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5.09 Practical problems with area, perimeter and circumference

Lesson

Perimeter in use

Once we know how to calculate the perimeter of a shape, or polygon, we can use this to solve real problems that we may come across.

The outside measurement can be used for lots of things, including measuring how much a new fence might cost.

Worked example

Question 1

A farmer is looking to enclose a rectangular field which is $200$200 ft wide and $500$500 ft long. The fencing is $\$1.25$$1.25 per foot. How much will it cost to enclose the field?

Think: To enclose the field, we need to go all around the perimeter of it. Once we know the amount of fencing we will need, we can multiply it by the cost per foot.

Do:

Let's first find the perimeter to find the total amount of fencing required.

$P$P $=$= $2l+2w$2l+2w

Perimeter of a rectangle

  $=$= $2\times500+2\times200$2×500+2×200

Filling in given length and width

  $=$= $1000+400$1000+400

Evaluating the multiplication

  $=$= $1400$1400

Evaluating the addition

The total perimeter of the field is $1400$1400 ft. Now let's find the cost.

If $1$1 ft costs $\$1.25$$1.25, how much will $1400$1400 ft cost?

Cost = $1.25\times1400$1.25×1400

It will cost $\$1750$$1750 to fence the whole field.

Reflect: Could you approach this problem in a different way?

 

Practice questions

Question 2

A rectangular athletics field is $140$140 meters long and $40$40 meters wide. How far, in km, will an athlete run by completing $6$6 laps of along the edge of the field?

Question 3

The length of a rectangle is twice its width and its perimeter is $30$30 cm.

  1. If the width of the rectangle is $x$x cm, write an expression for the perimeter of the rectangle in terms of $x$x.

  2. Use the expression obtained above to find the width ($x$x) of the rectangle.

  3. Find the length of the rectangle.

Question 4

A present is contained in a cube shaped box. Ribbon is wrapped around the present as shown.

  1. If the side length of the box is $10$10 cm, what is the shortest length of the ribbon needed to neatly go around the box without overlap?

  2. The bow requires $8$8 cm of ribbon. How much ribbon is needed altogether to wrap the box in this way?

  3. What is the total length of ribbon needed if we wrap the present with two lengths of ribbons, as shown below? (Assume a single bow is tied.)

 

Remember!

Once we know the perimeter of our shape, or even how to calculate the perimeter, we can calculate other things. No matter the shape, perimeter is always the sum of the outside lengths.

Perimeter of a rectangle = $2l+2w$2l+2w

 

The area of rectangles

We know how to calculate the area of something practical, when it's a rectangle. Maybe our desk, a rug, or the wall. What happens if we want the total area for more than one 'thing'. Or a bigger area?

In this video, we look at how to do those things, as well as checking that our approach is sound. You can skip past that section, if you are feeling confident.

Time to play

Now that we know we can multiply the area by the number of objects we have, you might like to play around with a tile here. Change the dimensions, which changes its area, and then change how many you have altogether. You can then use any unit of measurement. Just remember the area will be squared.

 

Practice questions

Question 5

A kitchen floor is tiled with the tiles shown in the picture. If $30$30 tiles are needed to tile the floor, what is the total area of the floor? Give your answer in square centimeters.

Question 6

Luke made a square mosaic that has side lengths of $3$3 meters. Luke decided to add a border to his mosaic, and now it has side lengths of $3.2$3.2 meters. By how much has the area of the mosaic increased?

 
Remember!

If ever you're not sure about differences in area, or increases in area, go back to basics. Look at the rectangles and work out the separate areas. You can always add and subtract, as you need to.

Area of rectangle = $lw$lw

 

The area of triangles

Now that we know how to calculate the area of triangles, we can use it  to work through actual problems we may have. That can include calculating the base or height, if we know the area of our triangle.

 

Practice questions

Question 7

Sharon has purchased a rectangular piece of fabric measuring $12$12 m in length and $7$7 m in width.

What is the area of the largest triangular piece she can cut out from it?

Question 8

A gutter running along the roof of a house has a cross-section in the shape of a triangle. If the area of the cross-section is $50$50 cm2, and the length of the base of the gutter is $10$10 cm, find the perpendicular height $h$h of the gutter.

Question 9

The faces on a 4-sided die are all triangular. Each face has a base length of $13$13 mm and a perpendicular height of $20$20 mm. What is the area of one face?

Remember!

Sometimes you can use the features of one shape to help solve problems with another.

Area of triangle = $\frac{1}{2}bh$12bh

 

Applications of circles

There are circles all around us from tires, to sports balls, to coins and more. If we are asked about distance around a circle or distance covered, we are likely looking at the circumference of the circle. However, if we are asked about the space inside, we are looking at the area. Remember our formulas:

Remember!

Circumference of a circle

$C=\pi d$C=πd

$C=2\pi r$C=2πr

Area of a circle

$A=\pi r^2$A=πr2

 

Worked example

Question 10

A door which is $32$32 inches wide and $80$80 inches tall drags slightly and is leaving scratch marks all over the floor. What area of floor can the door scratch? Use the value of $\pi$π from your calculator and give your answer to two decimal places.

Think: We have been given the height and width of the door, but the height is actually irrelevant. The width of the door is the radius of the semicircle of scratches the door is leaving on the floor. We need to find the area of a semi-circle which is half of the area of a circle.

Do: 

$A$A $=$= $\frac{1}{2}\pi r^2$12πr2

The formula for area of a semicircle

  $=$= $\frac{1}{2}\pi\times32^2$12π×322

Filling in given information

  $=$= $\frac{1}{2}\pi\times1024$12π×1024

Evaluating the square

  $=$= $512\pi$512π

Evaluating part of the product

  $=$= $1608.4954386$1608.4954386...

From our calculator

  $=$= $1608.50$1608.50

Rounding correctly to two decimal places

 

The area which can be scratched is $1608.50$1608.50 square inches.

Reflect: Is this a small area or a larger area?

 

Practice questions

Question 11

What is the circumference of the Ferris wheel?

Give your answer correct to one decimal place. Use the value of $\pi$π from your calculator in your calculations.

Question 12

A lion has escaped through a hole in the fencing at an animal sanctuary. The wall that it escaped through runs east to west for over $50$50 km in both directions from the hole.

The animal handling team has to set up a perimeter to recapture the lion. They know it could only have traveled at most $21$21 km in the time since it escaped.

  1. What length of fencing will they require to set up a semicircular perimeter? Round your answer to two decimal places.

  2. How much area will they have to search? Round your answer to two decimal places.

 

Outcomes

8.1

Compare and order real numbers

8.10

Solve area and perimeter problems, including practical problems, involving composite plane figures.

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