Question : The radius of the base of a hollow cone is 8 cm, and its height is 15 cm. A sphere of the largest radius is put inside the cone. What is the ratio of the radius of the base of a cone to the radius of a sphere?
Option 1: 5 : 3
Option 2: 4 : 1
Option 3: 2 : 1
Option 4: 7 : 3
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Correct Answer: 5 : 3
Solution : Given: The radius of the base of a hollow cone = 8 cm Height = 15 cm Now consider the problem in two dimensions. Area of the triangle = $\frac{1}{2}\times$ base $\times$ height. The inradius of the circle = $\frac{A}{s}$, where $A$ is the area of the triangle and $s$ is the semi–perimeter of the triangle. Using Pythagoras theorem, $(CB)^2=(BD)^2+(CD)^2$ $⇒(CB)^2=(8)^2+(15)^2$ $⇒(CB)^2=64+225$ $⇒(CB)^2=289$ $⇒(CB)=17$ cm Now, inradius = $\frac{A}{s}$ $A =\frac{1}{2}\times 15\times 16$ $= 120$ cm$^2$ $⇒s=\frac{17+16+17}{2}$ $⇒s= 25$ cm Inradius = $\frac{120}{25}=\frac{24}{5}$ The ratio of the radius of the base of a cone to the radius of a sphere is $8:\frac{24}{5}=5:3$ Hence, the correct answer is 5 : 3.
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Question : A hemisphere of lead of radius 4 cm is cast into a right circular cone of height 72 cm. What is the radius of the base of the cone?
Option 1: 1.63 cm
Option 2: 1.35 cm
Option 3: 1.33 cm
Option 4: 1.45 cm
Question : A solid metallic sphere of radius 12 cm is melted and recast into a cone having a diameter of the base of 12 cm. What is the height of the cone?
Option 1: 258 cm
Option 2: 192 cm
Option 3: 166 cm
Option 4: 224 cm
Question : A solid metallic sphere of radius 13 cm is melted and recast into a cone having a diameter of the base as 13 cm. What is the height of the cone?
Option 1: 246 cm
Option 2: 152 cm
Option 3: 174 cm
Option 4: 208 cm
Question : The curved surface area of a right circular cone of a base radius of 21 cm is 594 sq. cm. What is the slant height of the cone?
Option 1: 15 cm
Option 2: 11 cm
Option 3: 9 cm
Option 4: 6 cm
Question : A sphere has the same curved surface area as a cone, with a vertical height of 40 cm and a radius of 30 cm. The radius of the sphere is:
Option 1: $5 \sqrt 5 \text{ cm}$
Option 2: $5\sqrt 3\text{ cm}$
Option 3: $5\sqrt{15}\text{ cm}$
Option 4: $5\sqrt{10}\text{ cm}$
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