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CAT 2017 Slot 2 — QA Question 16

Basics of Mensuration/PrismEasy

The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 cm and 20 cm while the other two sides are equal in length. The perpendicular distance between the parallel sides of the trapezium is 12 cm. If the height of the pillar is 20 cm, then the total area, in sq cm, of all six surfaces of the pillar is

Answer & solution

Correct answer: 1480

  • A

    1300

  • B

    1340

  • 1480

  • D

    1520

Solution

Easy

The pillar is a prism with a trapezoidal cross-section. Find the slant sides of the trapezium, then add the four rectangular lateral faces (each side ×\times height 2020) to the two trapezoidal ends.

AB = 10 DC = 20 12 x=5 13
1

Horizontal overhang. The longer base exceeds the shorter by 20−10=1020-10=10, split equally on both sides since the slant sides are equal.

2x+10=20(base = overhang + top + overhang)⇒ x=5\begin{aligned} &2x + 10 = 20 \quad\text{(base = overhang + top + overhang)}\\ &\Rightarrow\ x = 5 \end{aligned}
2

Slant side. Each slant is the hypotenuse of a right triangle with legs 1212 (height) and 55.

slant=122+52=169=13\begin{aligned} &\text{slant} = \sqrt{12^2 + 5^2} = \sqrt{169} = 13 \end{aligned}
3

Lateral surface. Four rectangles, each (a side of the trapezium) ×\times height 2020. The four sides are 10, 20, 13, 1310,\ 20,\ 13,\ 13.

lateral=(10+20+13+13)×20⇒ =56×20=1120\begin{aligned} &\text{lateral} = (10+20+13+13)\times 20\\ &\Rightarrow\ = 56\times 20 = 1120 \end{aligned}
4

Two trapezoidal ends. Trapezium area is 12(sum of parallel sides)×height\tfrac12(\text{sum of parallel sides})\times\text{height}, taken twice.

2×12(10+20)×12=30×12=360\begin{aligned} &2\times \tfrac12(10+20)\times 12 = 30\times 12 = 360 \end{aligned}
5

Total surface area. Add lateral and the two ends.

Total=1120+360=1480\begin{aligned} &\text{Total} = 1120 + 360 = 1480 \end{aligned}
Total surface area=1480 sq cm\text{Total surface area} = 1480 \text{ sq cm}

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CAT 2017 Slot 2 QA Q16: The base of a vertical pillar with uniform cross section is a trapezium whose parallel sides are of lengths 10 — Solution | TheCATExam