Question

A storage tank consists of a circular cylinder with hemisphere adjoined on either end. If the external diameter of the cylinder be 1.4m and its length be 8m, find the cost of painting it on the outside at the rate of Rs.10 per \[{m^2}\]

Answer 1

Hint: We draw a rough diagram of the situation and calculate the radius of the cylinder and the hemisphere using the given diameter of the cylinder. Calculate the surface area of the storage tank by adding surface area of cylinder and curved surface area of a hemisphere. Substitute the values of radius and height and calculate the total surface area. Using a unitary method calculate the cost painting outside of the storage tank.
* Curved surface area of a cylinder with radius of base as ‘r’ and height as ‘h’ is given by \[2\pi rh\] and curved surface area of hemisphere with radius ‘r’ is given by \[2\pi {r^2}\].
* Unitary method helps us to calculate the value of multiple units by multiplying the value of a single unit to the number of units given.

Step-By-Step answer:
We are given a storage tank consisting of a circular cylinder and hemisphere adjoined at one end. We draw a diagram depicting the situation.
We are given the diameter of the of the cylinder is 1.4m and its length is 8m
Since radius is always half the diameter, we calculate the radius of the base of the cylinder.
Radius of circular base of cylinder, \[r = \dfrac{{1.4}}{2}\]
i.e. \[r = 0.7\]m
Height of the cylinder, \[h = 8\]m
We can calculate curved surface area of the cylinder using the formula \[2\pi rh\]
\[ \Rightarrow \]Curved surface area of cylinder \[ = 2 \times \dfrac{{22}}{7} \times 0.7 \times 8\]
Write values from decimal into fraction form
\[ \Rightarrow \]Curved surface area of cylinder \[ = 2 \times \dfrac{{22}}{7} \times \dfrac{7}{{10}} \times 8\]
Cancel same factors from numerator and denominator
\[ \Rightarrow \]Curved surface area of cylinder \[ = \dfrac{{352}}{{10}}\]
Convert fraction form into decimal form
\[ \Rightarrow \]Curved surface area of cylinder \[ = 35.2\]\[{m^2}\] … (1)
Now we know the hemisphere is made with the same circular base as the base of the cylinder, so the radius of the hemisphere will be the same as the radius of the cylinder.
We can calculate curved surface area of the cylinder using the formula \[2\pi rh\]
i.e. \[r = 0.7\]m
\[ \Rightarrow \]Curved surface area of hemisphere \[ = 2 \times \dfrac{{22}}{7} \times 0.7 \times 0.7\]
Write values from decimal into fraction form
\[ \Rightarrow \]Curved surface area of hemisphere \[ = 2 \times \dfrac{{22}}{7} \times \dfrac{7}{{10}} \times \dfrac{7}{{10}}\]
Cancel same factors from numerator and denominator
\[ \Rightarrow \]Curved surface area of hemisphere \[ = \dfrac{{308}}{{100}}\]
Convert fraction form into decimal form
\[ \Rightarrow \]Curved surface area of hemisphere \[ = 3.08\]\[{m^2}\] … (2)
So, the total curved surface area is sum of equation (1) and (2)
\[ \Rightarrow \]Curved surface area of storage tank\[ = \left( {35.2 + 3.08} \right)\]\[{m^2}\]
\[ \Rightarrow \]Curved surface area of storage tank\[ = 38.28\]\[{m^2}\]
Now we have to paint a 38.28 square meter area at a cost of Rs.10 per meter square.
Use a unitary method to calculate the cost of painting the outside of the tank by multiplying the curved surface area tank by cost of painting per unit square meter.
Cost of painting 1 sq.m \[ = Rs10\]
\[ \Rightarrow \]Cost of painting 38.28 sq.m \[ = Rs(38.28 \times 10)\]
\[ \Rightarrow \]Cost of painting 38.28 sq.m \[ = Rs382.8\]

\[\therefore \]Cost of painting outside of the storage tank is Rs.382.8.

Note: Students many times make mistakes while calculating the cost using unitary method as they leave the cost value in fractional terms. Keep in mind the value of cost should always be simplified and should never be in fraction form. Also, many students calculate the curved surface area of the hemisphere and write half of it as the hemisphere is half of the sphere. Keep in the mid hemisphere means half part of the sphere so we don’t need to take half again.