The value of Sara's new car decreases at a rate of 8% each year.

1.Write an exponential function to model the decrease in the car's value each month.
2.Write an exponential function to model the decrease in the car's value each week.
3.Write an exponential function to model the decrease in the car's value each day.
4.What relationship is there between the amount of decrease and the time interval measured?

1. V = Vo(1-r)^t.

Vo = Initial value of car.
r = 0.08/12mo. = 0.0067/mo.
t = Time in months.
V = Vo(1-0.0067)^1 = 0.9933Vo.

2. V = Vo(1-r)^t.
r = 0.08/52wks. = 0.00154/wk.
t = Time in weeks.
V = Vo(1-0.00154)^1 = 0.9985Vo.

3. V = Vo(1-r)^t.
r = 0.08/365da. = 0.000219/day.
V = Vo(1-0.000219)^1 = 0.9998Vo.

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1. To model the decrease in the car's value each month, we can use the following exponential function:

V(t) = V(0) * (1 - 0.08)^(t/12)

Where:
- V(t) represents the value of the car after t months.
- V(0) represents the initial value of the car.
- t represents the number of months.

2. To model the decrease in the car's value each week, we can modify the formula as follows:

V(t) = V(0) * (1 - 0.08)^(t/52)

Where:
- V(t) represents the value of the car after t weeks.
- V(0) represents the initial value of the car.
- t represents the number of weeks.

3. To model the decrease in the car's value each day, we can modify the formula again:

V(t) = V(0) * (1 - 0.08)^(t/365)

Where:
- V(t) represents the value of the car after t days.
- V(0) represents the initial value of the car.
- t represents the number of days.

4. The relationship between the amount of decrease and the time interval measured is that as the time interval decreases, the amount of decrease also decreases. In other words, the more frequently the car's value is updated, the smaller the decrease is in each update. This is because the exponent in the exponential function becomes smaller as the time interval gets smaller.

To answer the given questions, we need to write exponential functions to model the decrease in the car's value for different time intervals. We'll use the general form of an exponential function:

f(x) = a * (r^x)

where:
- f(x) represents the value of the car after x time intervals
- a represents the initial value (value at the beginning)
- r represents the rate of decrease (expressed as a decimal)

1. Exponential function for monthly decrease:
Here, we need to find the monthly rate of decrease based on the given annual rate of 8%. Since there are 12 months in a year, we divide the annual rate by 12. The monthly rate of decrease is 8% / 12 = 0.08 / 12 = 0.00667.
The exponential function for monthly decrease is: f(x) = a * (0.99333^x)

2. Exponential function for weekly decrease:
Similar to the monthly rate, we need to find the weekly rate of decrease. Since there are approximately 52 weeks in a year, we divide the annual rate by 52. The weekly rate of decrease is 8% / 52 = 0.08 / 52 = 0.00154.
The exponential function for weekly decrease is: f(x) = a * (0.99846^x)

3. Exponential function for daily decrease:
We need to find the daily rate of decrease based on the given annual rate. Since there are approximately 365 days in a year, we divide the annual rate by 365. The daily rate of decrease is 8% / 365 = 0.08 / 365 = 0.000219.
The exponential function for daily decrease is: f(x) = a * (0.999781^x)

4. Relationship between the amount of decrease and the time interval measured:
From the equations above, we can observe that as we decrease the time interval (from annual to monthly, weekly, and daily), the rate of decrease (represented by r) decreases. This means that as we measure the car's value over smaller time intervals, the rate of decrease becomes less significant, resulting in a slower decrease in value. In other words, the amount of decrease is inversely proportional to the time interval measured.