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The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.?
Verified Answer
The height of a cone is 60cm. A small cone is cut off at the top by a ...
Height of original cone = 60 cm
Let the radius of the original cone of original cone be r₁.
Let the radius of the smaller cone be r₂.
Let the height of the smaller cone be h.
By similarity criterion,
r₁ / r₂ = h / 60
1 / 64 Volume of the original cone = Volume of the small cone
1 / 64 X 1 /3 π r₁^2 60 = 1 / 3 π r₂^2 h
1 / 64 X r₁^260 = r₂^2 h
( r₁ / r₂ )^2 X h = 64 / 60
( h / 60 )^2 X h = 64 / 60
h^3 / 3600 = 64 / 60
h^3 = 64 X 60
h = ∛ 3840
h = 64 ∛60
h = 15.659 cm
Therefore, the height of the smaller cone is 15.659 cm.
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Most Upvoted Answer
The height of a cone is 60cm. A small cone is cut off at the top by a ...
Understanding the Problem
We have a cone with a height of 60 cm. A smaller cone is created by slicing off the top part of the original cone with a plane parallel to the base. The volume of this smaller cone is \( \frac{1}{64} \) of the original cone's volume.
Volume of a Cone
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( r \) is the radius
- \( h \) is the height
Volume Relationship
Let the height of the smaller cone be \( h_1 \) and the radius be \( r_1 \). The volume of the smaller cone can be expressed as:
\[
V_1 = \frac{1}{3} \pi r_1^2 h_1
\]
Since the volume of the smaller cone is \( \frac{1}{64} \) of the original cone's volume, we write:
\[
V_1 = \frac{1}{64} V
\]
Similarity of Cones
Because the section is parallel to the base, the smaller cone is similar to the original cone. Therefore, the ratios of their dimensions are equal:
\[
\frac{r_1}{r} = \frac{h_1}{h}
\]
Let \( h_1 = 60 - x \), where \( x \) is the height from the base at which the section is made.
Setting Up the Equation
Using the volume relationship and similarity:
\[
\frac{h_1^3}{h^3} = \frac{1}{64}
\]
\[
\frac{(60 - x)^3}{60^3} = \frac{1}{64}
\]
Taking the cube root gives:
\[
\frac{60 - x}{60} = \frac{1}{4}
\]
Solving for x
Cross-multiplying:
\[
4(60 - x) = 60
\]
\[
240 - 4x = 60
\]
\[
4x = 180 \implies x = 45
\]
Final Answer
The height from the base at which the section is made is:
(c) 45 cm
We have a cone with a height of 60 cm. A smaller cone is created by slicing off the top part of the original cone with a plane parallel to the base. The volume of this smaller cone is \( \frac{1}{64} \) of the original cone's volume.
Volume of a Cone
The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]
Where:
- \( r \) is the radius
- \( h \) is the height
Volume Relationship
Let the height of the smaller cone be \( h_1 \) and the radius be \( r_1 \). The volume of the smaller cone can be expressed as:
\[
V_1 = \frac{1}{3} \pi r_1^2 h_1
\]
Since the volume of the smaller cone is \( \frac{1}{64} \) of the original cone's volume, we write:
\[
V_1 = \frac{1}{64} V
\]
Similarity of Cones
Because the section is parallel to the base, the smaller cone is similar to the original cone. Therefore, the ratios of their dimensions are equal:
\[
\frac{r_1}{r} = \frac{h_1}{h}
\]
Let \( h_1 = 60 - x \), where \( x \) is the height from the base at which the section is made.
Setting Up the Equation
Using the volume relationship and similarity:
\[
\frac{h_1^3}{h^3} = \frac{1}{64}
\]
\[
\frac{(60 - x)^3}{60^3} = \frac{1}{64}
\]
Taking the cube root gives:
\[
\frac{60 - x}{60} = \frac{1}{4}
\]
Solving for x
Cross-multiplying:
\[
4(60 - x) = 60
\]
\[
240 - 4x = 60
\]
\[
4x = 180 \implies x = 45
\]
Final Answer
The height from the base at which the section is made is:
(c) 45 cm
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The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.?
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The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.?.
The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The height of a cone is 60cm. A small cone is cut off at the top by a plane parallel to the base and its volume is 1/64th the volume of original cone. The height from the base at which the section is made is: (a) 15cm (b) 30cm (c) 45cm (d) 7√2 Please explain with answer.?.
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