The Value { V $}$ Of A Certain Automobile That Is { T $}$ Years Old Can Be Modeled By $ V(t) = 14,949(0.82)^t $. According To The Model, When Will The Car Be Worth Each Of The Following Amounts?(a) $ $ 6000 $(b)

Introduction

In the world of finance and economics, mathematical models play a crucial role in predicting the value of assets over time. One such model is used to determine the value of a certain automobile that is $t$ years old. The model is given by the equation $V(t) = 14,949(0.82)^t$, where $V(t)$ represents the value of the automobile at time $t$. In this article, we will explore this model and use it to determine when the car will be worth specific amounts.

The Mathematical Model

The mathematical model used to determine the value of the automobile is given by the equation $V(t) = 14,949(0.82)^t$. This equation represents the value of the automobile at time $t$, where $t$ is measured in years. The equation is a simple exponential function, where the base is 0.82 and the exponent is $t$. The coefficient 14,949 represents the initial value of the automobile.

Solving for Specific Values

To determine when the car will be worth specific amounts, we need to solve the equation $V(t) = 14,949(0.82)^t$ for $t$. This involves isolating $t$ on one side of the equation. We can do this by using logarithms to rewrite the equation in a more manageable form.

(a) $V(t) = \$6000$

To determine when the car will be worth $ $6000$, we need to solve the equation $6000 = 14,949(0.82)^t$ for $t$. We can start by dividing both sides of the equation by 14,949 to get:

$$\frac{6000}{14,949} = (0.82)^t$$

Next, we can take the logarithm of both sides of the equation to get:

$$\log\left(\frac{6000}{14,949}\right) = t \log(0.82)$$

Now, we can solve for $t$ by dividing both sides of the equation by $\log(0.82)$:

$$t = \frac{\log\left(\frac{6000}{14,949}\right)}{\log(0.82)}$$

Using a calculator to evaluate the expression, we get:

$$t \approx 10.32$$

Therefore, according to the model, the car will be worth $ $6000$ when it is approximately 10.32 years old.

(b) $V(t) = \$3000$

To determine when the car will be worth $ $3000$, we need to solve the equation $3000 = 14,949(0.82)^t$ for $t$. We can start by dividing both sides of the equation by 14,949 to get:

$$\frac{3000}{14,949} = (0.82)^t$$

Next, we can take the logarithm of both sides of the equation to get:

$$\log\left(\frac{3000}{14,949}\right) = t \log(0.82)$$

Now, we can solve for $t$ by dividing both sides of the equation by $\log(0.82)$:

$$t = \frac{\log\left(\frac{3000}{14,949}\right)}{\log(0.82)}$$

Using a calculator to evaluate the expression, we get:

$$t \approx 11.93$$

Therefore, according to the model, the car will be worth $ $3000$ when it is approximately 11.93 years old.

(c) $V(t) = \$2000$

To determine when the car will be worth $ $2000$, we need to solve the equation $2000 = 14,949(0.82)^t$ for $t$. We can start by dividing both sides of the equation by 14,949 to get:

$$\frac{2000}{14,949} = (0.82)^t$$

Next, we can take the logarithm of both sides of the equation to get:

$$\log\left(\frac{2000}{14,949}\right) = t \log(0.82)$$

Now, we can solve for $t$ by dividing both sides of the equation by $\log(0.82)$:

$$t = \frac{\log\left(\frac{2000}{14,949}\right)}{\log(0.82)}$$

Using a calculator to evaluate the expression, we get:

$$t \approx 13.54$$

Therefore, according to the model, the car will be worth $ $2000$ when it is approximately 13.54 years old.

(d) $V(t) = \$1000$

To determine when the car will be worth $ $1000$, we need to solve the equation $1000 = 14,949(0.82)^t$ for $t$. We can start by dividing both sides of the equation by 14,949 to get:

$$\frac{1000}{14,949} = (0.82)^t$$

Next, we can take the logarithm of both sides of the equation to get:

$$\log\left(\frac{1000}{14,949}\right) = t \log(0.82)$$

Now, we can solve for $t$ by dividing both sides of the equation by $\log(0.82)$:

$$t = \frac{\log\left(\frac{1000}{14,949}\right)}{\log(0.82)}$$

Using a calculator to evaluate the expression, we get:

$$t \approx 16.15$$

Therefore, according to the model, the car will be worth $ $1000$ when it is approximately 16.15 years old.

Conclusion

$$$$$$$$

Q: What is the mathematical model used to determine the value of the automobile?

A: The mathematical model used to determine the value of the automobile is given by the equation $V(t) = 14,949(0.82)^t$, where $V(t)$ represents the value of the automobile at time $t$.

Q: How does the model take into account the depreciation of the automobile over time?

A: The model takes into account the depreciation of the automobile over time by using an exponential function with a base of 0.82. This means that the value of the automobile decreases by a factor of 0.82 each year.

Q: What is the initial value of the automobile according to the model?

A: The initial value of the automobile according to the model is $14,949.

Q: How can I use the model to determine the value of the automobile at a specific time?

A: To use the model to determine the value of the automobile at a specific time, you can plug in the value of $t$ into the equation $V(t) = 14,949(0.82)^t$ and solve for $V(t)$.

Q: What are some common applications of the model?

A: Some common applications of the model include:

• Determining the value of a used car
• Calculating the depreciation of a car over time
• Estimating the value of a car at a specific age
• Comparing the value of different cars at different ages

Q: Can I use the model to determine the value of other types of assets?

A: While the model is specifically designed to determine the value of an automobile, it can be adapted to determine the value of other types of assets that depreciate over time. However, the model may need to be modified to take into account the specific characteristics of the asset.

Q: How accurate is the model?

A: The accuracy of the model depends on the specific assumptions and data used to create it. In general, the model is most accurate when used to estimate the value of an automobile over a short period of time (e.g. 1-5 years). However, the model may become less accurate as the time period increases.

Q: Can I use the model to make predictions about the future value of the automobile?

A: Yes, the model can be used to make predictions about the future value of the automobile. However, it's essential to keep in mind that the model is based on historical data and may not take into account future events or changes in the market.

Q: How can I modify the model to take into account other factors that affect the value of the automobile?

A: To modify the model to take into account other factors that affect the value of the automobile, you can add additional variables to the equation or use a more complex model that incorporates multiple factors.

Q: Can I use the model to compare the value of different automobiles?

A: Yes, the model can be used to compare the value of different automobiles. However, it's essential to ensure that the model is used consistently and that the same assumptions and data are used for each comparison.

Q: How can I use the model to determine the optimal time to sell or trade in the automobile?

A: To use the model to determine the optimal time to sell or trade in the automobile, you can plug in the current value of the automobile and the desired value into the equation and solve for $t$. This will give you an estimate of the time it will take for the automobile to reach the desired value.

Conclusion

In this article, we have answered some of the most frequently asked questions about the value of a certain automobile. We have discussed the mathematical model used to determine the value of the automobile, how the model takes into account the depreciation of the automobile over time, and how to use the model to determine the value of the automobile at a specific time. We have also discussed some common applications of the model, its accuracy, and how to modify the model to take into account other factors that affect the value of the automobile.