Trapezoidal Rule

Measurement, STD2 M1 2021 HSC 12 MC

A block of land is represented by the shaded region on the number plane. All measurements are in kilometres.

Which of the following is the approximation for the area of this block of land in square kilometres, using two applications of the trapezoidal rule?

99
19.8
39.6
72

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`B`

Show Worked Solution

`\text{Solution 1}`

`text(Area)`	`≈ 6/2(1.2 +2) + 6/2(2+1.4)`
	`≈ 3(3.2) + 3(3.4)`
	`≈ 19.8\ text(km)^2`

`\text{Solution 2}`

`\text{Area}`	`≈ \frac{6}{2} (1.2 + 2 \times 2 + 1.4)`
	`≈ 19.8 \ \text{km}^2`

`=> B`

Measurement, STD2 M1 2014 HSC 28d*

An aerial diagram of a swimming pool is shown.

The swimming pool is a standard length of 50 metres but is not in the shape of a rectangle.

In the diagram of the swimming pool, the five widths are measured to be:

`CD = 21.88\ text(m)`

`EF = 25.63\ text(m)`

`GH = 31.88\ text(m)`

`IJ = 36.25\ text(m)`

`KL = 21.88\ text(m)`

Use four applications of the Trapezoidal Rule to calculate the surface area of the pool. (2 marks)

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The average depth of the pool is 1.2 m

Calculate the approximate volume of the swimming pool, in litres. (1 mark)

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`1445.5\ text(m)^2`
`1\ 734\ 600\ text(L)`

Show Worked Solution

a. `\text{Strategy 1}`

`text(Surface Area of pool)`

`~~ 12.5/2(21.88 + 25.63) + 12.5/2(25.63+31.88)+12.5/2(31.88+36.25) …`

`+ 12.5/2(36.25 + 21.88)`

`~~ 1445.5\ text(m)^2`

`\text{Strategy 2}`

`text(Surface Area of pool)`

`~~ 12.5/2[21.88 + 2(25.63 + 31.88 + 36.25) + 21.88]`

`~~ 1445.5\ text(m)^2`

♦ Mean mark (b) 50%.

b.	`V`	`= Ah`
		`~~ 1445.5 xx 1.2`
		`~~ 1734.6\ text(m)^3`
		`~~ 1\ 734\ 600\ text(L)`

Measurement, STD2 M1 2015 HSC 28c*

Three equally spaced cross-sectional areas of a vase are shown.

Use the Trapezoidal rule to find the approximate capacity of the vase in litres. (3 marks)

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`3.3\ text(litres)`

Show Worked Solution

`text(Solution 1)`

`V`	`≈ 15/2(45 + 180) + 15/2(180 + 35)`
	`≈ 15/2(225 + 215)`
	`≈ 3300\ text{mL (1 cm³ = 1 mL)}`
	`~~3.3\ text(L)`

`text(Solution 2)`

`V≈ 15/2(45 + 2 xx 180 + 35)~~3.3\ text(L)`

Measurement, STD2 M1 EQ-Bank 11 MC

The diagram represents a field.

What is the area of the field, using four applications of the Trapezoidal’s rule?

105 m²
136 m²
210 m²
420 m²

Show Answers Only

`A`

Show Worked Solution

`text(Solution 1)`

`text(Area)`	`~~ 3/2(6 + 7) + 3/2(7 + 12) + 3/2(12 + 8) + 3/2(8 + 10)`
	`~~ 3/2(13 + 19 + 20 + 18)`
	`~~ 105\ text(m)^2`

`text(Solution 2)`

\begin{array} {|l|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & 0 & 3 & 6 & 9 & 12 \\
\hline
\rule{0pt}{2.5ex} \text{height} \rule[-1ex]{0pt}{0pt} & \ \ \ 6\ \ \ & \ \ \ 7\ \ \ & \ \ 12\ \ & \ \ \ 8\ \ \ & \ \ 10\ \ \\
\hline
\rule{0pt}{2.5ex} \text{weight} \rule[-1ex]{0pt}{0pt} & 1 & 2 & 2 & 2 & 1 \\
\hline
\end{array}

`text(Area)`	`~~ 3/2(6 + 2 xx 7 + 2 xx 12 + 2 xx 8 + 10)`
	`~~ 3/2(70)`
	`~~ 105\ text(m)^2`

`=> A`

Measurement, STD2 M1 EQ-Bank 5 MC

The shaded region represents a block of land bounded on one side by a road.

What is the approximate area of the block of land, using the Trapezoidal rule?

720 m²
880 m²
1140 m²
1440 m²

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`A`

Show Worked Solution

`text(Area)`	`≈ 20/2(23 + 15) + 20/2(15 + 19)`
	`≈ 10(38) + 10(34)`
	`≈ 720\ text(m²)`

`=> A`

Measurement, STD2 M1 EQ-Bank 22

The scale diagram shows the aerial view of a block of land bounded on one side by a road. The length of the block, `AB`, is known to be 90 metres.

Calculate the approximate area of the block of land, using three applications of the Trapezoidal rule. (3 marks)

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`text{5175 m}^2`

Show Worked Solution

`text(Solution 1)`

`text(6 cm → 90 metres)`

` text(1 cm → 15 metres)`

`text(Height) = 2 xx 15 = 30\ text(metres)`

`text(Area)`	`~~ 30/2(75 + 60) + 30/2(60 + 45) + 30/2(45 + 60)`
	`~~ 15(135 + 105 + 105)`
	`~~ 5175\ text(m)^2`

`text(Solution 2)`

`text(After converting from scale:)`

`text(Area)~~ 30/2(75 + 2 xx 60 + 2 xx 45 + 60)~~ 5175\ text(m)^2`

Measurement, STD2 M1 2018 HSC 28a

A field is bordered on one side by a straight road and on the other side by a river, as shown. Measurements are taken perpendicular to the road every 7.5 metres along the road.

Use four applications of the Trapeziodal rule to find an approximation to the area of the field. Answer to the nearest square metre. (3 marks)

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Show Answers Only

`242\ text{m}^2`

Show Worked Solution

`text(Strategy 1)`

`A`	`~~ 7.5/2(8.8 + 7.1) + 7.5/2(7.1 + 9.8) + 7.5/2(9.8 + 8.5) + 7.5/2(8.5 + 4.9)`
	`~~ 241.875`
	`~~ 242\ text{m}^2\ \text{(nearest m}^2\text{)}`

`text(Strategy 2)`

`A`	`~~ 7.5/2(8.8 + 2 xx 7.1 + 2 xx 9.8 + 2 xx 8.5 + 4.9)`
	`~~ 242\ text{m}^2\ \text{(nearest m}^2\text{)}`

Measurement, STD2 M1 EQ-Bank 21

A farmer wants to estimate the area of an irregular shaped paddock.

What is the estimated area of the land using the Trapezoidal Rule? (2 marks)

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`370\ text(m)^2`

Show Worked Solution

`text(Solution 1)`

`text(Height) = 20 div 4 = 5\ text(m)`

`text(Area)`	`~~ 5/2 (28 + 18) + 5/2 (18 + 17) + 5/2 (17 + 16) + 5/2 (16 + 18)`
	`~~ 370\ text(m)^2`

`text(Solution 2)`

`x`	`0`	`5`	`10`	`15`	`20`
`text(height)`	`28`	`18`	`17`	`16`	`18`
`text(weight)`	`1`	`2`	`2`	`2`	`1`

`text(Area)`	`~~ h/2 [28 + 2 (18 + 17 + 16) + 18]`
	`~~ 5/2 xx 148`
	`~~ 370\ text(m)^2`

Measurement, STD2 M1 2013 HSC 15a*

The diagram shows the front of a tent supported by three vertical poles. The poles are 1.2 m apart. The height of each outer pole is 1.5 m, and the height of the middle pole is 1.8 m. The roof hangs between the poles.

The front of the tent has area `A\ text(m²)`.

Use the trapezoidal rule to estimate `A`. (2 marks)

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Explain whether the trapezoidal rule give a greater or smaller estimate of `A`? (1 mark)

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`3.96\ text(m²)`
`text(The trapezoidal rule assumes a straight line between)`

`text(all points and therefore would estimate a greater)`

`text(area than the actual area of the tent front.)`

Show Worked Solution

a.	`A`	`~~ h/2 [y_0 + 2y_1 + y_2]`
		`~~ 1.2/2 [1.5 + (2 xx 1.8) + 1.5]`
		`~~ 0.6 [6.6]`
		`~~ 3.96\ text(m²)`

b. `text(The trapezoidal rule assumes a straight line between)`

`text(all points and therefore would estimate a greater)`

`text(area than the actual area of the tent front.)`