Question 9
A petrol container is in the form of a central cylindrical portion 5.0 m long with a hemispherical section surmounted on each end. If the diameters of the hemisphere and cylinder are both 1.2 m determine the capacity of the tank in litres. (1 litre = 1000 cm3).
Find the height and radius of a closed cylinder of volume 125 cm3 which has the least surface area.
To find the capacity of the tank, we first need to calculate the volumes of the cylindrical portion and the two hemispherical sections separately, and then add them up.
Volume of cylindrical portion = πr^2h = π(0.6)^2(5.0) = 1.8π m^3
Volume of each hemispherical section = (2/3)πr^3 = (2/3)π(0.6)^3 = 0.288π m^3
Total volume of tank = 1.8π + 0.288π + 0.288π = 2.376π m^3
Converting to litres: 1 m^3 = 1000 litres
Therefore, capacity = 2.376π * 1000 = 2376π litres ≈ 7486.65 litres
Now, for the closed cylinder with the least surface area,
Let the height of the cylinder be h and the radius be r.
Volume of the cylinder = πr^2h = 125 cm^3
Therefore, h = 125 / (πr^2)
Surface area of the cylinder = 2πrh + 2πr^2
Substitute the value of h in terms of r into the surface area formula:
Surface area = 2πr(125 / (πr^2)) + 2πr^2
Surface area = 250 / r + 2πr^2
To find the minimum surface area we will differentiate it with respect to r and find the critical points:
d(surface area) / dr = -250 / r^2 + 4πr
Setting this equal to 0 and finding the critical points:
-250 / r^2 + 4πr = 0
250 = 4πr^3
r^3 = 250 / (4π)
r ≈ 2.8 cm
Now, plug this value back into the equation for surface area to find the minimum surface area.