage, \( 0.24\times100 = 24\% \) growth rate.

Step6: Calculate for 0.33

\( r=0.33 - 1=-0.67 \). As a percentage, \( - 0.67\times100=-67\% \) (67% decay rate).

Step7: Calculate for 0.55

\( r=0.55 - 1=-0.45 \). As a percentage, \( - 0.45\times100=-45\% \) (45% decay rate).

Step8: Calculate for 1.51

\( r=1.51 - 1 = 0.51 \). As a percentage, \( 0.51\times100=51\% \) growth rate.

Step9: Calculate for 0.90

\( r=0.90 - 1=-0.10 \). As a percentage, \( - 0.10\times100=-10\% \) (10% decay rate).

Step10: Calculate for 0.75

\( r=0.75 - 1=-0.25 \). As a percentage, \( - 0.25\times100=-25\% \) (25% decay rate).

Step11: Calculate for 1.36

\( r=1.36 - 1=0.36 \). As a percentage, \( 0.36\times100 = 36\% \) growth rate.

For the savings account problem:

Step1: Identify the formula and values

The formula is \( y = 500(1.02)^t \), where \( t = 8 \) years, principal \( P = 500 \), rate \( r=0.02 \) (2% APR), time \( t = 8 \).

Step2: Calculate \( (1.02)^8 \)

Using a calculator, \( (1.02)^8\approx1.171659 \).

Step3: Calculate the amount \( y \)

\( y=500\times1.171659\approx585.83 \).

Answer:

Growth/Decay Rates:

1. \( 17\% \) growth
2. \( 3\% \) growth
3. \( 13\% \) decay
4. \( 24\% \) growth
5. \( 67\% \) decay
6. \( 45\% \) decay
7. \( 51\% \) growth
8. \( 10\% \) decay
9. \( 25\% \) decay
10. \( 36\% \) growth

Savings Account Amount:

After 8 years, you would have approximately \(\$585.83\)