Math / Algebra

QuestionSuppose that the future price $p(t)$ of a certain item is given by the following exponential function. In this function, $p(t)$ is measured in dollars and $t$ is the number of years from today. $p(t)=2000(1.039)^{t}$
Find the initial price of the item. sॉI
Does the function represent growth or decay? growth decay By what percent does the price change each year? $\square$ \%

Studdy Solution

STEP 1

What is this asking? We're given a formula that tells us the price of an item in the future, and we need to figure out the starting price, whether the price goes up or down over time, and by how much it changes each year. Watch out! Don't mix up growth and decay!
A growth factor greater than 1 means the price is going up.
Also, remember that the percentage change isn't the growth factor itself, but how much bigger or smaller it is than 1.

STEP 2

1. Find the Initial Price
2. Determine Growth or Decay
3. Calculate the Annual Percentage Change

STEP 3

The initial price is the price at the very beginning, which means at time $t = 0$ .
So, we need to plug in $t = 0$ into our price function $p(t) = 2000(1.039)^{t}$ .

STEP 4

Let's do the calculation: $p(0) = 2000(1.039)^{0}$ Remember, anything raised to the power of 0 is 1 (except 0 itself, but that's not what we have here!). So, $(1.039)^0 = 1$ .

STEP 5

That makes our calculation super easy: $p(0) = 2000 \cdot 1 = 2000$ So the initial price is $\$2000$ !

STEP 6

Our function is $p(t) = 2000(1.039)^{t}$ .
The key here is the growth factor, which is 1.039.
Since $1.039 > 1$ , the price is increasing over time.

STEP 7

If the growth factor were less than 1 (like 0.95), then we'd have decay, but that's not the case here.
Because our factor is greater than 1, we have growth!

STEP 8

The growth factor is 1.039.
To find the percentage change, we need to see how much bigger this is than 1.
We can do this by subtracting 1: $1.039 - 1 = 0.039$

STEP 9

Now, we convert this to a percentage by multiplying by 100%: $0.039 \cdot 100\% = 3.9\%$ So, the price increases by 3.9% each year!

STEP 10

The initial price of the item is $\$2000$ .
The function represents growth.
The price changes by 3.9% each year.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Find the Initial Price
2. Determine Growth or Decay
3. Calculate the Annual Percentage Change

STEP 3

The initial price is the price at the very beginning, which means at time $t = 0$ .
So, we need to plug in $t = 0$ into our price function $p(t) = 2000(1.039)^{t}$ .

STEP 4

Let's do the calculation: $p(0) = 2000(1.039)^{0}$ Remember, anything raised to the power of 0 is 1 (except 0 itself, but that's not what we have here!). So, $(1.039)^0 = 1$ .

STEP 5

That makes our calculation super easy: $p(0) = 2000 \cdot 1 = 2000$ So the initial price is $\$2000$ !

STEP 6

Our function is $p(t) = 2000(1.039)^{t}$ .
The key here is the growth factor, which is 1.039.
Since $1.039 > 1$ , the price is increasing over time.

STEP 7

If the growth factor were less than 1 (like 0.95), then we'd have decay, but that's not the case here.
Because our factor is greater than 1, we have growth!

STEP 8

The growth factor is 1.039.
To find the percentage change, we need to see how much bigger this is than 1.
We can do this by subtracting 1: $1.039 - 1 = 0.039$

STEP 9

Now, we convert this to a percentage by multiplying by 100%: $0.039 \cdot 100\% = 3.9\%$ So, the price increases by 3.9% each year!

STEP 10

The initial price of the item is $\$2000$ .
The function represents growth.
The price changes by 3.9% each year.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Find the Initial Price2. Determine Growth or Decay3. Calculate the Annual Percentage Change

STEP 3

The **initial price** is the price at the very beginning, which means at time t=0t = 0t=0.So, we need to **plug in** t=0t = 0t=0 into our price function p(t)=2000(1.039)tp(t) = 2000(1.039)^{t}p(t)=2000(1.039)t.

STEP 4

Let's **do the calculation**: p(0)=2000(1.039)0p(0) = 2000(1.039)^{0}p(0)=2000(1.039)0 Remember, *anything* raised to the power of 0 is 1 (except 0 itself, but that's not what we have here!). So, (1.039)0=1(1.039)^0 = 1(1.039)0=1.

STEP 5

That makes our calculation super easy: p(0)=2000⋅1=2000p(0) = 2000 \cdot 1 = 2000p(0)=2000⋅1=2000 So the **initial price** is $2000\$2000$2000!

STEP 6

Our function is p(t)=2000(1.039)tp(t) = 2000(1.039)^{t}p(t)=2000(1.039)t.The **key** here is the **growth factor**, which is 1.039.Since 1.039>11.039 > 11.039>1, the price is increasing over time.

STEP 7

If the **growth factor** were *less than* 1 (like 0.95), then we'd have decay, but that's not the case here.Because our factor is greater than 1, we have **growth**!

STEP 8

The **growth factor** is 1.039.To find the **percentage change**, we need to see how much bigger this is than 1.We can do this by subtracting 1: 1.039−1=0.0391.039 - 1 = 0.0391.039−1=0.039

STEP 9

Now, we **convert this to a percentage** by multiplying by 100%: 0.039⋅100%=3.9%0.039 \cdot 100\% = 3.9\%0.039⋅100%=3.9% So, the price increases by **3.9%** each year!

STEP 10

The initial price of the item is $2000\$2000$2000.The function represents growth.The price changes by 3.9% each year.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Find the Initial Price
2. Determine Growth or Decay
3. Calculate the Annual Percentage Change

The initial price is the price at the very beginning, which means at time $t = 0$ .
So, we need to plug in $t = 0$ into our price function $p(t) = 2000(1.039)^{t}$ .

Let's do the calculation: $p(0) = 2000(1.039)^{0}$ Remember, anything raised to the power of 0 is 1 (except 0 itself, but that's not what we have here!). So, $(1.039)^0 = 1$ .

That makes our calculation super easy: $p(0) = 2000 \cdot 1 = 2000$ So the initial price is $\$2000$ !

Our function is $p(t) = 2000(1.039)^{t}$ .
The key here is the growth factor, which is 1.039.
Since $1.039 > 1$ , the price is increasing over time.

If the growth factor were less than 1 (like 0.95), then we'd have decay, but that's not the case here.
Because our factor is greater than 1, we have growth!

The growth factor is 1.039.
To find the percentage change, we need to see how much bigger this is than 1.
We can do this by subtracting 1: $1.039 - 1 = 0.039$

Now, we convert this to a percentage by multiplying by 100%: $0.039 \cdot 100\% = 3.9\%$ So, the price increases by 3.9% each year!

The initial price of the item is $\$2000$ .
The function represents growth.
The price changes by 3.9% each year.