Question 12 of 13advanced🔧 ApplyNumerical5 marks

A solid consisting of a right circular cone of height 120 cm and radius 60 cm standing on a hemisphere of radius 60 cm is placed upright in a right circular cylinder full of water such that it touches the bottom. Find the volume of water left in the cylinder, if the radius of the cylinder is 60 cm and its height is 180 cm.

Correct Answer

The volume of water left in the cylinder is 1,131,42847 cm31,131,428 \frac{4}{7} \text{ cm}^3 (or approximately 1.131×106 cm31.131 \times 10^6 \text{ cm}^3).

Exercise: EXERCISE 12.2 | Q: 7 | (Page Page 10)
For More Understanding

Explanation

The problem is solved by calculating the volume of the cylinder and the volume of the solid (comprising the cone and hemisphere), then subtracting the volume of the solid from the volume of the cylinder. This follows the logic presented in Example 7 of the provided text, where the volume of the toy is subtracted from the volume of the circumscribing cylinder. The formula for the volume of a cylinder is πr2h\pi r^2 h, for a cone is 13πr2h\frac{1}{3} \pi r^2 h, and for a hemisphere is 23πr3\frac{2}{3} \pi r^3. The value of π\pi is taken as 227\frac{22}{7} as instructed in the exercise header.

Solution Steps

  1. Step 1: Identify the dimensions from the question. Radius of the cone, hemisphere, and cylinder (rr) = 60 cm. Height of the cone (hh) = 120 cm. Height of the cylinder (HH) = 180 cm.

  2. Step 2: Calculate the Volume of the Solid (Cone + Hemisphere). Volume of cone = 13πr2h=13π(60)2(120)\frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (60)^2 (120). Volume of hemisphere = 23πr3=23π(60)3\frac{2}{3} \pi r^3 = \frac{2}{3} \pi (60)^3. Total Volume of solid = π(60)2[1203+2×603]=π(3600)[40+40]=π(3600)(80)=288,000π cm3\pi (60)^2 [\frac{120}{3} + \frac{2 \times 60}{3}] = \pi (3600) [40 + 40] = \pi (3600)(80) = 288,000 \pi \text{ cm}^3.

  3. Step 3: Calculate the Volume of the Cylinder. Volume = πr2H=π(60)2(180)=π(3600)(180)=648,000π cm3\pi r^2 H = \pi (60)^2 (180) = \pi (3600)(180) = 648,000 \pi \text{ cm}^3.

  4. Step 4: Calculate the Volume of water left. Volume of water left = Volume of cylinder - Volume of solid. =648,000π−288,000π=360,000π cm3= 648,000 \pi - 288,000 \pi = 360,000 \pi \text{ cm}^3.

  5. Step 5: Substitute π=227\pi = \frac{22}{7} (as per Exercise 12.2 instructions). Volume = 360,000×227=7,920,0007=1,131,42847 cm3360,000 \times \frac{22}{7} = \frac{7,920,000}{7} = 1,131,428 \frac{4}{7} \text{ cm}^3.

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