Question

The value of a Kobe Bryant Rookie cardcan be approximated by the modelwhere t is the number of years since thecard was released.a. Tell whether the modelrepresents exponential growth orexponential decay.b. Identify the annual percentincrease or decrease in the valueof the card.C. Estimate the year when the cardvalue is over $250000.

Answer 1

Given the model:

`y=250(1.55)^t`

The model above represents the value of a pair of OFF-white Jordan, where t is the number of years since the shoe was released.

Let's answer the following questions.

• a) Let's determine if the model represents an exponential growth or decay.

Apply the exponential function:

`y=a(b)^x`

When b is greater than 1, we can say the function reppresents exponential grwoth.

When b is less than 1, the funcrion represents exponential decay.

Here, b = 1.55 is greater than 1.

Since b is greater than 1, the given model represents an exponential growth.

• b) Take the exponential growth function:

`y=a(1+r)^x`

Where:

r = percentage increase.

Since the model is an exponential growth, we are to find the annual percent increase.

To find the annula percentage increase, we have:

`\begin{gathered} y=250(1.55)^t \\ y=250(1+(1.55-1)^t \\ \\ \\ y=250(1+0.55)^t \\ \\ r=0.55\Longrightarrow\text{ 55\%} \end{gathered}`

Therefore, the annual percent increase (r) is = 55%

• c) To find when the resale value of the shoe will be over 2000, substitute 2000 for y and solve for t.

Thus, we have:

`\begin{gathered} y=250(1.55)^t \\ \\ 2000<250(1.55)^t \end{gathered}`

Let's solve for t.

Divide both sides by 250

`\begin{gathered} (2000)/(250)<(250(1.55)^t)/(250)^{} \\ \\ 8<1.55^t \end{gathered}`

Take the natural logarithm of both sides:

Therefore, the shoe's resale value will be over $2000 after 5 years

ANSWER:

• a) Exponential growth

,

• b) 55%

,

• c) 5 years