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A hollow sphere of internal and external radius 3 cm and 5 cm respectively is melted into a solid right circular cone of diameter 8 cm. The height of the cone is
- a)24.5 cm
- b)25.5 cm
- c)22.5 cm
- d)23.5 cm
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
A hollow sphere of internal and external radius 3 cm and 5 cm respecti...
We know that the formula of the volume of a hollow sphere is
4π(R3 – r3)
Here, R = external radius and r = internal radius
A hollow sphere of internal and external radius 3 cm and 5 cm respectively
So the volume of the hollow sphere = [4×π×(53 – 33)]/3 cc
Now, after melting this sphere, we will get a right circular cone, which’s diameter is 8 cm
So, radius of that cone = 8/2 cm = 4 cm
We know that the formula of the volume of a right circular cone is πr2h/3
Here, r is the radius of the cone and h is the height of the cone
From the question,
we can make the equation,
π × 42 × h/3 = [4×π×(53 – 33)]/3
⇒ 4h = 98
⇒ h = 24.5
So, the height of the cone is 24.5 cm
4π(R3 – r3)
Here, R = external radius and r = internal radius
A hollow sphere of internal and external radius 3 cm and 5 cm respectively
So the volume of the hollow sphere = [4×π×(53 – 33)]/3 cc
Now, after melting this sphere, we will get a right circular cone, which’s diameter is 8 cm
So, radius of that cone = 8/2 cm = 4 cm
We know that the formula of the volume of a right circular cone is πr2h/3
Here, r is the radius of the cone and h is the height of the cone
From the question,
we can make the equation,
π × 42 × h/3 = [4×π×(53 – 33)]/3
⇒ 4h = 98
⇒ h = 24.5
So, the height of the cone is 24.5 cm
Most Upvoted Answer
A hollow sphere of internal and external radius 3 cm and 5 cm respecti...
Understanding the Problem
To find the height of the cone formed by melting a hollow sphere, we first need to calculate the volume of the hollow sphere and then use that volume to find the height of the cone.
Volume of the Hollow Sphere
The volume \( V \) of a hollow sphere can be calculated using the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
Where:
- \( R \) is the external radius (5 cm)
- \( r \) is the internal radius (3 cm)
Calculating the volumes:
- External volume \( V_{external} = \frac{4}{3} \pi (5^3) \)
- Internal volume \( V_{internal} = \frac{4}{3} \pi (3^3) \)
Now substituting the values:
- \( V_{external} = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \) cm³
- \( V_{internal} = \frac{4}{3} \pi (27) = \frac{108}{3} \pi = 36 \pi \) cm³
So, the volume of the hollow sphere is:
\[ V_{hollow} = V_{external} - V_{internal} = \frac{500}{3} \pi - 36 \pi = \frac{500 - 108}{3} \pi = \frac{392}{3} \pi \text{ cm}^3 \]
Volume of the Cone
The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- Diameter of the cone = 8 cm → Radius \( r = 4 \) cm
Setting the volume of the cone equal to the volume of the hollow sphere:
\[ \frac{392}{3} \pi = \frac{1}{3} \pi (4^2) h \]
Simplifying:
\[ \frac{392}{3} = \frac{1}{3} (16) h \]
Cancelling \( \pi \) and multiplying through by 3:
\[ 392 = 16h \]
Thus:
\[ h = \frac{392}{16} = 24.5 \text{ cm} \]
Conclusion
The height of the cone is \( \text{24.5 cm} \), which corresponds to option (a).
To find the height of the cone formed by melting a hollow sphere, we first need to calculate the volume of the hollow sphere and then use that volume to find the height of the cone.
Volume of the Hollow Sphere
The volume \( V \) of a hollow sphere can be calculated using the formula:
\[ V = \frac{4}{3} \pi (R^3 - r^3) \]
Where:
- \( R \) is the external radius (5 cm)
- \( r \) is the internal radius (3 cm)
Calculating the volumes:
- External volume \( V_{external} = \frac{4}{3} \pi (5^3) \)
- Internal volume \( V_{internal} = \frac{4}{3} \pi (3^3) \)
Now substituting the values:
- \( V_{external} = \frac{4}{3} \pi (125) = \frac{500}{3} \pi \) cm³
- \( V_{internal} = \frac{4}{3} \pi (27) = \frac{108}{3} \pi = 36 \pi \) cm³
So, the volume of the hollow sphere is:
\[ V_{hollow} = V_{external} - V_{internal} = \frac{500}{3} \pi - 36 \pi = \frac{500 - 108}{3} \pi = \frac{392}{3} \pi \text{ cm}^3 \]
Volume of the Cone
The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Where:
- Diameter of the cone = 8 cm → Radius \( r = 4 \) cm
Setting the volume of the cone equal to the volume of the hollow sphere:
\[ \frac{392}{3} \pi = \frac{1}{3} \pi (4^2) h \]
Simplifying:
\[ \frac{392}{3} = \frac{1}{3} (16) h \]
Cancelling \( \pi \) and multiplying through by 3:
\[ 392 = 16h \]
Thus:
\[ h = \frac{392}{16} = 24.5 \text{ cm} \]
Conclusion
The height of the cone is \( \text{24.5 cm} \), which corresponds to option (a).
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A hollow sphere of internal and external radius 3 cm and 5 cm respectively is melted into a solid right circular cone of diameter 8 cm. The height of the cone isa)24.5 cmb)25.5 cmc)22.5 cmd)23.5 cmCorrect answer is option 'A'. Can you explain this answer?
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