Question

The outer diameter of a gas cylinder is 20 cm. Find the cost of the painting of its outer curved surface at ₹ 4 per square centimetre if the height of the cylinder is 70 cm.

Answer 1

Hint: Here, we will first find the curved surface area of the gas cylinder. Then, we will use multiplication to find the cost of painting the curved surface area. The cost of painting the outer curved surface is the product of the area of the outer curved surface in \[{\rm{c}}{{\rm{m}}^2}\] and the cost per square centimetre.

Formula Used: We will use the following formulas to solve the question:
1.The radius of the base of a cylinder, \[r = \dfrac{d}{2}\], where \[d\] is the diameter of the base of the cylinder.
2.The curved surface area of a cylinder, \[C.S.A. = 2\pi rh\], where \[r\] is the radius of the base of the cylinder and \[h\] is the height of the cylinder.

Complete step-by-step answer:
We will use the formula for curved surface area of a cylinder, and then find the cost of painting it.
First, we will draw the diagram of a cylinder.

We know that the radius of the base of a cylinder is half of its diameter, that is \[r = \dfrac{d}{2}\].
Therefore, we get the outer radius of the gas cylinder as \[r = \dfrac{{20}}{2} = 10\] cm.
Now, we will find the area of the outer curved surface using the height and outer radius of the gas cylinder.
The curved surface area of a cylinder is given by the formula \[C.S.A. = 2\pi rh\], where \[r\] is the radius of the base of the cylinder and \[h\] is the height of the cylinder.
Substituting \[r = 10\] cm and \[h = 70\]cm in the formula, we get
\[ \Rightarrow C.S.A. = 2\pi \left( {10} \right)\left( {70} \right)\]
Substituting \[\pi = \dfrac{{22}}{7}\] in the expression, we get
\[ \Rightarrow C.S.A. = 2 \times \dfrac{{22}}{7} \times 10 \times 70\]
Simplifying the expression, we get
\[ \Rightarrow C.S.A. = 2 \times 22 \times 10 \times 10\]
Multiplying the terms in the expression, we get
\[ \Rightarrow C.S.A. = 4400{\rm{ c}}{{\rm{m}}^2}\]
\[\therefore\] The area of the outer curved surface of the gas cylinder is 4400 \[{\rm{c}}{{\rm{m}}^2}\].
Now, we will find the cost of painting the outer curved surface of the gas cylinder.
We will multiply the area of the outer curved surface in \[{\rm{c}}{{\rm{m}}^2}\] to the cost per square centimetre to find the cost of painting the outer curved surface.
The cost of painting is ₹4 per square centimetre.
Therefore, we get
Cost of painting the outer curved surface \[ = 4400 \times 4\]
Multiplying the terms, we get
Cost of painting the outer curved surface \[ = 17,600\] rupees
\[\therefore\] The cost of the painting of its outer curved surface at ₹ 4 per square centimetre is ₹ 17,600.

Note: For solving the question we need to remember the formula for curved surface area of a cylinder. A common mistake we can make is to use the formula for total surface area \[2\pi r\left( {r + h} \right)\], to calculate the outer curved surface area of the gas cylinder. This will give us incorrect answers.