Question

The value of a particular item can be modeled by
P(t) = P0(a)t
where P is in dollars and t is the number of years since the item was purchased. Suppose the value of the item increases 5% each year and the item was purchased for $20.

(a) Write a formula for P(t) according to the model.
P(t) =

(b) How fast is the value of the item increasing when t = 25 years? Round your answer to two decimal places.

Answer 1

Answer:

$3.29

Step-by-step explanation:

The value of item increases 5% each year and the item was purchased for $20.00.

A. We have to write a formula for P(t) according to the model:

$P(t)=P_o(1+r)^t$ Where, $P_o$ is the initial price, 'r' is the rate of interest. Here in our case it is compounded annually.  

So the formula is:

$P(t)=P_o(1+r)^t$

B. Now we need to find how fast is the value of item increasing when t=25 years.

$P(t)=P_o(1+r)^t$

$P(t)=P_o(1+.05)^t$

$P(t)=P_o(1.05)^t$

Differentiating the function we get the rate of change:

$P'(t)=20 \times (1.05)^t\times ln(1.05)$ (since $P_o=20$)

$P'(t)=20\times 0.0487\times (1.05)^t=0.974(1.05)^t$

Putting t=25, we get:

$P'(25)=0.974(1.05)^{25}=0.974\times3.386=3.29$

So the value of item is increasing by $3.29  each year.

Answer 2

The required price after 25 years is $67.73

The required model will be  P(t) = 20(1.05)^t

Exponential functions

The standard exponential function is expressed as:

P(t) = P0(a)^t

Given the following

• P0 = $20
• a = 0.05
• t = number of years

Since there is an increment yearly, the required model will be

P(t) = 20(1.05)^t

If t = 25, the required pricee will be calculated as:

P(t) = 20(1.05)^25

P(25) = $67.73

Hence the required price after 25 years is $67.73

Learn more one exponential function here: https://brainly.com/question/12940982