A gold ingot in the shape of a cylinder is melted and the resulting molten metal molded into a few identical conical ingots. If the height of each cone is half the height of the original cylinder and the area of the circular base of each cone is one ﬁfth that of the circular base of the cylinder, then how many conical ingots can be made?a) 60b) 10c) 30d) 20e) 40Correct answer is option 'C'. Can you explain this answer?

Question

Accepted Answer

Let the radius of the cylinder be and height be (as shown).

Let the dimensions of the cone be and (as shown).

It is given that h_c = h/2

Area of base of cone A_c = π × r²_c

Area of base of cylinder A = π × r²

Given, A_c = 1/5 × A

⇒π x r²_c = 1 / 5 x π x r²

⇒ r² / r²_c = 5

Now we know that the volume of cylinder = total volume of cones

Let the number of cones be n.

So, volume of cylinder = n x volume of each cone

⇒ V = n × V_c

⇒ π × r² × h = n × 1/3 × π × r²_c × hc

⇒ r² × h = n × 1/3 × r²_c× hc

⇒ r² / r²_c x h / h_c x 3 = n

⇒ 5 × 2 × 3 = n

⇒ 30 = n

Thus, 30 conical ingots can be made.

Answer

To solve this problem, we need to consider the relationship between the volume and surface area of a cylinder and a cone.

Let's assume the original cylinder has a height of h and a radius of r. The volume of the cylinder is given by Vcylinder = πr^2h, and the surface area of the cylinder is given by Acylinder = 2πrh.

According to the problem, the height of each cone is half the height of the cylinder, so the height of each cone is h/2. The radius of each cone can be determined by using the fact that the area of the circular base of each cone is one fifth that of the circular base of the cylinder. Therefore, the radius of each cone is √(1/5) times the radius of the cylinder, which is (√1/5)r.

Let's assume we can make n identical conical ingots from the molten metal. The volume of each cone is given by Vcone = (1/3)π(√1/5r)^2(h/2) = (1/15)πr^2(h/2), and the surface area of each cone is given by Acone = π(√1/5r)(√1/5r + √(h/2)^2) = (2/5)πr(√(1/5) + √(h/2)^2).

Since the molten metal is molded into a few identical conical ingots, the total volume of the ingots should be equal to the volume of the original cylinder. Therefore, we can write the equation:

n(Vcone) = Vcylinder
n((1/15)πr^2(h/2)) = πr^2h

Simplifying the equation, we get:
n(h/2) = 15h
n = 30

Therefore, we can make 30 conical ingots from the molten metal. Hence, the correct answer is option C) 30.

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