Studdy AI Logo
Math  /  Algebra

QuestionA house was valued at $125,000\$ 125,000 in the year 1990 . The value appreciated to $160,000\$ 160,000 by the year 2001. A) What was the annual growth rate between 1990 and 2001? r=r= □\square Round the growth rate to 4 decimal places. B) What is the correct answer to part A written in percentage form? r=□%r=\square \% C) Assume that the house value continues to grow by the same percentage. What will the value equal in the year 2005? value =$=\$ □\square

Studdy Solution

STEP 1

1. The house value in 1990 was 125,000.<br/>2.Thehousevaluein2001was125,000.<br />2. The house value in 2001 was 160,000.
3. We need to calculate the annual growth rate between 1990 and 2001.
4. The growth rate should be rounded to 4 decimal places.
5. The growth rate should also be expressed in percentage form.
6. We assume the growth rate continues to be the same for future calculations.

STEP 2

1. Calculate the annual growth rate between 1990 and 2001.
2. Convert the growth rate to percentage form.
3. Calculate the house value in 2005 assuming the same growth rate.

STEP 3

To find the annual growth rate, use the formula for compound annual growth rate (CAGR):
\[ r = \left( \frac{\text{Final Value}}{\text{Initial Value}} \right)^{\frac{1}{n}} - 1 $
where the Final Value is 160,000,theInitialValueis160,000, the Initial Value is 125,000, and n n is the number of years between 1990 and 2001, which is 11 years.
\[ r = \left( \frac{160,000}{125,000} \right)^{\frac{1}{11}} - 1 $

STEP 4

Calculate the ratio:
\[ \frac{160,000}{125,000} = 1.28 $
Now substitute back into the formula:
\[ r = 1.28^{\frac{1}{11}} - 1 $

STEP 5

Calculate 1.28111 1.28^{\frac{1}{11}} :
\[ 1.28^{\frac{1}{11}} \approx 1.0219 $
Now subtract 1:
\[ r \approx 1.0219 - 1 = 0.0219 $
Round to 4 decimal places:
\[ r \approx 0.0219 $

STEP 6

Convert the growth rate to percentage form by multiplying by 100:
\[ r = 0.0219 \times 100 = 2.19\% $

STEP 7

To find the value of the house in 2005, use the formula for future value with compound interest:
\[ \text{Future Value} = \text{Initial Value} \times (1 + r)^n $
where the Initial Value is $160,000, \( r = 0.0219 \), and \( n \) is the number of years from 2001 to 2005, which is 4 years.
\[ \text{Future Value} = 160,000 \times (1 + 0.0219)^4 $

STEP 8

Calculate (1+0.0219)4 (1 + 0.0219)^4 :
\[ (1 + 0.0219)^4 \approx 1.0906 $
Now calculate the future value:
\[ \text{Future Value} \approx 160,000 \times 1.0906 = 174,496 $
Round to the nearest dollar:
\[ \text{Future Value} = \$174,496 $
The annual growth rate is r=0.0219 r = 0.0219 or 2.19% 2.19\% , and the value of the house in 2005 is \$174,496.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord