If the rate of inflation is $$ 3.9 \% $$ 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. $$ p(t)=1200(1.039)^{t} $$ Find the current price of the item and the price 8 years from today. Round your answers to the nearest dollar as necessary. Current price: Price 8 years from today:

Question

UpStudy AI · Accepted Answer

To find the current price of the item and the price 8 years from today, we need to calculate the values of $$ p(0) $$ and $$ p(8) $$ using the given exponential function $$ p(t) = 1200(1.039)^{t} $$.

1. Current price:
   To find the current price, we substitute $$ t = 0 $$ into the function:
   $$ p(0) = 1200(1.039)^{0} $$
   Since any number raised to the power of 0 is 1, the current price is:
   $$ p(0) = 1200 $$

2. Price 8 years from today:
   To find the price 8 years from today, we substitute $$ t = 8 $$ into the function:
   $$ p(8) = 1200(1.039)^{8} $$
   We can calculate this value using a calculator or by simplifying the expression:
   $$ p(8) = 1200(1.039)^{8} $$

Now, let's calculate the values of $$ p(0) $$ and $$ p(8) $$ to find the current price and the price 8 years from today.
Calculate the value by following steps:
- step0: Calculate:
  $$1200\times 1.039^{0}$$
- step1: Evaluate the power:
  $$1200\times 1$$
- step2: Multiply:
  $$1200$$
Calculate or simplify the expression $$ 1200*(1.039)^8 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$1200\times 1.039^{8}$$
- step1: Convert the expressions:
  $$1200\left(\frac{1039}{1000}\right)^{8}$$
- step2: Simplify:
  $$1200\times \frac{1039^{8}}{1000^{8}}$$
- step3: Rewrite the expression:
  $$200\times 6\times \frac{1039^{8}}{1000^{8}}$$
- step4: Rewrite the expression:
  $$200\times 6\times \frac{1039^{8}}{200^{8}\times 5^{8}}$$
- step5: Reduce the numbers:
  $$6\times \frac{1039^{8}}{200^{7}\times 5^{8}}$$
- step6: Rewrite the expression:
  $$2\times 3\times \frac{1039^{8}}{200^{7}\times 5^{8}}$$
- step7: Rewrite the expression:
  $$2\times 3\times \frac{1039^{8}}{8^{7}\times 25^{7}\times 5^{8}}$$
- step8: Rewrite the expression:
  $$2\times 3\times \frac{1039^{8}}{2^{21}\times 25^{7}\times 5^{8}}$$
- step9: Reduce the numbers:
  $$3\times \frac{1039^{8}}{2^{20}\times 25^{7}\times 5^{8}}$$
- step10: Multiply the fractions:
  $$\frac{3\times 1039^{8}}{2^{20}\times 5^{22}}$$
The current price of the item is $1200, and the price 8 years from today is approximately $1629.69.
The current price of the item is $1200, and the price 8 years from today is approximately $1630.

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Economics Questions