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A hemispherical bowl is made of brass, 0.25 cm thickness. The inner radius of the bowl is 5 cm. Find the ratio of the outer surface area to the inner surface area.

Answer
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Hint: First of all, find the outer radius of the bowl by adding the inner radius and thickness of the bowl. Now, find the surface area of the bowl by using 2πr2 and substitute ‘r’ as the outer radius and inner radius to find the outer area and inner area and take the ratio of it.

Complete step-by-step solution -
Here, we are given a hemispherical bowl made of brass having 0.25 cm thickness. Its inner radius is 5 cm. We have to find the ratio of its outer surface area to the inner surface area. Let us, first of all, draw the side and front view of the hemispherical bowl made of brass to visualize the question.

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Here, we are given that the inner radius of the hemispherical bowl, (r1)=5 cm
The thickness of hemispherical bowl = 0.25 cm
So, we get the outer radius of the hemispherical bowl, (r2)= Inner radius of the hemispherical bowl + Thickness of the hemispherical bowl.
= 5 cm + 0.25 cm
r2=5.25 cm
Now, we know that the surface area of the hemispherical bowl =2πr2 where ‘r’ is the radius of the bowl. So, we get the inner surface area of the hemispherical bowl,
Hi=2π(r1)2
By substituting the value of r1=5 cm in the above equation, we get,
Hi=2π(5)2 cm2=2π×25 cm2Hi=50π cm2......(i)
Also, we get the outer surface area of the hemispherical bowl,
Ho=2π(r2)2
By substituting the value of r2=5.25 cm in the above equation, we get,
Ho=2π(5.25)2 cm2=2π×(27.5625) cm2Ho=55.125π cm2......(ii)
So, we get the ratio of the outer surface area of the bowl to the inner surface area of the bowl as,
Outer Surface area of the hemispherical bowlInner surface area of the hemispherical bowl=HoHi
By substituting the value of Ho and Hi from equation (ii) and (i) respectively, we get,
Outer Surface area of the hemispherical bowlInner surface area of the hemispherical bowl=55.125π cm250π cm2
Hence, we get the ratio of the outer surface area of the bowl to the inner surface area of the bowl as 1.1025.

Note: By looking at the hemisphere and surface area, some students make this mistake of writing the surface area as 3πr2 in this wrong question. Students must note that the bowl is an open hemispherical container. So, we use the formula for the surface area as 2πr2 where ‘r’ is the radius of the hemisphere. Also, note that the ratio of the same quantity is a dimensionless quantity. So here, we get the final answer without any dimensions because the quantities at numerator and denominator are the same and that is the surface area.