CAT Quantitative Aptitude Questions | CAT Geometry - Mensuration Problems

CAT Questions | CAT Mensuration Questions | Spheres

This question is from CAT Mensuration in CAT Geometry. In this Question, A sphere is cut into two pieces and its cumulative surface area increases. We need to find the height. Mensuration Questions in the CAT exam can be solved with practical approach and thought process rather than just knowing theorems from CAT Geometry. Even if one is not completely prepared for CAT Geometry exam, he or she should be able to do well in Mensuration problems tested by the CAT Exam. Knowing Mensuration formulas is important. Make sure you master Mensuration questions for CAT exam. Cutting a sphere into two equal parts increases its surface area by 50%, what will be the increase if we cut it in some other fashion?

Question 2: A sphere of radius r is cut by a plane at a distance of h from its center, thereby breaking this sphere into two different pieces. The cumulative surface area of these two pieces is 25% more than that of the sphere. Find h.

  1. \\frac{r}{√2}\\)
  2. \\frac{r}{√3}\\)
  3. \\frac{r}{√5}\\)
  4. \\frac{r}{√6}\\)

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Explanatory Answer

Method of solving this CAT Question from Geometry - Mensuration: Visualising the question and understanding of the surface area formula is the key to answer.

Area = 4πr2
CAT : Geometry: Mensuration; Sphere
Cumulative area of the two pieces = 25%
more than that of sphere.
Area of 2 pieces = 1.25 × 4π2 = 5πr2
Extra area = πr2
Extra area = area of two new circles that are now created circles.
Area each new circle = \\frac{πr^2}{2}\\)

Let radius of new circle be r1.
Now, πr12 = \\frac{πr^2}{2}\\)
r1 = \\frac{r}{√2}\\) CAT : Geometry: Mensuration; Sphere
Now, r1 , h and r form a right angled triangle.
h2 + r12 = r2
h2 + (\\frac{r}{√2}\\))2 = r2
h = \\frac{r}{√2}\\)
Choice (A) is therefore, the correct answer.

The question is " The cumulative surface area of these two pieces is 25% more than that of the sphere. Find h. "

Hence, the answer is \\frac{r}{√2}\\)

Choice A is the correct answer.


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