In construction, calculating surface area is a part of planning, for example, calculating how much materials to buy, as well as determining costs. Similar calculations are required in manufacturing and design based professions.

Let's look at the following worked question which applies the concept of surface area in real life.

Examples

Example 1

Ivan is building a storage chest in the shape of a rectangular prism. The chest will be 55\operatorname{cm} long, 41\operatorname{cm} deep, and 39\operatorname{cm} high. Find the surface area of the chest.

Worked Solution

Create a strategy

A rectangular prism has three pairs of identical faces.

To find the surface area of a rectangular prism, we want to add the areas of all these faces.

Apply the idea

The net of a rectangular prism is made up of three pairs of equal rectangles, with each pair representing one possible pair of dimensions of the prism.

Since this rectangular prism has dimensions of 55 cm , 41 cm and 39 cm, the pairs of dimensions for the faces of the net will be 55\times 39, 41\times 39 and 55\times 41.

\displaystyle \text{Surface area}	\displaystyle =	\displaystyle 2\times 55\times 39 + 2\times 41\times 39 + 2\times 55 \times 41 \text{ cm}^2
	\displaystyle =	\displaystyle 4290+3198+4510	Evaluate the products.
	\displaystyle =	\displaystyle 11\,998\text{ cm}^2	Evaluate the sum.

The surface area of the chest is 11\,998\text{ cm}^2.

Idea summary

We can use the concept of surface area in real-world problems.

The surface area of a prism is the sum of the areas of all its faces.

Problem solving with volume

Whether you want to find out the volume of a swimming pool so you know how much water it can hold or find out the available space that can be occupied by a van or a truck in a gargage, the concept of volume is used often in daily life.

The volume of a three dimensional shape is the amount of space that the shape takes up.

With a given volume and some of the dimensions provided, we can also determine the missing dimensions of a solid.

Examples

Example 2

A nesting box in a shape of a rectangular prism with height of 83 centimeters, width of 54 centimeters and length of d centimeters.

This wild animal house is made out of plywood.

If the nesting box needs to have a volume of 129\,978 \text{ cm}^3 , a height of 83 cm and front width of 54 cm, find the depth of the box.

Worked Solution

Create a strategy

The volume is found by multiplying the height, width and length. Working backwards to obtain the length, we will divide the volume by the product of the other dimensions.

Apply the idea

\displaystyle \text{Depth}	\displaystyle =	\displaystyle 129\,978 \text{ cm}^3\div (83\times 54) \text{ cm}^2
	\displaystyle =	\displaystyle 129\,978\div 4482	Evaluate the product.
	\displaystyle =	\displaystyle 29\text{ cm}	Evaluate the quotient.

The depth of the nesting box is 29 \text{ cm}.

Idea summary

The volume of a three dimensional shape is the amount of space that is taken up by the shape.

Dividing the volume by the product of other dimensions gives the unknown dimension.

Outcomes

6.G.A.2

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas v = lwh and = bhv to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

6.G.A.4

Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

7.08 Problem solving with volume and surface area

Introduction

Ideas

Problem solving with surface area

Examples

Example 1

Problem solving with volume

Examples

Example 2

Outcomes

6.G.A.2

6.G.A.4

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