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Math  /  Algebra

QuestionEvaluating an exponentall
If the rate of inflation is 3.9%3.9 \% per year, the future price p(t)p(t) (in dollars) of a certain item can be modeled by the following exponential function, where tt is the number of years from today. p(t)=600(1.039)tp(t)=600(1.039)^{t}
Find the current price of the item and the price 9 years from today. Round your answers to the nearest dollar as necessary.

Studdy Solution

STEP 1

1. The future price p(t) p(t) is given by the function p(t)=600(1.039)t p(t) = 600(1.039)^t .
2. The rate of inflation is 3.9% 3.9\% per year.
3. We need to find the current price of the item, which is p(0) p(0) .
4. We need to find the price of the item 9 years from today, which is p(9) p(9) .

STEP 2

1. Evaluate the function at t=0 t = 0 to find the current price.
2. Evaluate the function at t=9 t = 9 to find the price 9 years from today.
3. Round the results to the nearest dollar.

STEP 3

Evaluate p(t) p(t) at t=0 t = 0 to find the current price:
p(0)=600(1.039)0 p(0) = 600(1.039)^0

STEP 4

Simplify the expression:
Since any number to the power of 0 is 1, we have:
p(0)=600×1=600 p(0) = 600 \times 1 = 600
The current price of the item is 600 600 dollars.

STEP 5

Evaluate p(t) p(t) at t=9 t = 9 to find the price 9 years from today:
p(9)=600(1.039)9 p(9) = 600(1.039)^9

STEP 6

Calculate (1.039)9 (1.039)^9 using a calculator:
(1.039)9≈1.40255 (1.039)^9 \approx 1.40255

STEP 7

Multiply by 600 to find p(9) p(9) :
p(9)=600×1.40255≈841.53 p(9) = 600 \times 1.40255 \approx 841.53

STEP 8

Round the results to the nearest dollar:
The current price is already a whole number: 600 600 .
The price 9 years from today, rounded to the nearest dollar, is 842 842 .
The current price of the item is 600 \boxed{600} dollars, and the price 9 years from today is 842 \boxed{842} dollars.

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