Question

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The value of a brand new car is $27,000 and the value depreciates 23% every year. Write a function to represent the value of the car after t years, where the monthly rate of change can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage rate of change per month, to the nearest hundredth of a percent.

Answer 1

Answer and Explanation:

The value of a brand new car,

P

=

$

28000

The value depreciates every year,

r

=

19

%

Time,

t

=

1

4

Number of compounding period (Compounded quarterly),

n

=

4

Write a function to represent the value of the car after

t

years:

A

=

P

(

1

−

r

n

)

n

×

t

∴

A

=

28000

(

1

−

19

%

4

)

4

t

Let us evaluate the value of the car after one quarter:

A

=

28000

(

1

−

19

%

4

)

4

t

=

28000

(

1

−

19

100

4

)

4

×

1

4

=

28000

(

1

−

0.19

4

)

4

×

1

4

=

28000

(

1

−

0.19

4

)

1

=

28000

(

1

−

0.19

4

)

=

28000

(

1

−

0.0475

)

=

28000

(

0.9525

)

∴

A

=

26670

Therefore, the value of the car after one quarter is

$

26670

.

To calculate the percentage rate of change per quarter:

Percentage rate

=

P

−

A

P

×

100

%

=

28000

−

26670

28000

×

100

%

=

1330

28000

×

100

%

=

70

⋅

19

70

⋅

400

×

100

%

=

19

400

×

100

%

=

0.0475

×

100

%

∴

Percentage rate of change per quarter

=

4.75

%

Hence, the percentage rate of change per quarter is

4.75

%

.

Answer 2

To calculate the value of the car after t years, use the formula V(t) = P(1 - r)^t, where P is the initial value of the car, r is the rate of depreciation per year, and t is the number of years. The monthly rate of change can be found by dividing the annual rate by 12. In this case, the monthly rate of change is approximately 1.92% per month.

Step-by-step explanation:

To calculate the value of the car after t years, we need to use the formula:

V(t) = P(1 - r)t

Where:

• V(t) represents the value of the car after t years
• P represents the initial value of the car, which is $27,000
• r represents the rate of depreciation per year, which is 23% or 0.23
• t represents the number of years

So, the function to represent the value of the car after t years would be:

V(t) = 27000 * (1 - 0.23)t

To find the monthly rate of change, we need to convert the annual rate to a monthly rate. There are 12 months in a year, so we divide the annual rate by 12:

Monthly rate of change = 0.23 / 12 = 0.019167

Therefore, the monthly rate of change is approximately 0.0192 or 1.92% per month.