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A right circular cone of maximum volume is cut from a solid right circular cylinder of metal and the volume of the remaining cylinder is 1078 cm3. If the ratio between radius and height of the cylinder is 2: 3, then what was the total surface area of the cylinder initially?
- a)770 cm2
- b)750 cm2
- c)775 cm2
- d)778 cm2
- e)772 cm2
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A right circular cone of maximum volume is cut from a solid right circ...
Let the radius and height of the cylinder be ‘2x’ cm and ‘3x’ cm, respectively.
So, for maximum volume, radius and height of the cone should be ‘2x’ cm and ‘3x’ cm, respectively.
From the question:
x3 = 42.875
x = 3.5 cm
Radius of the cylinder = 2 x 3.5 = 7cm
Height of the cylinder = 3 x 3.5 = 10.5 cm
So, total surface area of the cylinder initially:
2 x (22/7) x 7 x (7 + 10.5) = 770 cm2
So, for maximum volume, radius and height of the cone should be ‘2x’ cm and ‘3x’ cm, respectively.
From the question:
x3 = 42.875
x = 3.5 cm
Radius of the cylinder = 2 x 3.5 = 7cm
Height of the cylinder = 3 x 3.5 = 10.5 cm
So, total surface area of the cylinder initially:
2 x (22/7) x 7 x (7 + 10.5) = 770 cm2
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Community Answer
A right circular cone of maximum volume is cut from a solid right circ...
To solve this problem, we need to consider the relationship between the cone and the cylinder and use the given information to find the volume and surface area of the cylinder.
Let's assume the radius of the cylinder is 2x and the height is 3x (according to the given ratio).
1. Finding the volume of the cone:
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Since the cone is cut from the cylinder, the height of the cone will be equal to the height of the cylinder, which is 3x.
Let's assume the radius of the cone is r1 and the height is h1.
According to similar triangles, we can write:
r1/r = h1/h
Substituting the values, we get:
r1/(2x) = h1/(3x)
r1 = (2/3)h1
The volume of the cone is given as the difference between the volume of the cylinder and the volume of the remaining cylinder, which is 1078 cm^3.
So, we have:
V_cone = V_cylinder - V_remaining_cylinder
(1/3)πr1^2h1 = π(2x)^2(3x) - 1078
(1/3)π[(2/3)h1]^2h1 = π(4x^2)(3x) - 1078
(1/3)π(4/9)h1^3 = 12πx^3 - 1078
(4/27)h1^3 = 12x^3 - 1078/π
2. Finding the volume of the cylinder:
The volume of a cylinder is given by the formula V = πr^2h.
Substituting the values, we get:
V_cylinder = π(2x)^2(3x)
V_cylinder = 12πx^3
3. Finding the surface area of the cylinder:
The surface area of a cylinder is given by the formula A = 2πrh + 2πr^2.
Substituting the values, we get:
A_cylinder = 2π(2x)(3x) + 2π(2x)^2
A_cylinder = 12πx^2 + 8πx^2
A_cylinder = 20πx^2
Now, we have two equations:
(4/27)h1^3 = 12x^3 - 1078/π
A_cylinder = 20πx^2
To find the maximum volume of the cone, we need to find the maximum value of h1. We can do this by differentiating the equation (4/27)h1^3 = 12x^3 - 1078/π with respect to h1 and equating it to zero.
Differentiating, we get:
(4/9)h1^2 = 0
Solving for h1, we get:
h1 = 0
Since h1 cannot be zero, there is no maximum volume of the cone. This means that the cylinder is already a maximum volume cylinder.
Therefore, the total surface area of the cylinder initially is given by A_cylinder =
Let's assume the radius of the cylinder is 2x and the height is 3x (according to the given ratio).
1. Finding the volume of the cone:
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius and h is the height.
Since the cone is cut from the cylinder, the height of the cone will be equal to the height of the cylinder, which is 3x.
Let's assume the radius of the cone is r1 and the height is h1.
According to similar triangles, we can write:
r1/r = h1/h
Substituting the values, we get:
r1/(2x) = h1/(3x)
r1 = (2/3)h1
The volume of the cone is given as the difference between the volume of the cylinder and the volume of the remaining cylinder, which is 1078 cm^3.
So, we have:
V_cone = V_cylinder - V_remaining_cylinder
(1/3)πr1^2h1 = π(2x)^2(3x) - 1078
(1/3)π[(2/3)h1]^2h1 = π(4x^2)(3x) - 1078
(1/3)π(4/9)h1^3 = 12πx^3 - 1078
(4/27)h1^3 = 12x^3 - 1078/π
2. Finding the volume of the cylinder:
The volume of a cylinder is given by the formula V = πr^2h.
Substituting the values, we get:
V_cylinder = π(2x)^2(3x)
V_cylinder = 12πx^3
3. Finding the surface area of the cylinder:
The surface area of a cylinder is given by the formula A = 2πrh + 2πr^2.
Substituting the values, we get:
A_cylinder = 2π(2x)(3x) + 2π(2x)^2
A_cylinder = 12πx^2 + 8πx^2
A_cylinder = 20πx^2
Now, we have two equations:
(4/27)h1^3 = 12x^3 - 1078/π
A_cylinder = 20πx^2
To find the maximum volume of the cone, we need to find the maximum value of h1. We can do this by differentiating the equation (4/27)h1^3 = 12x^3 - 1078/π with respect to h1 and equating it to zero.
Differentiating, we get:
(4/9)h1^2 = 0
Solving for h1, we get:
h1 = 0
Since h1 cannot be zero, there is no maximum volume of the cone. This means that the cylinder is already a maximum volume cylinder.
Therefore, the total surface area of the cylinder initially is given by A_cylinder =
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A right circular cone of maximum volume is cut from a solid right circular cylinder of metal and the volume of the remaining cylinder is 1078 cm3. If the ratio between radius and height of the cylinder is 2: 3, then what was the total surface area of the cylinder initially?a)770 cm2b)750 cm2c)775 cm2d)778 cm2e)772 cm2Correct answer is option 'A'. Can you explain this answer? for Banking Exams 2025 is part of Banking Exams preparation. The Question and answers have been prepared according to the Banking Exams exam syllabus. Information about A right circular cone of maximum volume is cut from a solid right circular cylinder of metal and the volume of the remaining cylinder is 1078 cm3. If the ratio between radius and height of the cylinder is 2: 3, then what was the total surface area of the cylinder initially?a)770 cm2b)750 cm2c)775 cm2d)778 cm2e)772 cm2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Banking Exams 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A right circular cone of maximum volume is cut from a solid right circular cylinder of metal and the volume of the remaining cylinder is 1078 cm3. If the ratio between radius and height of the cylinder is 2: 3, then what was the total surface area of the cylinder initially?a)770 cm2b)750 cm2c)775 cm2d)778 cm2e)772 cm2Correct answer is option 'A'. Can you explain this answer?.
A right circular cone of maximum volume is cut from a solid right circular cylinder of metal and the volume of the remaining cylinder is 1078 cm3. If the ratio between radius and height of the cylinder is 2: 3, then what was the total surface area of the cylinder initially?a)770 cm2b)750 cm2c)775 cm2d)778 cm2e)772 cm2Correct answer is option 'A'. Can you explain this answer? for Banking Exams 2025 is part of Banking Exams preparation. The Question and answers have been prepared according to the Banking Exams exam syllabus. Information about A right circular cone of maximum volume is cut from a solid right circular cylinder of metal and the volume of the remaining cylinder is 1078 cm3. If the ratio between radius and height of the cylinder is 2: 3, then what was the total surface area of the cylinder initially?a)770 cm2b)750 cm2c)775 cm2d)778 cm2e)772 cm2Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for Banking Exams 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A right circular cone of maximum volume is cut from a solid right circular cylinder of metal and the volume of the remaining cylinder is 1078 cm3. If the ratio between radius and height of the cylinder is 2: 3, then what was the total surface area of the cylinder initially?a)770 cm2b)750 cm2c)775 cm2d)778 cm2e)772 cm2Correct answer is option 'A'. Can you explain this answer?.
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