Question

It costs ₹2200 to paint the inner curved surface of a cylindrical vessel 10 m deep if the cost of painting is at the rate of ₹20 per \[{m^2}\] . Find
1) The radius of the base
2) Inner CSA of the vessel
3) Capacity of the vessel

Answer 1

Hint:
Here, we will use the total cost of painting and rate of painting to find the inner curved surface area of a cylindrical vessel. Then by using the inner cylindrical vessel formula we will find the radius of the base of a cylindrical vessel. Using the radius and height and the formula of the volume of a cylindrical vessel, we will find the capacity of the vessel.

Formula Used:
We will use the following formulas:
1) Total Cost of Painting \[ = \] Inner Curved Surface Area \[ \times \] Rate of Painting
2) Inner Curved Surface Area of a cylindrical vessel is given by the formula \[CSA = 2\pi rh\]
3) Volume of a cylinder is given by the formula \[V = \pi {r^2}h\] where \[r\] is the radius of the cylindrical vessel and \[h\] is the height of the cylindrical vessel.

Complete step by step solution:
We are given that the cost to paint the inner curved surface of a cylindrical vessel of 10 m deep is ₹2200 and the cost of painting is at the rate of ₹20 per \[{m^2}\].

We know that the total cost of painting is found by the inner curved surface of a cylindrical vessel multiplied by the rate of painting.
Total Cost of Painting \[ = \] Inner Curved Surface Area \[ \times \] Rate of Painting
By substituting 2200 for the total cost of painting and 20 for rate of painting in the above equation, we get
\[ \Rightarrow 2200 = \] Inner Curved Surface Area \[ \times 20\]
Dividing both sides by 20, we get
\[ \Rightarrow \] Inner Curved Surface Area \[ = \dfrac{{2200}}{{20}}\]
\[ \Rightarrow \] Inner Curved Surface Area \[ = 110{m^2}\]
Now, we will find the radius of the vessel by using the inner curved surface area of the Cylindrical Vessel formula.
Let \[r\] be the radius of the cylinder and \[h\] is the height of the cylinder where \[h = 10m\] .
Substituting the formula \[CSA = 2\pi rh\] in the above equation, we get
\[ \Rightarrow 2\pi rh = 110\]
By substituting the radius and the height of the cylinder in the above equation, we get
\[ \Rightarrow 2 \times \dfrac{{22}}{7} \times r \times 10 = 110\]
By rewriting the equation, we get
\[ \Rightarrow r = 110 \times \dfrac{7}{{22}} \times \dfrac{1}{{10}} \times \dfrac{1}{2}\]
Multiplying the terms, we get
\[ \Rightarrow r = \dfrac{7}{4}\]
Dividing 7 by 4, we get
\[ \Rightarrow r = 1.75m\]
Now, we will find the capacity of the inner cylindrical vessel using the volume of the cylinder.
By substituting the radius and the height of the cylinder in the formula \[V = \pi {r^2}h\], we get
\[V = \dfrac{{22}}{7} \times \dfrac{7}{4} \times \dfrac{7}{4} \times {\rm{10}}\]
Multiplying the terms, we get
\[ \Rightarrow V = 96.25{\rm{ }}{m^3}\]

Therefore, the radius of the base of a cylindrical vessel is \[1.75m\], the inner curved surface area of the cylindrical vessel is \[110{m^2}\] and the volume of a cylindrical vessel is \[96.25{m^3}\]

Note:
We know that Volume is defined as the quantity of the substance that can be contained in an enclosed curve or a container. We should know that the capacity of a cylindrical vessel is equal to the volume of a cylindrical vessel. We should convert the given dimensions of liters to the cubic centimeters. We should know the conversion of units. Depth is also known as the height of the cylinder.