If the rate of inflation is $$ 2.5 \% $$ 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. $$ p(t)=1500(1.025)^{t} $$ Find the current price of the item and the price 10 years from today. Round your answers to the nearest dollar as necessary. Current price: Price 10 years from today: $$ \left.\$ \prod\right] $$

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UpStudy AI · Accepted Answer

To find the current price of the item and the price 10 years from today, we need to calculate the values of $$ p(t) $$ for $$ t = 0 $$ and $$ t = 10 $$ respectively.

Given the exponential function:
$$ p(t) = 1500(1.025)^{t} $$

1. Current price (at $$ t = 0 $$):
$$ p(0) = 1500(1.025)^{0} $$
$$ p(0) = 1500 $$

2. Price 10 years from today (at $$ t = 10 $$):
$$ p(10) = 1500(1.025)^{10} $$

Now, let's calculate the values of $$ p(0) $$ and $$ p(10) $$ to find the current price and the price 10 years from today.
Calculate the value by following steps:
- step0: Calculate:
  $$1500\times 1.025^{0}$$
- step1: Evaluate the power:
  $$1500\times 1$$
- step2: Multiply:
  $$1500$$
Calculate or simplify the expression $$ 1500*(1.025)^10 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1500\times 1.025^{10}$$
- step1: Convert the expressions:
  $$1500\left(\frac{41}{40}\right)^{10}$$
- step2: Simplify:
  $$1500\times \frac{41^{10}}{40^{10}}$$
- step3: Rewrite the expression:
  $$20\times 75\times \frac{41^{10}}{40^{10}}$$
- step4: Rewrite the expression:
  $$20\times 75\times \frac{41^{10}}{20^{10}\times 2^{10}}$$
- step5: Reduce the numbers:
  $$75\times \frac{41^{10}}{20^{9}\times 2^{10}}$$
- step6: Rewrite the expression:
  $$25\times 3\times \frac{41^{10}}{20^{9}\times 2^{10}}$$
- step7: Rewrite the expression:
  $$25\times 3\times \frac{41^{10}}{5^{9}\times 4^{9}\times 2^{10}}$$
- step8: Rewrite the expression:
  $$5^{2}\times 3\times \frac{41^{10}}{5^{9}\times 4^{9}\times 2^{10}}$$
- step9: Reduce the numbers:
  $$3\times \frac{41^{10}}{5^{7}\times 4^{9}\times 2^{10}}$$
- step10: Multiply the fractions:
  $$\frac{3\times 41^{10}}{5^{7}\times 4^{14}}$$
- step11: Calculate:
  $$\frac{3\times 41^{10}}{80^{7}}$$
The current price of the item is $1500, and the price 10 years from today is approximately $1920.13.
The current price of the item is \$1500, and the price 10 years from today is approximately \$1920.

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