Answer :
To find the average value of the yacht over its first 14 years of use, we'll follow these steps:
1. Understand the Function: The value of the yacht after [tex]\( t \)[/tex] years is given by the function [tex]\( V(t) = 250000 e^{-0.16 t} \)[/tex]. This represents an exponentially decaying value.
2. Formula for Average Value: The average value [tex]\( V_{\text{ave}} \)[/tex] over a time interval from [tex]\( t = a \)[/tex] to [tex]\( t = b \)[/tex] is calculated using the formula:
[tex]\[
V_{\text{ave}} = \frac{1}{b-a} \int_{a}^{b} V(t) \, dt
\][/tex]
For this problem, [tex]\( a = 0 \)[/tex] and [tex]\( b = 14 \)[/tex], so:
[tex]\[
V_{\text{ave}} = \frac{1}{14-0} \int_{0}^{14} 250000 e^{-0.16 t} \, dt
\][/tex]
3. Calculate the Integral: The definite integral of the function from 0 to 14 gives the total accumulated value of the yacht over these years. This involves calculating:
[tex]\[
\int_{0}^{14} 250000 e^{-0.16 t} \, dt
\][/tex]
The result of this integration is approximately [tex]\( 1,396,158.59 \)[/tex].
4. Calculate the Average Value: Divide the result of the integral by the length of the interval (14 years) to find the average value over this period:
[tex]\[
V_{\text{ave}} = \frac{1}{14} \times 1,396,158.59 \approx 99,725.61
\][/tex]
Thus, the average value of the yacht over its first 14 years of use is approximately [tex]\(\$99,725.61\)[/tex].
1. Understand the Function: The value of the yacht after [tex]\( t \)[/tex] years is given by the function [tex]\( V(t) = 250000 e^{-0.16 t} \)[/tex]. This represents an exponentially decaying value.
2. Formula for Average Value: The average value [tex]\( V_{\text{ave}} \)[/tex] over a time interval from [tex]\( t = a \)[/tex] to [tex]\( t = b \)[/tex] is calculated using the formula:
[tex]\[
V_{\text{ave}} = \frac{1}{b-a} \int_{a}^{b} V(t) \, dt
\][/tex]
For this problem, [tex]\( a = 0 \)[/tex] and [tex]\( b = 14 \)[/tex], so:
[tex]\[
V_{\text{ave}} = \frac{1}{14-0} \int_{0}^{14} 250000 e^{-0.16 t} \, dt
\][/tex]
3. Calculate the Integral: The definite integral of the function from 0 to 14 gives the total accumulated value of the yacht over these years. This involves calculating:
[tex]\[
\int_{0}^{14} 250000 e^{-0.16 t} \, dt
\][/tex]
The result of this integration is approximately [tex]\( 1,396,158.59 \)[/tex].
4. Calculate the Average Value: Divide the result of the integral by the length of the interval (14 years) to find the average value over this period:
[tex]\[
V_{\text{ave}} = \frac{1}{14} \times 1,396,158.59 \approx 99,725.61
\][/tex]
Thus, the average value of the yacht over its first 14 years of use is approximately [tex]\(\$99,725.61\)[/tex].
