D AY 154 – A PPLICATION OF VOLUME OF PYRAMID AND CONES
I NTRODUCTION 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.
V OCABULARY 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 𝝆𝒔 𝟑 × 𝒊. Pyramid has a polygon base. In this presentation we will discuss some of the applications of volume of cones and pyramids.
Making containers. 1. 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 of popcorns missing in the container.
Solution 2 𝑗𝑜 3 𝑗𝑜 6 in 3 𝑗𝑜 𝑠 6 3 From similarity 2 = 𝑠 2 𝑠 = 6 × 3 = 1 𝑗𝑜 3 × 3.142 × 2 2 × 6 = 25.136 𝑗𝑜 3 1 Volume of the container = 1 Volume of the popcorns in the container = 3 × 3.142 × 1 2 × 3 = 3.142 𝑗𝑜 3 Volume of missing popcorns = 25.136 𝑗𝑜 3 − 3.142 𝑗𝑜 3 = 21.99 𝑗𝑜 3
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 of 0.3 𝑗𝑜 using gemstone. Find the volume of gemstone that can make 1000 jewels.
Solution 1 Volume of one jewel = 3 (0.25 × 0.25) × 0.3 0.00625 𝑗𝑜 3 Volume of 1000 jewels = 0.00625 1000 = 6.25 𝑗𝑜 3
3. Estimating the volume of conical piles It might not be possible to get the exact volume of piles of sand or soil. However, we can use the idea of volume of 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. 1 3 3.142 × 2.5 2 × 4 ≈ 26.18 𝑔𝑢 3 volume ≈
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.
A NSWERS TO HOMEWORK 6.335 𝑗𝑜 3
THE END
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