Given the model:
![y=250(1.55)^t](https://tex.z-dn.net/?f=y%3D250%281.55%29%5Et)
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](https://tex.z-dn.net/?f=y%3Da%28b%29%5Ex)
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](https://tex.z-dn.net/?f=y%3Da%281%2Br%29%5Ex)
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}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20y%3D250%281.55%29%5Et%20%5C%5C%20y%3D250%281%2B%281.55-1%29%5Et%20%5C%5C%20%20%5C%5C%20%20%5C%5C%20y%3D250%281%2B0.55%29%5Et%20%5C%5C%20%20%5C%5C%20r%3D0.55%5CLongrightarrow%5Ctext%7B%2055%5C%25%7D%20%5Cend%7Bgathered%7D)
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](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20y%3D250%281.55%29%5Et%20%5C%5C%20%20%5C%5C%202000%3C250%281.55%29%5Et%20%5Cend%7Bgathered%7D)
Let's solve for t.
Divide both sides by 250
![\begin{gathered} \frac{2000}{250}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cfrac%7B2000%7D%7B250%7D%3C%5Cfrac%7B250%281.55%29%5Et%7D%7B250%7D%5E%7B%7D%20%5C%5C%20%20%5C%5C%208%3C1.55%5Et%20%5Cend%7Bgathered%7D)
Take the natural logarithm of both sides:
![\begin{gathered} \ln 8Divide both sides by ln1.55:[tex]\begin{gathered} \frac{\ln 8}{\ln 1.55}4.7\approx5 \\ \\ \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cln%208Divide%20both%20sides%20by%20%20ln1.55%3A%5Btex%5D%5Cbegin%7Bgathered%7D%20%5Cfrac%7B%5Cln%208%7D%7B%5Cln%201.55%7D%3C%5Cfrac%7Bt%5Cln1.55%7D%7B%5Cln1.55%7D%20%5C%5C%20%20%5C%5C%20%5Cfrac%7B%5Cln%208%7D%7B%5Cln%201.55%7D%5Cfrac%7B%5Cln8%7D%7B%5Cln1.55%7D%3D%5Cfrac%7B2.079%7D%7B0.438%7D%20%5C%5C%20%20%5C%5C%20t%3E4.7%5Capprox5%20%5C%5C%20%20%5C%5C%20%20%5Cend%7Bgathered%7D)
Therefore, the shoe's resale value will be over $2000 after 5 years
ANSWER:
• a) Exponential growth
,
• b) 55%
,
• c) 5 years