The formula $$ \mathrm{S}=\mathrm{C}(1+r)^{\mathrm{t}} $$ models inflation, where $$ \mathrm{C}= $$ the value today, $$ \mathrm{r}= $$ the annual inflation rate (in decimal form), and $$ \mathrm{S}= $$ the inflated value $$ t $$ years from now. If the inflation rate is $$ 5 \% $$, how much will a house now worth $$ \$ 177,000 $$ be worth in 10 years? Round your answer to the nearest dollar. The house will be worth $$ \$ \square $$. (Round to the nearest dollar as needed.)

Question

UpStudy AI · Accepted Answer

Given the formula for inflation $$ S = C(1 + r)^t $$, where:
- $$ C $$ is the value today,
- $$ r $$ is the annual inflation rate in decimal form,
- $$ S $$ is the inflated value $$ t $$ years from now.

We are given:
- The inflation rate $$ r = 5\% = 0.05 $$ (in decimal form),
- The value today $$ C = \$177,000 $$,
- The number of years $$ t = 10 $$.

We need to find the inflated value of the house in 10 years.

Substitute the given values into the formula:
$$ S = 177,000(1 + 0.05)^{10} $$

Now, we can calculate the inflated value of the house in 10 years.
Calculate the value by following steps:
- step0: Calculate:
  $$177000\left(1+0.05\right)^{10}$$
- step1: Add the numbers:
  $$177000\times 1.05^{10}$$
- step2: Convert the expressions:
  $$177000\left(\frac{21}{20}\right)^{10}$$
- step3: Simplify:
  $$177000\times \frac{21^{10}}{20^{10}}$$
- step4: Rewrite the expression:
  $$20\times 8850\times \frac{21^{10}}{20^{10}}$$
- step5: Reduce the numbers:
  $$8850\times \frac{21^{10}}{20^{9}}$$
- step6: Rewrite the expression:
  $$10\times 885\times \frac{21^{10}}{20^{9}}$$
- step7: Rewrite the expression:
  $$10\times 885\times \frac{21^{10}}{10^{9}\times 2^{9}}$$
- step8: Reduce the numbers:
  $$885\times \frac{21^{10}}{10^{8}\times 2^{9}}$$
- step9: Rewrite the expression:
  $$5\times 177\times \frac{21^{10}}{10^{8}\times 2^{9}}$$
- step10: Rewrite the expression:
  $$5\times 177\times \frac{21^{10}}{5^{8}\times 2^{8}\times 2^{9}}$$
- step11: Reduce the numbers:
  $$177\times \frac{21^{10}}{5^{7}\times 2^{8}\times 2^{9}}$$
- step12: Multiply the fractions:
  $$\frac{177\times 21^{10}}{2^{17}\times 5^{7}}$$
The house will be worth approximately $288,314.35 in 10 years, rounded to the nearest dollar.
The house will be worth \$288,314 in 10 years.

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Pre Algebra Questions