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Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.?
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Let I be the purchase value of an equipment and V(t) be the value afte...
Scrap Value of Equipment
Differential Equation: dv(t)/dt = -k(T-t)
Explanation:
To find the scrap value of the equipment, we need to determine the value V(T) at the end of its total life T years.
Step 1: Solving the Differential Equation:
The given differential equation dv(t)/dt = -k(T-t) represents the rate at which the value of the equipment depreciates over time. To solve this equation, we can separate the variables and integrate both sides.
∫(1/V) dV = ∫-k(T-t) dt
Applying the integration, we get:
ln|V| = -k(Tt - (t^2)/2) + C
Where C is the constant of integration.
Step 2: Applying Initial Condition:
To determine the value of C, we can apply the initial condition. Initially, when t = 0, the value of the equipment is the purchase value I. Thus, we have:
ln|I| = -k(0 - (0^2)/2) + C
ln|I| = C
Therefore, the equation becomes:
ln|V| = -k(Tt - (t^2)/2) + ln|I|
Step 3: Finding the Value at the End of Total Life:
To find the value V(T) at the end of the total life T years, we substitute t = T into the equation:
ln|V(T)| = -k(TT - (T^2)/2) + ln|I|
ln|V(T)| = -k(T^2)/2 + ln|I|
Exponentiating both sides of the equation:
|V(T)| = e^(-k(T^2)/2) * |I|
Step 4: Determining the Scrap Value:
The scrap value of the equipment is the value at the end of its total life, which is V(T). Therefore, the scrap value V(T) can be expressed as:
Scrap Value (V(T)) = e^(-k(T^2)/2) * Purchase Value (I)
Conclusion:
The scrap value of the equipment, V(T), can be calculated using the equation V(T) = e^(-k(T^2)/2) * I, where I is the purchase value of the equipment, k is a positive constant representing the depreciation rate, and T is the total life of the equipment in years.
Differential Equation: dv(t)/dt = -k(T-t)
Explanation:
To find the scrap value of the equipment, we need to determine the value V(T) at the end of its total life T years.
Step 1: Solving the Differential Equation:
The given differential equation dv(t)/dt = -k(T-t) represents the rate at which the value of the equipment depreciates over time. To solve this equation, we can separate the variables and integrate both sides.
∫(1/V) dV = ∫-k(T-t) dt
Applying the integration, we get:
ln|V| = -k(Tt - (t^2)/2) + C
Where C is the constant of integration.
Step 2: Applying Initial Condition:
To determine the value of C, we can apply the initial condition. Initially, when t = 0, the value of the equipment is the purchase value I. Thus, we have:
ln|I| = -k(0 - (0^2)/2) + C
ln|I| = C
Therefore, the equation becomes:
ln|V| = -k(Tt - (t^2)/2) + ln|I|
Step 3: Finding the Value at the End of Total Life:
To find the value V(T) at the end of the total life T years, we substitute t = T into the equation:
ln|V(T)| = -k(TT - (T^2)/2) + ln|I|
ln|V(T)| = -k(T^2)/2 + ln|I|
Exponentiating both sides of the equation:
|V(T)| = e^(-k(T^2)/2) * |I|
Step 4: Determining the Scrap Value:
The scrap value of the equipment is the value at the end of its total life, which is V(T). Therefore, the scrap value V(T) can be expressed as:
Scrap Value (V(T)) = e^(-k(T^2)/2) * Purchase Value (I)
Conclusion:
The scrap value of the equipment, V(T), can be calculated using the equation V(T) = e^(-k(T^2)/2) * I, where I is the purchase value of the equipment, k is a positive constant representing the depreciation rate, and T is the total life of the equipment in years.
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Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.?
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Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.?.
Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let I be the purchase value of an equipment and V(t) be the value after it has been used for `t' years. The value V(t) depreciate at a rate given by differential equation dv(t)/dt=-k(T-t),where k>0 is a constant and T is the total life in years of equipment. The scrap value V(T) of equipment is.?.
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