If the rate of inflation is $$ 3.7 \% $$ per year, the future price $$ p(t) $$ (in dollars) of a certain item can be modeled by the following exponential function, where $$ t $$ is the number of years from today. $$ p(t)=3000(1.037)^{t} $$ Find the current price of the item and the price 9 years from today. Round your answers to the nearest dollar as necessary. Current price: 9 years from today:

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UpStudy AI · Accepted Answer

To find the current price of the item and the price 9 years from today, we need to calculate the values of $$ p(0) $$ and $$ p(9) $$ using the given exponential function.

Given exponential function:
$$ p(t) = 3000(1.037)^{t} $$

1. Current price (at $$ t = 0 $$):
$$ p(0) = 3000(1.037)^{0} $$
$$ p(0) = 3000 $$

2. Price 9 years from today (at $$ t = 9 $$):
$$ p(9) = 3000(1.037)^{9} $$

Now, let's calculate the values of $$ p(0) $$ and $$ p(9) $$ using the given function.
Calculate the value by following steps:
- step0: Calculate:
  $$3000\times 1.037^{0}$$
- step1: Evaluate the power:
  $$3000\times 1$$
- step2: Multiply:
  $$3000$$
Calculate or simplify the expression $$ 3000*(1.037)^9 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$3000\times 1.037^{9}$$
- step1: Convert the expressions:
  $$3000\left(\frac{1037}{1000}\right)^{9}$$
- step2: Simplify:
  $$3000\times \frac{1037^{9}}{1000^{9}}$$
- step3: Rewrite the expression:
  $$1000\times 3\times \frac{1037^{9}}{1000^{9}}$$
- step4: Reduce the numbers:
  $$3\times \frac{1037^{9}}{1000^{8}}$$
- step5: Multiply:
  $$\frac{3\times 1037^{9}}{1000^{8}}$$
The current price of the item is $3000, and the price 9 years from today is approximately $4160.35.
Current price: $3000, Price 9 years from today: $4160

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