The word "geometry" comes from the Greek words "geo" meaning earth and "metron" meaning measurement!
1
A 14 m deep well with inner diameter 10 m is dug and the earth taken out is evenly spread all around the well to form an embankment of width 5 m. Find the height of the embankment.
📏 Step 1: Find the volume of earth dug out
The well is cylindrical in shape. Volume of cylinder = πr²h
A cylindrical glass with diameter 20 cm has water to a height of 9 cm. A small cylindrical metal of radius 5 cm and height 4 cm is immersed completely. Calculate the raise of the water in the glass?
A conical container is fully filled with petrol. The radius is 10m and the height is 15 m. If the container can release the petrol through its bottom at the rate of 25 cu. meter per minute, in how many minutes the container will be emptied. Round off your answer to the nearest minute.
📏 Step 1: Find volume of conical container
Volume of cone = (1/3)πr²h
r = 10 m, h = 15 m
Volume = (1/3) × π × (10)² × 15
= (1/3) × π × 100 × 15 = 500π m³
≈ 500 × 3.1416 ≈ 1570.8 m³
The container will be emptied in approximately 63 minutes.
5
A right angled triangle whose sides are 6 cm, 8 cm and 10 cm is revolved about the sides containing the right angle in two ways. Find the difference in volumes of the two solids so formed.
🔄 Step 1: Understand the two rotations
When rotated about different sides, we get different cones:
Case 1: Rotate about 6 cm side
Case 2: Rotate about 8 cm side
The ratio of volumes is √3 : 4 (after simplifying).
9
The outer and the inner surface areas of a spherical copper shell are 576π cm² and 324π cm² respectively. Find the volume of the material required to make the shell.
📏 Step 1: Find outer and inner radii
Surface area = 4πr²
Outer: 576π = 4πR² ⇒ R² = 144 ⇒ R = 12 cm
Inner: 324π = 4πr² ⇒ r² = 81 ⇒ r = 9 cm
A container open at the top is in the form of a frustum of a cone of height 16 cm with radii of its lower and upper ends are 8 cm and 20 cm respectively. Find the cost of milk which can completely fill a container at the rate of ₹40 per litre.