Question 3 of 9 , step 1 of 1 The balance owed on your credit card triples from $$ \$ 700 $$ to $$ \$ 2100 $$ in 9 months. If the balance is growing linearly then it would take 53.4 months to reach $$ \$ 9000 $$. If, on the other hand, the balance is growing exponentially, $$ f(x)=700(1+0.13)^{x} $$ where $$ x $$ represents the number of months, what would the balance be after 53.4 months? Round your answer to the nearest cent. Answer How to enter your answer (opens in new window) Keyboard Shortcuts

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UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$700\left(1+0.13\right)^{53.4}$$
- step1: Add the numbers:
  $$700\times 1.13^{53.4}$$
- step2: Convert the expressions:
  $$700\left(\frac{113}{100}\right)^{\frac{267}{5}}$$
- step3: Simplify:
  $$700\times \frac{113^{\frac{267}{5}}}{100^{\frac{267}{5}}}$$
- step4: Multiply:
  $$\frac{700\times 113^{\frac{267}{5}}}{100^{\frac{267}{5}}}$$
- step5: Simplify:
  $$\frac{700\times 113^{53}\sqrt[5]{113^{2}}}{100^{\frac{267}{5}}}$$
- step6: Simplify:
  $$\frac{700\times 113^{53}\sqrt[5]{113^{2}}}{100^{53}\sqrt[5]{100^{2}}}$$
- step7: Rewrite the expression:
  $$\frac{700\times 113^{53}\sqrt[5]{113^{2}}}{100^{53}\sqrt[5]{10000}}$$
- step8: Multiply by the Conjugate:
  $$\frac{700\times 113^{53}\sqrt[5]{113^{2}}\times \sqrt[5]{10000^{4}}}{100^{53}\sqrt[5]{10000}\times \sqrt[5]{10000^{4}}}$$
- step9: Simplify:
  $$\frac{700\times 113^{53}\sqrt[5]{113^{2}}\times 10^{3}\sqrt[5]{10}}{100^{53}\sqrt[5]{10000}\times \sqrt[5]{10000^{4}}}$$
- step10: Multiply the numbers:
  $$\frac{700000\times 113^{53}\sqrt[5]{127690}}{100^{53}\sqrt[5]{10000}\times \sqrt[5]{10000^{4}}}$$
- step11: Multiply the numbers:
  $$\frac{700000\times 113^{53}\sqrt[5]{127690}}{10^{110}}$$
- step12: Rewrite the expression:
  $$\frac{100000\times 7\times 113^{53}\sqrt[5]{127690}}{10^{110}}$$
- step13: Rewrite the expression:
  $$\frac{10^{5}\times 7\times 113^{53}\sqrt[5]{127690}}{10^{110}}$$
- step14: Reduce the fraction:
  $$\frac{7\times 113^{53}\sqrt[5]{127690}}{10^{105}}$$
To find the balance on the credit card after 53.4 months using the exponential growth formula $$ f(x) = 700(1 + 0.13)^{x} $$, we will substitute $$ x = 53.4 $$ into the equation.

1. **Substituting the value of $$ x $$**:
   $$
   f(53.4) = 700(1 + 0.13)^{53.4}
   $$

2. **Calculating $$ (1 + 0.13)^{53.4} $$**:
   - First, calculate $$ 1 + 0.13 = 1.13 $$.
   - Then raise $$ 1.13 $$ to the power of $$ 53.4 $$.

3. **Multiplying by 700**:
   - Finally, multiply the result by 700 to find the balance.

After performing the calculations, we find:
$$
f(53.4) \approx 478064.97
$$

Thus, the balance after 53.4 months would be approximately **$478,064.97**.
After 53.4 months, the balance would be approximately \$478,064.97.

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