3 Area of the base vertical height. Since the cone has a circular - - PowerPoint PPT Presentation

DAY 154 – APPLICATION OF

VOLUME OF PYRAMID AND CONES

INTRODUCTION

We always come across cone-shaped objects in real life such as Christmas trees, sand piles, funnels and lamp shades. On the other hand, pyramid-shaped objects abound in real life. Some roofs, monuments, and perfume bottles are pyramid shaped. The volume of the pyramid and conically shaped objects is always depended on the area of their base and their vertical height. Thus, it is important to understand how the side if these features can be determined. In this lesson, we will discuss the applications of the volume of pyramids and cones.

VOCABULARY

Pyramid This is a three dimensional solid with a triangular, rectangular, triangular or any other polygonal base and triangular faces which meet at a single apex. Cone This a three dimensional solid with a circular base and a curved surface that results in a single apex.

Cones and pyramids do not have uniform cross

• sections. Their volumes are given by the formula

1 3 Area of the base × vertical height.

Since the cone has a circular base, a cone with a base radius 𝑠 and a vertical height ℎ has a volume

𝟐 𝟒 𝝆𝒔𝟑 × 𝒊.

Pyramid has a polygon base. In this presentation we will discuss some of the applications of volume of cones and pyramids.

1.

Making containers. In most cases, ice creams and popcorns are sold in conical containers. The volume of ice cream contained in the container depends on the height and the base area of the container. Perfumes are also sold in pyramid-shaped containers to be attractive to the customer. Example 1 A trader sells popcorns in conical containers with a height of 6 𝑗𝑜 and a radius 2 𝑗𝑜. A container contains popcorns up to a height of 3 𝑗𝑜. Calculate the volume

• f popcorns missing in the container.

Solution From similarity

6 2 = 3 𝑠

𝑠 =

2 6 × 3 = 1 𝑗𝑜

Volume of the container =

1 3 × 3.142 × 22 × 6 = 25.136 𝑗𝑜3

Volume of the popcorns in the container =

1 3 × 3.142 ×

12 × 3 = 3.142 𝑗𝑜3 Volume of missing popcorns = 25.136 𝑗𝑜3 − 3.142 𝑗𝑜3 = 21.99 𝑗𝑜3

2 𝑗𝑜

3 𝑗𝑜 𝑠

3 𝑗𝑜

6 in

• 2. Making jewels and ornaments

Most ornaments and jewels such as earrings are conical and pyramid shaped. The volume of each jewels determines the amount of raw material to be used in making them. Example 2 A company makes pyramid shaped jewels with a square base measuring 0.25 𝑗𝑜 by 0.25 𝑗𝑜 and a height

• f 0.3 𝑗𝑜 using gemstone. Find the volume of

gemstone that can make 1000 jewels.

Solution Volume of one jewel =

1 3 (0.25 × 0.25) × 0.3

0.00625 𝑗𝑜3 Volume of 1000 jewels = 0.00625 1000 = 6.25 𝑗𝑜3

• 3. Estimating the volume of conical piles

It might not be possible to get the exact volume of piles

• f sand or soil. However, we can use the idea of volume
• f a cone to approximate their volume. If a sand pile

4 𝑔𝑢 heigh has a diameter of 5 𝑔𝑢, is volume can be calculated as follows. volume ≈

1 3 3.142 × 2.52 × 4 ≈ 26.18 𝑔𝑢3

• 4. Building monuments

Most monuments are build in form of pyramids. The volume of concrete using in building the monument depends the base area of the monument and the vertical height. Washington pyramid and Egyptian pyramids are practical examples of pyramid monuments.

HOMEWORK A company that manufactures perfumes packs them in triangular based pyramids with sides 3.2 𝑗𝑜, 3.2 𝑗𝑜 and 2.6 𝑗𝑜. The height of perfume bottle is 5 𝑗𝑜. Calculate the volume of the perfume bottle.

ANSWERS TO HOMEWORK

6.335 𝑗𝑜3

THE END