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CAT 2008 — QA Question 21

Basics of Mensuration/PrismEasy
Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the cone with one of the flat surface resting on the base of the cone. Find the largest possible total surface area (in sq. cm) of the cylinder.

Answer & solution

  • 100π3

  • B

    80π3

  • C

    120π7

  • D

    130π9

  • E

    110π7

Solution

As shown in the figure, ABC is the cross section of a cone of height 10 cm and radius of base 4 cm.

PQRS is the cross section of a cylinder which needs to be fitted inside the cone such that one of the flat faces of the cylinder (represented by RS) coincides with the base of the cone (represented by BC).

∵ ∆ABD and ∆PBR are similar triangles.

∴ 104=h4-r

∴ h = 10 - 2.5 × r

Total surface area of the cylinder = S = 2πr × (h + r) = 2πr × (10 - 2.5 × r + r) cm2

⇒ S = πr(20 - 3r) cm2   ...(1)

Now consider two terms 3r and (20 - 3r)

Now, 3r+(20-3r)2 ≥ 3r×(20-3r) [AM ≥ GM]

⇒ 10 ≥ 3r×(20-3r)

⇒ 3r(20 - 3r) ≤ 100

∴ r(20 - 3r) ≤ 100/3

From (1), we have

⇒ S = πr(20 - 3r) ≤ π × 100/3

Hence, option (a).

CAT 2008 QA Q21: Consider a right circular cone of base radius 4 cm and height 10 cm. A cylinder is to be placed inside the con — Solution | TheCATExam