A cylindrical candle of diameter 8cm and height 7cm is lit. After burning, a hemispherical
depression of diameter 4cm is left in the candle. Find the volume and surface area of this
partially burnt candle.
volume = πr^2 h = 112π
area = 2πr^2 + 2πrh = 88π
hemisphere:
volume = 2/3 πr^3 = 16π/3
area = 4πr^2 = 16π
so the remaining candle has
volume = 112π - 16π/3
original volume minus the hemisphere
area = πr^2 + 2πrh + 2πr^2 + (16π-4π)
that is, the original bottom + lateral + (top - hole)
Wtf
To find the volume of the partially burnt candle, we need to calculate the volume of the original cylinder and subtract the volume of the hemispherical depression.
1. Volume of the original cylinder:
The formula for the volume of a cylinder is given by V = π * r^2 * h, where r is the radius of the cylinder and h is the height.
Given that the diameter of the candle is 8 cm, the radius (r) is half of the diameter, which is 4 cm. The height (h) is given as 7 cm.
Therefore, the volume of the original cylinder is V_cylinder = π * 4^2 * 7.
2. Volume of the hemispherical depression:
The formula for the volume of a hemisphere is V = (2/3) * π * r^3, where r is the radius of the hemisphere.
Since the diameter of the depression is 4 cm, the radius (r) is half of the diameter, which is 2 cm.
Therefore, the volume of the hemispherical depression is V_hemisphere = (2/3) * π * 2^3.
3. Volume of the partially burnt candle:
To calculate the volume of the partially burnt candle, we subtract the volume of the hemispherical depression from the volume of the original cylinder:
V_partial_burnt_candle = V_cylinder - V_hemisphere.
After substituting the values, you can evaluate the expression to find the volume of the partially burnt candle.
To find the surface area of the partially burnt candle, we need to calculate the surface area of the original cylinder and subtract the surface area of the hemispherical depression.
1. Surface area of the original cylinder:
The formula for the surface area of a cylinder is given by A = 2πr^2 + 2πr * h.
Given that the diameter of the candle is 8 cm, the radius (r) is half of the diameter, which is 4 cm. The height (h) is given as 7 cm.
Therefore, the surface area of the original cylinder is A_cylinder = 2π * 4^2 + 2π * 4 * 7.
2. Surface area of the hemispherical depression:
The formula for the surface area of a hemisphere is A = 2πr^2, where r is the radius of the hemisphere.
Since the diameter of the depression is 4 cm, the radius (r) is half of the diameter, which is 2 cm.
Therefore, the surface area of the hemispherical depression is A_hemisphere = 2π * 2^2.
3. Surface area of the partially burnt candle:
To calculate the surface area of the partially burnt candle, we subtract the surface area of the hemispherical depression from the surface area of the original cylinder:
A_partial_burnt_candle = A_cylinder - A_hemisphere.
After substituting the values, you can evaluate the expression to find the surface area of the partially burnt candle.
cylinder:
volume = πr^2 h = 112π
area = 2πr^2 + 2πrh = 88π
hemisphere:
volume = 2/3 πr^3 = 16π/3
area = 4πr^2 = 16π
so the remaining candle has
volume = 112π - 16π/3
original volume minus the hemisphere
area = πr^2 + 2πrh + 2πr^2 + (16π-4π)
that is, the original bottom + lateral + (top - hole)