Question

The value today of the other car is 15000. The value of the car increases exponentially by x% each year.

Answer 1

The future value of the car after y years, increasing exponentially by x% each year, is given by the formula
`\( \text{Future Value} = \text{Current Value} * (1 + (x)/(100))^y \).`

Step-by-step explanation:

Exponential growth is modeled using the formula
`\( \text{Future Value} = \text{Current Value} * (1 + (x)/(100))^y \)`, where the current value of the car is $15,000, x is the annual growth rate in percentage, and y is the number of years into the future.

The formula utilizes the compound interest concept, with
`\((1 + (x)/(100))\)`representing the growth factor. Each year, the car's value is multiplied by this growth factor, resulting in exponential growth. The exponent y denotes the number of years the growth is applied.

For instance, if the car experiences a 5% annual exponential growth for 3 years, the calculation would be
`\( \text{Future Value} = 15,000 * (1 + (5)/(100))^3 \)`. This yields the future value of the car after 3 years.

It's crucial to note that exponential growth implies a compounding effect, making the value increase at an accelerating rate. This contrasts with linear growth, where the value increases by a fixed amount each year.

The exponential growth model is commonly used in finance and investment to predict future values based on a consistent percentage growth rate.

Complete Question :

What is the future value of the car after y years if it continues to increase exponentially by x% each year, considering its current value is 15,000?

Answer 2

If the value of the car increases exponentially by x% each year, and its present value is $15,000, the future value after one year can be calculated using the formula for exponential growth: Future Value = Present Value × (1 + Rate of Growth)^Time. Therefore, the future value after one year would be $15,000 × (1 + x/100).

Step-by-step explanation:

Exponential growth represents a situation where a quantity increases by a fixed percentage over a given period. In this case, if the car's value increases exponentially by x% each year, after one year, the future value can be calculated using the formula for exponential growth. The formula for exponential growth is Future Value = Present Value × (1 + Rate of Growth)^Time, where the Rate of Growth is expressed as a decimal (x/100).

Substituting the given present value of $15,000 and the rate of growth as x% (which is x/100 as a decimal) into the exponential growth formula for one year, the future value can be expressed as $15,000 × (1 + x/100). This formula allows for the determination of the car's value after one year, considering the annual exponential increase of x%.

This exponential growth formula is fundamental in understanding how a quantity, in this case, the car's value, increases over time by a certain percentage annually. Utilizing the formula provides a straightforward method to calculate the future value based on the current value and the rate of growth, allowing for projections of the car's value in subsequent years based on the given percentage increase (x%) per year.