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5.07 The trapezoidal rule

Worksheet
Estimate areas using the trapezoidal rule
1

The following pieces of land have straight boundaries on the east, west and south borders and follows a creek at the north. For each of the following:

i

Approximate Area 1 using one application of the trapezoidal rule.

ii

Approximate Area 2 using one application of the trapezoidal rule.

iii

Hence, find the approximate area of the piece of land.

iv

Is the actual area of the land greater than or less than this calculated area?

a
b
2

Use one application of the trapezoidal rule to approximate the following areas. Give your answer in square centimetres.

a
b
3

Consider the following shape:

a

Find the area of the shape using one application of the trapezoidal rule. Give your answer in square metres.

b

Find the area of the shape using two applications of the trapezoidal rule. Give your answer in square metres.

c

Which approximation is more accurate?

4

Use the trapezoidal rule to approximate the area of the garden in square metres.

5

The diagram shows the cross-section of a river at the exact point where it feeds into a dam. The depths of the river are marked at 2 \text{ m} intervals.

Using three applications of the trapezoidal rule, approximate the area of the cross-section of the river to one decimal place.

6

Use four applications of the trapezoidal rule to approximate the area of the cross-section of the following river:

7

The following shape has measurements given in metres:

Use the trapezoidal rule to find the area in hectares \left( 1 \text{ ha} = 10\,000 \text{ m}^{2} \right). Round your answer to two decimal places.

8

A river has its depths marked out at equal intervals of 9 \text{ m}. The depths are given by the following measurements in metres:

\ 0,\, 12,\, 14,\, 17,\, 5,\, 0

Find the approximate area of the cross section of the river.

9

A garden is 49 \text{ m} long. At 7 \text{ m} intervals, the width of the garden was given by the following measurements in metres:

\ 0,\, 2.9,\, 5.2,\, 6.6,\, 5.6,\, 4.3,\, 3,\, 2.5

Using the trapezoidal rule, approximate the area of the garden. Give your answer in square metres, correct to two decimal places.

10

The elevation values of a mountain are recorded at equal intervals of 250 \text{ m}. The heights are shown in the diagram:

Find the approximate area of the cross section of the mountain.

Estimate volumes using the trapezoidal rule
11

The following piece of land has straight boundaries on the east, west and south borders and follows a creek at the north.

The land has been divided into two sections so we can use the trapezoidal rule to approximate the area.

a

Find the approximate area of the piece of land by using two applications of the trapezoidal rule. Give your answer in square metres.

b

During a heavy storm, 35.2 \text{ mm} of rain fell. Find the volume of water that lands on this property in cubic metres.Give your answer correct to the nearest cubic metre.

12

The following piece of land has straight boundaries on the east, west and south borders and follows a creek at the north.

The land has been divided into two sections so we can use the trapezoidal rule to approximate the area.

a

Find the approximate area of the piece of land by using two applications of the trapezoidal rule. Give your answer in square metres.

b

During a heavy storm, 15.5 \text{ mm} of rain fell. Find the volume of water that lands on this property. Give your answer correct to the nearest cubic metre.

13

A surveyor provided the following diagram with measurements for a property she was mapping out:

a

Find the approximate total area of the property by using three applications of the trapezoidal rule. Give your answer in square metres.

b

The average weekly rainfall is 34 \text{ mm}. Find the total volume of water that falls on the land in cubic metres. Give your answer in cubic metre, correct to two decimal places.

14

A surveyor provided the following diagram with measurements for a property she was mapping out:

a

Find the approximate total area of the property by using three applications of the trapezoidal rule. Give your answer in square metres.

b

The surveyor is reading a meteorological report that lists the average monthly rainfall in the region. According to the report, August sees 13.8 \text{ cm} of rainfall on average.

Find the volume of rainfall that the surveyor can expect to fall over the property next August. Give your answer to the nearest cubic metre.

15

The following diagram shows the measurements of a plot of land:

a

Find the approximate total area of the block of land by using four applications of the trapezoidal rule. Give your answer in square metres, correct to one decimal place.

b

The average weekly rainfall is 42.2 \text{ mm}. Find the total volume of water that falls on the land in cubic metres. Give your answer correct to one decimal place.

16

A large company recently purchased a block of land to build a warehouse on. The following is a surveyor's diagram of the block:

a

Find the approximate total area of the block of land by using five applications of the trapezoidal rule. Give your answer in square metres, correct to one decimal place.

b

The average annual rainfall is 900 \text{ mm}. Find the total volume of water that falls on the land in cubic metres. Give your answer correct to two decimal places.

17

Consider the following cross section of a river:

a

Use the trapezoidal rule to approximate the area of the cross section. Give your answer in square metres, correct to one decimal place.

b

The river flows at 0.6 \text{ m/s}. Approximate the volume of water that passes through the cross section after 9 seconds. Give your answer in cubic metres, correct to two decimal places.

18

Consider the following tent:

a

Use the trapezoidal rule to find the area of the front of the tent. Give your answer in square metres.

b

If the tent is 8 \text{ m} deep, approximate the volume of the tent. Give your answer in cubic metres.

19

The diagram shows the cross-section of a river at the exact point where it feeds into a dam. The depths of the river are marked at 2 \text{ m} intervals.

a

Using three applications of the trapezoidal rule, approximate the area of the cross-section of the river to one decimal place.

b

Water flows down the river at a rate of 3.4 \text{ m/s}. Find the volume of water that flows into the dam each day. Round your answer to the nearest cubic metre.

20

Five measurements across a pool were taken at 3 \text{ m} intervals:

a

Using four applications of the trapezoidal rule, find the area of surface of the pool. Give your answer in square metres, correct to one decimal place.

b

The pool is a constant 2 \text{ m} deep. Find the capacity of the pool. Round your answer to the nearest kilolitre.

21

Consider the driveway with the following measurements:

a

Use two applications of the trapezoidal rule to find the area of the driveway. Round your answer to the nearest square metre.

b

Find the approximate cost of sealing the driveway if sealer costs \$45 per litre and 1 \text{ L} covers 11 \text{ m}^{2}.

22

A section of the ground is removed for a road to be built. At every 7 \text{ m}, the area of the cross sections are measured in square metres, which are given below:

\ 0,\, 2.8,\, 3.3,\, 4.4,\, 4.1,\, 2.3,\, 0

Using the trapezoidal rule, approximate the volume of the ground removed. Give your answer in cubic metres, correct to one decimal place.

23

A boat sits partly in the water. At every 2 \text{ m}, the area of the cross section of the boat submerged underwater is measured in square metres, which is given below:

\ 1.94,\, 2.98,\, 3.3,\, 4.13,\, 2.63,\, 1.14,\, 0.86

Using the trapezoidal rule, approximate the volume of the submerged part of the boat. Give your answer in cubic metres, correct to two decimal places.

24

A children’s toy company is creating couches that are filled with foam in the shape shown:

a

Use the trapezoidal rule to approximate the area of the cross-section of the couch.

b

To minimise the cost of production, the company calculates that each couch should use 1\,553\,510 \text{ cm}^{3} of foam to fill the couch. Find the length of each couch to the nearest centimetre.

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MS11-4

performs calculations in relation to two-dimensional and three-dimensional figures

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