A video game costs $$ \$ 60 $$ today. It the inflation rate is $$ 4.75 \% $$ and is compounded continuously how much will this same video game cost in 5 years? Round your answer to the nearest cent (hundredth).

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UpStudy AI · Accepted Answer

To find the cost of the video game in 5 years with continuous compounding, we can use the formula for continuous compounding:

$$ A = P \times e^{r \times t} $$

Where:
- $$ A $$ is the amount after time $$ t $$,
- $$ P $$ is the principal amount (initial cost),
- $$ r $$ is the annual interest rate (in decimal form),
- $$ t $$ is the time in years.

Given:
- Initial cost of the video game today: $$ P = \$60 $$
- Inflation rate: $$ r = 4.75\% = 0.0475 $$ (in decimal form)
- Time: $$ t = 5 $$ years

Substitute the values into the formula:

$$ A = 60 \times e^{0.0475 \times 5} $$

Now, we can calculate the cost of the video game in 5 years using the formula for continuous compounding.
Calculate the value by following steps:
- step0: Calculate:
  $$60e^{0.0475\times 5}$$
- step1: Multiply the numbers:
  $$60e^{0.2375}$$
- step2: Convert the expressions:
  $$60e^{\frac{19}{80}}$$
- step3: Rewrite the expression:
  $$60\sqrt[80]{e^{19}}$$
The cost of the video game in 5 years with continuous compounding is approximately $76.08. Rounding to the nearest cent (hundredth), the cost will be $76.08.
The video game will cost approximately \$76.08 in 5 years.

A video game costs today. It the
inflation rate is and is compounded
continuously how much will this same
video game cost in 5 years?
Round your answer to the nearest cent (hundredth).

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A video game costs today. It the inflation rate is and is compounded continuously how much will this same video game cost in 5 years? Round your answer to the nearest cent (hundredth).