A metallic right circular cylinder is15cm high and the diameter of the base is 14cm it is melted and recasted into another cylinder with radius 4cm find its height and curved surface area
To find the height of the new cylinder after the casting, we can use the principle of conservation of volume. The volume of the original cylinder is equal to the volume of the new cylinder.
1. Volume of the original cylinder:
The volume of a cylinder is given by the formula V = πr^2h, where r is the radius and h is the height. In this case, the radius of the original cylinder is half the diameter, so r = 14/2 = 7 cm. The height is given as 15 cm.
V1 = π(7^2)(15)
2. Volume of the new cylinder:
The radius of the new cylinder is given as 4 cm. To find the height, we rearrange the formula for volume:
V2 = π(4^2)(h2)
Since the volume of the original and new cylinder are the same, we can set up the following equation:
V1 = V2
π(7^2)(15) = π(4^2)(h2)
Now we can solve for h2, the height of the new cylinder:
(7^2)(15) = (4^2)(h2)
h2 = (7^2)(15) / (4^2)
Calculate h2:
h2 = (49)(15) / (16)
h2 = 735 / 16
h2 = 45.9375 cm (rounded to 4 decimal places)
Therefore, the height of the new cylinder is approximately 45.9375 cm.
To find the curved surface area of the new cylinder, we can use the formula:
CSA = 2πrh,
where r is the radius of the new cylinder and h is its height.
Substituting the given values:
r = 4 cm
h = 45.9375 cm
CSA = 2π(4)(45.9375)
CSA = 364.875π cm^2 (rounded to 3 decimal places)
Therefore, the curved surface area of the new cylinder is approximately 364.875π cm^2.
To find the height and curved surface area of the recast cylinder, we can use the principle of conservation of volume.
1. First, let's calculate the volume of the original cylinder using its height and diameter.
The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height.
Given:
Height (h) of the original cylinder = 15 cm
Diameter (d) of the base of the original cylinder = 14 cm
We can calculate the radius (r) of the original cylinder using the formula r = d/2:
r = 14 cm / 2 = 7 cm
Now we can calculate the volume (V1) of the original cylinder:
V1 = π(7 cm)^2 * 15 cm = 3.14 * 49 cm^2 * 15 cm ≈ 10882.2 cm^3
2. Next, let's calculate the radius of the recast cylinder.
Given:
Radius (r2) of the recast cylinder = 4 cm
We can now calculate the volume (V2) of the recast cylinder using V2 = V1.
V2 = π(4 cm)^2 * h2, where h2 is the height of the recast cylinder.
Since V2 = V1, we can substitute the values to get:
π(4 cm)^2 * h2 = 10882.2 cm^3
Simplifying the equation:
16π cm^2 * h2 = 10882.2 cm^3
h2 = 10882.2 cm^3 / (16π cm^2)
h2 ≈ 10882.2 cm^3 / 50.2655 cm^2 ≈ 216.8 cm
Therefore, the height of the recast cylinder is approximately 216.8 cm.
3. Finally, let's calculate the curved surface area (CSA) of the recast cylinder.
The formula for the CSA of a cylinder is CSA = 2πrh, where r is the radius and h is the height.
Given:
Radius (r2) of the recast cylinder = 4 cm
Height (h2) of the recast cylinder = 216.8 cm
We can now calculate the CSA of the recast cylinder:
CSA = 2π(4 cm)(216.8 cm) = 2π(4 cm)(216.8 cm) ≈ 3451.6 cm^2
Therefore, the curved surface area of the recast cylinder is approximately 3451.6 cm^2.
volume=PI*r^2 h=PI*7^2*15
new volume is the same...
PI*7^2*15=PI*4^2*h
height= 49*15/16 cm
Surface area: PI*14*49*15/16