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If the radius of a right circular cone is increased by p% without increasing its height, then what is the percentage increase in the volume of the cone?
- a)p2
- b)2p2
- c)p2/100
- d)p(2 + p/100)
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If the radius of a right circular cone is increased by p% without incr...
Suppose the radius and height of the cone is r and h cm respectively; π
∴ Volume of the cone 
After increasing the radius by p%;
New radius = (1 + p/100) × r
∴ New volume of the cone = 1/3 × π × (1 + p/100)2 × r2h
∴ Increase in the volume of the cone = (1/3 × π × (1 + p/100)2 × r2h – 1/3 × π × r2h)
⇒ π /3 × r2h[1 + p2/10000 + 2p/100 – 1]
⇒ π/3 × r2h × p/100 × [p/100 + 2]
∴ Percentage increase in the volume
After increasing the radius by p%;
New radius = (1 + p/100) × r
∴ New volume of the cone = 1/3 × π × (1 + p/100)2 × r2h
∴ Increase in the volume of the cone = (1/3 × π × (1 + p/100)2 × r2h – 1/3 × π × r2h)
⇒ π /3 × r2h[1 + p2/10000 + 2p/100 – 1]
⇒ π/3 × r2h × p/100 × [p/100 + 2]
∴ Percentage increase in the volume
Most Upvoted Answer
If the radius of a right circular cone is increased by p% without incr...
Solution:
Let's assume the original radius of the cone is 'r' and the original height is 'h'.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h
If the radius is increased by p%, then the new radius would be (r + (p/100)r) = (1 + p/100)r.
Since the height is not changing, the new volume of the cone would be:
V' = (1/3) * π * (1 + p/100)^2 * r^2 * h
To find the percentage increase in volume, we need to calculate the difference between the new volume and the original volume, and then express it as a percentage of the original volume.
Percentage increase in volume = ((V' - V)/V) * 100
Let's calculate this step by step.
Step 1:
V' = (1/3) * π * (1 + p/100)^2 * r^2 * h
Step 2:
V' - V = [(1/3) * π * (1 + p/100)^2 * r^2 * h] - [(1/3) * π * r^2 * h]
= (1/3) * π * r^2 * h * [(1 + p/100)^2 - 1]
Step 3:
V' - V = (1/3) * π * r^2 * h * [1 + 2(p/100) + (p/100)^2 - 1]
= (1/3) * π * r^2 * h * (2(p/100) + (p/100)^2)
Step 4:
(V' - V)/V = [(1/3) * π * r^2 * h * (2(p/100) + (p/100)^2)] / [(1/3) * π * r^2 * h]
= (2(p/100) + (p/100)^2)
Step 5:
Percentage increase in volume = ((V' - V)/V) * 100
= [(2(p/100) + (p/100)^2)] * 100
= 2p + (p^2/100)
Therefore, the percentage increase in the volume of the cone is p(2 + p/100), which is represented by option 'D'.
Let's assume the original radius of the cone is 'r' and the original height is 'h'.
The volume of a cone is given by the formula:
V = (1/3) * π * r^2 * h
If the radius is increased by p%, then the new radius would be (r + (p/100)r) = (1 + p/100)r.
Since the height is not changing, the new volume of the cone would be:
V' = (1/3) * π * (1 + p/100)^2 * r^2 * h
To find the percentage increase in volume, we need to calculate the difference between the new volume and the original volume, and then express it as a percentage of the original volume.
Percentage increase in volume = ((V' - V)/V) * 100
Let's calculate this step by step.
Step 1:
V' = (1/3) * π * (1 + p/100)^2 * r^2 * h
Step 2:
V' - V = [(1/3) * π * (1 + p/100)^2 * r^2 * h] - [(1/3) * π * r^2 * h]
= (1/3) * π * r^2 * h * [(1 + p/100)^2 - 1]
Step 3:
V' - V = (1/3) * π * r^2 * h * [1 + 2(p/100) + (p/100)^2 - 1]
= (1/3) * π * r^2 * h * (2(p/100) + (p/100)^2)
Step 4:
(V' - V)/V = [(1/3) * π * r^2 * h * (2(p/100) + (p/100)^2)] / [(1/3) * π * r^2 * h]
= (2(p/100) + (p/100)^2)
Step 5:
Percentage increase in volume = ((V' - V)/V) * 100
= [(2(p/100) + (p/100)^2)] * 100
= 2p + (p^2/100)
Therefore, the percentage increase in the volume of the cone is p(2 + p/100), which is represented by option 'D'.
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