Solved on Mar 06, 2024

Write a function to model exponential decay. Given a car costing $55,800 that depreciates 16% annually, find its value next year.

STEP 1

Assumptions
1. The initial cost of the family car is $55,800.
2. The annual depreciation rate is 16%.
3. The car was purchased 3 years ago.
4. The value of the car decreases exponentially.
5. We are interested in the value of the car 1 year from now, which is 4 years after the purchase.

STEP 2

To model the exponential decay of the car's value, we use the formula for exponential decay:
$V(t) = V_0 \cdot (1 - r)^t$
where: - $V(t)$ is the value of the car at time $t$ , - $V_0$ is the initial value of the car, - $r$ is the annual depreciation rate (as a decimal), - $t$ is the time in years since the car was purchased.

STEP 3

Convert the annual depreciation rate from a percentage to a decimal.
$16\% = 0.16$

STEP 4

Substitute the given values into the exponential decay formula:
$V(t) = \$55,800 \cdot (1 - 0.16)^t$

STEP 5

Since we want to find the value of the car next year, which is 4 years after the purchase, we set $t = 4$ .
$V(4) = \$55,800 \cdot (1 - 0.16)^4$

STEP 6

Calculate the value inside the parentheses:
$1 - 0.16 = 0.84$

STEP 7

Raise 0.84 to the power of 4 to find the decay factor after 4 years:
$(0.84)^4$

STEP 8

Calculate the decay factor:
$(0.84)^4 \approx 0.49787136$

STEP 9

Multiply the initial value of the car by the decay factor to find the value of the car after 4 years:
$V(4) = \$55,800 \cdot 0.49787136$

STEP 10

Calculate the value of the car after 4 years:
$V(4) \approx \$55,800 \cdot 0.49787136 \approx \$27,779.24$
The car will be worth approximately $27,779.24 next year.

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Write a function to model exponential decay. Given a car costing $55,800 that depreciates 16% annually, find its value next year.

STEP 1

Assumptions
1. The initial cost of the family car is $55,800.
2. The annual depreciation rate is 16%.
3. The car was purchased 3 years ago.
4. The value of the car decreases exponentially.
5. We are interested in the value of the car 1 year from now, which is 4 years after the purchase.

STEP 2

STEP 3

Convert the annual depreciation rate from a percentage to a decimal.
$16\% = 0.16$

STEP 4

Substitute the given values into the exponential decay formula:
$V(t) = \$55,800 \cdot (1 - 0.16)^t$

STEP 5

Since we want to find the value of the car next year, which is 4 years after the purchase, we set $t = 4$ .
$V(4) = \$55,800 \cdot (1 - 0.16)^4$

STEP 6

Calculate the value inside the parentheses:
$1 - 0.16 = 0.84$

STEP 7

Raise 0.84 to the power of 4 to find the decay factor after 4 years:
$(0.84)^4$

STEP 8

Calculate the decay factor:
$(0.84)^4 \approx 0.49787136$

STEP 9

Multiply the initial value of the car by the decay factor to find the value of the car after 4 years:
$V(4) = \$55,800 \cdot 0.49787136$

STEP 10

Calculate the value of the car after 4 years:
$V(4) \approx \$55,800 \cdot 0.49787136 \approx \$27,779.24$
The car will be worth approximately $27,779.24 next year.

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Write a function to model exponential decay. Given a car costing $55,800 that depreciates 16% annually, find its value next year.

STEP 1

Assumptions1. The initial cost of the family car is $55,800.2. The annual depreciation rate is 16%.3. The car was purchased 3 years ago.4. The value of the car decreases exponentially.5. We are interested in the value of the car 1 year from now, which is 4 years after the purchase.

STEP 2

STEP 3

Convert the annual depreciation rate from a percentage to a decimal.16%=0.1616\% = 0.1616%=0.16

STEP 4

Substitute the given values into the exponential decay formula:V(t)=$55,800⋅(1−0.16)tV(t) = \$55,800 \cdot (1 - 0.16)^tV(t)=$55,800⋅(1−0.16)t

STEP 5

Since we want to find the value of the car next year, which is 4 years after the purchase, we set t=4t = 4t=4.V(4)=$55,800⋅(1−0.16)4V(4) = \$55,800 \cdot (1 - 0.16)^4V(4)=$55,800⋅(1−0.16)4

STEP 6

Calculate the value inside the parentheses:1−0.16=0.841 - 0.16 = 0.841−0.16=0.84

STEP 7

Raise 0.84 to the power of 4 to find the decay factor after 4 years:(0.84)4(0.84)^4(0.84)4

STEP 8

Calculate the decay factor:(0.84)4≈0.49787136(0.84)^4 \approx 0.49787136(0.84)4≈0.49787136

STEP 9

Multiply the initial value of the car by the decay factor to find the value of the car after 4 years:V(4)=$55,800⋅0.49787136V(4) = \$55,800 \cdot 0.49787136V(4)=$55,800⋅0.49787136

STEP 10

Calculate the value of the car after 4 years:V(4)≈$55,800⋅0.49787136≈$27,779.24V(4) \approx \$55,800 \cdot 0.49787136 \approx \$27,779.24V(4)≈$55,800⋅0.49787136≈$27,779.24The car will be worth approximately $27,779.24 next year.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Assumptions
1. The initial cost of the family car is $55,800.
2. The annual depreciation rate is 16%.
3. The car was purchased 3 years ago.
4. The value of the car decreases exponentially.
5. We are interested in the value of the car 1 year from now, which is 4 years after the purchase.

Convert the annual depreciation rate from a percentage to a decimal.
$16\% = 0.16$

Substitute the given values into the exponential decay formula:
$V(t) = \$55,800 \cdot (1 - 0.16)^t$

Since we want to find the value of the car next year, which is 4 years after the purchase, we set $t = 4$ .
$V(4) = \$55,800 \cdot (1 - 0.16)^4$

Calculate the value inside the parentheses:
$1 - 0.16 = 0.84$

Raise 0.84 to the power of 4 to find the decay factor after 4 years:
$(0.84)^4$

Calculate the decay factor:
$(0.84)^4 \approx 0.49787136$

Multiply the initial value of the car by the decay factor to find the value of the car after 4 years:
$V(4) = \$55,800 \cdot 0.49787136$

Calculate the value of the car after 4 years:
$V(4) \approx \$55,800 \cdot 0.49787136 \approx \$27,779.24$
The car will be worth approximately $27,779.24 next year.