Determine the $n$-intercept of the graph of the car value equation $V=-1385 n+18,790$. What is the practical significance of the $n$-intercept? Be sure to include units.
a. The car is worth $1385 after 14 years.
b. After 14 years, the car is worth $0.
c. The car is worth $14.
Answer (2)
Set the car's value, V , to 0 in the equation V = − 1385 n + 18790 .
Solve for n : 0 = − 1385 n + 18790 .
Calculate n = 1385 18790 = 13.566787... ≈ 13.57 .
The n -intercept is approximately 13.57 years, meaning the car's value is $0 after about 14 years. 14
Explanation
Understanding the Problem We are given the car value equation V = − 1385 n + 18790 , where V is the value of the car in dollars and n is the number of years of ownership. We need to find the n -intercept of this equation and explain its practical significance.
Finding the n-intercept The n -intercept is the value of n when V = 0 . So, we set V = 0 in the equation: 0 = − 1385 n + 18790
Solving for n Now, we solve for n :
1385 n = 18790 n = 1385 18790 n = 13.566787... Rounding up to the nearest whole number, we get n = 14 .
Interpreting the n-intercept The n -intercept is approximately 13.57 years. However, since the car's value is likely assessed at the end of each year, we can round this value to 14 years. This means that after 14 years, the car's value is approximately $0 .
Practical Significance The practical significance of the n -intercept is that it represents the number of years it takes for the car's value to depreciate to zero, according to the given linear depreciation model. In this case, it takes approximately 14 years for the car to have no value.
Examples
Understanding depreciation is crucial in personal finance. For example, when buying a new car, knowing how quickly it will lose value helps in making informed decisions about financing, insurance, and long-term ownership. The n -intercept, in this context, tells you when the car's value theoretically reaches zero, which can influence decisions about when to sell or trade it in. This concept is also applicable in business for asset management and tax planning, where depreciation affects the book value of assets and taxable income. By understanding the rate of depreciation, individuals and businesses can better plan their financial strategies.
The n -intercept of the equation V = − 1385 n + 18790 is found to be approximately 14 years, meaning the car's value reaches zero after this time. This indicates that after 14 years of ownership, the car has no value left according to the model. Thus, the correct answer is b: After 14 years, the car is worth $0.
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