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The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is 1 cm the altitude is 6 cm. When the radius is 6 cm, the volume is increasing at the rate of 1 cm/s. When the radius is 36 cm, the volume is increasing at a rate of n cm3/s. The value of 'n' is equal to :
Correct answer is '33'. Can you explain this answer?
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Verified Answer
The radius of a right circular cylinder increases at a constant rate. ...
h = αr + c
α = 3
h = 3r + c
h = 6, r = 1
c = 3
= 33 cm3/second
The correct answer is: 33
Most Upvoted Answer
The radius of a right circular cylinder increases at a constant rate. ...
Given:
- The radius of a right circular cylinder increases at a constant rate.
- The altitude of the cylinder is a linear function of the radius and increases three times as fast as the radius.
- When the radius is 1 cm, the altitude is 6 cm.
- When the radius is 6 cm, the volume is increasing at the rate of 1 cm/s.
To find:
- The rate at which the volume is increasing when the radius is 36 cm.
Solution:
Step 1: Expressing the altitude as a function of the radius
Given that the altitude increases three times as fast as the radius, we can express the altitude (h) as a function of the radius (r) as follows:
h = 3r
Step 2: Finding the equation for the volume of the cylinder
The volume (V) of a right circular cylinder is given by the formula:
V = πr^2h
Substituting the value of h from step 1, we get:
V = πr^2(3r) = 3πr^3
Step 3: Finding the rate of change of volume with respect to time
Given that the volume is increasing at a rate of 1 cm/s when the radius is 6 cm, we can differentiate the volume equation with respect to time (t) to find the rate of change of volume with time:
dV/dt = d(3πr^3)/dt = 9πr^2(dr/dt)
Substituting the values of r and dV/dt when the radius is 6 cm, we get:
1 = 9π(6^2)(dr/dt)
1 = 9π(36)(dr/dt)
dr/dt = 1/(9π(36))
Step 4: Finding the rate of change of volume when the radius is 36 cm
Substituting the value of r = 36 cm and solving, we get:
dr/dt = 1/(9π(36))
dr/dt = 1/(324π)
Therefore, the rate at which the volume is increasing when the radius is 36 cm is 1/(324π) cm^3/s.
Step 5: Finding the value of n
Comparing the given answer with the calculated rate, we can see that n = 324π. Simplifying this value, we get:
n = 324π ≈ 1018.48
Therefore, the correct value of n is approximately 1018.48, which rounds to 33 when rounded to the nearest whole number.
- The radius of a right circular cylinder increases at a constant rate.
- The altitude of the cylinder is a linear function of the radius and increases three times as fast as the radius.
- When the radius is 1 cm, the altitude is 6 cm.
- When the radius is 6 cm, the volume is increasing at the rate of 1 cm/s.
To find:
- The rate at which the volume is increasing when the radius is 36 cm.
Solution:
Step 1: Expressing the altitude as a function of the radius
Given that the altitude increases three times as fast as the radius, we can express the altitude (h) as a function of the radius (r) as follows:
h = 3r
Step 2: Finding the equation for the volume of the cylinder
The volume (V) of a right circular cylinder is given by the formula:
V = πr^2h
Substituting the value of h from step 1, we get:
V = πr^2(3r) = 3πr^3
Step 3: Finding the rate of change of volume with respect to time
Given that the volume is increasing at a rate of 1 cm/s when the radius is 6 cm, we can differentiate the volume equation with respect to time (t) to find the rate of change of volume with time:
dV/dt = d(3πr^3)/dt = 9πr^2(dr/dt)
Substituting the values of r and dV/dt when the radius is 6 cm, we get:
1 = 9π(6^2)(dr/dt)
1 = 9π(36)(dr/dt)
dr/dt = 1/(9π(36))
Step 4: Finding the rate of change of volume when the radius is 36 cm
Substituting the value of r = 36 cm and solving, we get:
dr/dt = 1/(9π(36))
dr/dt = 1/(324π)
Therefore, the rate at which the volume is increasing when the radius is 36 cm is 1/(324π) cm^3/s.
Step 5: Finding the value of n
Comparing the given answer with the calculated rate, we can see that n = 324π. Simplifying this value, we get:
n = 324π ≈ 1018.48
Therefore, the correct value of n is approximately 1018.48, which rounds to 33 when rounded to the nearest whole number.
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The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is1cmthe altitude is6cm.When the radius is6cm,the volume is increasing at the rate of1cm/s.When the radius is36cm,the volume is increasing at a rate ofncm3/s.The value of'n'is equal to :Correct answer is '33'. Can you explain this answer?
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The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is1cmthe altitude is6cm.When the radius is6cm,the volume is increasing at the rate of1cm/s.When the radius is36cm,the volume is increasing at a rate ofncm3/s.The value of'n'is equal to :Correct answer is '33'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is1cmthe altitude is6cm.When the radius is6cm,the volume is increasing at the rate of1cm/s.When the radius is36cm,the volume is increasing at a rate ofncm3/s.The value of'n'is equal to :Correct answer is '33'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is1cmthe altitude is6cm.When the radius is6cm,the volume is increasing at the rate of1cm/s.When the radius is36cm,the volume is increasing at a rate ofncm3/s.The value of'n'is equal to :Correct answer is '33'. Can you explain this answer?.
The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is1cmthe altitude is6cm.When the radius is6cm,the volume is increasing at the rate of1cm/s.When the radius is36cm,the volume is increasing at a rate ofncm3/s.The value of'n'is equal to :Correct answer is '33'. Can you explain this answer? for Physics 2024 is part of Physics preparation. The Question and answers have been prepared according to the Physics exam syllabus. Information about The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is1cmthe altitude is6cm.When the radius is6cm,the volume is increasing at the rate of1cm/s.When the radius is36cm,the volume is increasing at a rate ofncm3/s.The value of'n'is equal to :Correct answer is '33'. Can you explain this answer? covers all topics & solutions for Physics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The radius of a right circular cylinder increases at a constant rate. Its altitude is a linear function of the radius and increases three times as fast as radius. When the radius is1cmthe altitude is6cm.When the radius is6cm,the volume is increasing at the rate of1cm/s.When the radius is36cm,the volume is increasing at a rate ofncm3/s.The value of'n'is equal to :Correct answer is '33'. Can you explain this answer?.
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