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A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in
- a)6 month s
- b)9 month s
- c)12 month s
- d)infinite time
Correct answer is option 'A'. Can you explain this answer?
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A spherical naphthalene ball exposed to the atmosphere loses volume at...
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Problem Analysis:
We are given that the volume of a naphthalene ball decreases at a rate proportional to its instantaneous surface area due to evaporation. We need to determine the time it takes for the ball to completely evaporate.
Given:
Initial diameter of the ball = 2 cm
Final diameter of the ball = 1 cm
Time taken for the diameter to reduce from 2 cm to 1 cm = 3 months
Solution:
Step 1: Calculate the initial and final surface area of the ball:
The surface area of a sphere is given by the formula: A = 4πr², where r is the radius of the sphere.
Initial radius (r1) = initial diameter / 2 = 2 cm / 2 = 1 cm
Initial surface area (A1) = 4π(1 cm)² = 4π cm²
Final radius (r2) = final diameter / 2 = 1 cm / 2 = 0.5 cm
Final surface area (A2) = 4π(0.5 cm)² = π cm²
Step 2: Calculate the rate of volume loss:
We are given that the rate of volume loss is proportional to the instantaneous surface area. This can be represented by the equation:
dV/dt = -kA
Where dV/dt is the rate of volume loss, k is the proportionality constant, and A is the surface area.
Step 3: Solve the differential equation:
Since the rate of volume loss is proportional to the surface area, we can write the differential equation as:
dV/V = -kA dt
Integrating both sides of the equation gives:
ln(V) = -k∫A dt
Since the surface area is changing with time, we need to express A in terms of t. Since the surface area is proportional to the square of the radius, we have:
A = 4πr² = 4π(1 - kt)²
Substituting this into the equation and integrating gives:
ln(V) = -k∫4π(1 - kt)² dt
Simplifying and solving the integral gives:
ln(V) = -k(8πt - 6πkt² + 2πk²t³/3) + C
Where C is the constant of integration.
Step 4: Find the time for complete evaporation:
To find the time it takes for the ball to completely evaporate, we need to find the time at which the volume becomes zero.
When the volume becomes zero, ln(V) = ln(0) = -∞. Therefore, we can write:
-∞ = -k(8πt - 6πkt² + 2πk²t³/3) + C
Since -∞ is not a valid value, we must have k(8πt - 6πkt² + 2πk²t³/3) - C = 0.
Substituting the values of t = 0, V = V0 (initial volume), we get:
k(0) - C = 0
Therefore, C = 0.
Now we can rewrite the equation as
We are given that the volume of a naphthalene ball decreases at a rate proportional to its instantaneous surface area due to evaporation. We need to determine the time it takes for the ball to completely evaporate.
Given:
Initial diameter of the ball = 2 cm
Final diameter of the ball = 1 cm
Time taken for the diameter to reduce from 2 cm to 1 cm = 3 months
Solution:
Step 1: Calculate the initial and final surface area of the ball:
The surface area of a sphere is given by the formula: A = 4πr², where r is the radius of the sphere.
Initial radius (r1) = initial diameter / 2 = 2 cm / 2 = 1 cm
Initial surface area (A1) = 4π(1 cm)² = 4π cm²
Final radius (r2) = final diameter / 2 = 1 cm / 2 = 0.5 cm
Final surface area (A2) = 4π(0.5 cm)² = π cm²
Step 2: Calculate the rate of volume loss:
We are given that the rate of volume loss is proportional to the instantaneous surface area. This can be represented by the equation:
dV/dt = -kA
Where dV/dt is the rate of volume loss, k is the proportionality constant, and A is the surface area.
Step 3: Solve the differential equation:
Since the rate of volume loss is proportional to the surface area, we can write the differential equation as:
dV/V = -kA dt
Integrating both sides of the equation gives:
ln(V) = -k∫A dt
Since the surface area is changing with time, we need to express A in terms of t. Since the surface area is proportional to the square of the radius, we have:
A = 4πr² = 4π(1 - kt)²
Substituting this into the equation and integrating gives:
ln(V) = -k∫4π(1 - kt)² dt
Simplifying and solving the integral gives:
ln(V) = -k(8πt - 6πkt² + 2πk²t³/3) + C
Where C is the constant of integration.
Step 4: Find the time for complete evaporation:
To find the time it takes for the ball to completely evaporate, we need to find the time at which the volume becomes zero.
When the volume becomes zero, ln(V) = ln(0) = -∞. Therefore, we can write:
-∞ = -k(8πt - 6πkt² + 2πk²t³/3) + C
Since -∞ is not a valid value, we must have k(8πt - 6πkt² + 2πk²t³/3) - C = 0.
Substituting the values of t = 0, V = V0 (initial volume), we get:
k(0) - C = 0
Therefore, C = 0.
Now we can rewrite the equation as
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A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in a)6 month s b)9 month s c)12 month s d)infinite timeCorrect answer is option 'A'. Can you explain this answer?
Question Description
A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in a)6 month s b)9 month s c)12 month s d)infinite timeCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in a)6 month s b)9 month s c)12 month s d)infinite timeCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in a)6 month s b)9 month s c)12 month s d)infinite timeCorrect answer is option 'A'. Can you explain this answer?.
A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in a)6 month s b)9 month s c)12 month s d)infinite timeCorrect answer is option 'A'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in a)6 month s b)9 month s c)12 month s d)infinite timeCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A spherical naphthalene ball exposed to the atmosphere loses volume at a rate proportional to its instantaneous surface area due to evaporation. If the initial diameter of the ball is 2 cm and the diameter reduces to 1 cm after 3 months, the ball completely evaporates in a)6 month s b)9 month s c)12 month s d)infinite timeCorrect answer is option 'A'. Can you explain this answer?.
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