SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Measurement, STD2 M1 2023 HSC 24

The diagram shows the cross-section of a wall across a creek. 
 


 
  1. Use two applications of the trapezoidal rule to estimate the area of the cross-section of the wall.   (2 marks)

  2. The wall has a uniform thickness of 0.80 m. The weight of 1 m³ of concrete is 3.52 tonnes.  
  3. How many tonnes of concrete are in the wall? Give the answer to two significant figures.   (3 marks)

Show Answers Only

a.    `18\ text{m}^2`

b.    `text{51 tonnes}`

Show Worked Solution

a.    `h=8.0/2=4`

`A` `~~4/2(1.9+2.7) + 8/2(2.7+1.7)`  
  `~~2(4.6)+2(4.4)~~18\ text{m}^2`  

 
b.   `V_text{wall}=18 xx 0.8=14.4\ text{m}^3`

`text{Mass of concrete}` `=14.4 xx 3.52=50.688`  
  `=51\ text{tonnes (2 sig.fig.)}`  
♦ Mean mark (b) 46%.

Measurement, STD2 M1 2022 HSC 32

The diagram shows a park consisting of a rectangle and a semicircle. The semicircle has a radius of 100 m. The dimensions of the rectangle are 200 m and 250 m.

A lake occupies a section of the park as shown. The rest of the park is a grassed section. Some measurements from the end of the grassed section to the edge of the lake are also shown.
 

  1. Using two applications of the trapezoidal rule, calculate the approximate area of the grassed section.   (2 marks)

  2. Hence calculate the approximate area of the lake, to the nearest square metre.   (2 marks)

Show Answers Only

a.   `35\ 500\ text{m}^2`

b.   `30\ 208\ text{m}^2`

Show Worked Solution

a.   `h=100\ text{m}`

`A` `~~h/2(x_1+x_2)+h/2(x_2+x_3)`  
  `~~100/2(160+150)+100/2(150+250)`  
  `~~50xx310+50xx400~~35\ 500\ text{m}^2`  

  

b.    `text{Total Area}` `=\ text{Area of Rectangle + Area of Semi-Circle}`
    `= (250 xx 200) + 1/2 xx pi xx 100^2`
    `=65\ 707.96\ text{m}^2`

 
`text{Area of Lake} =65\ 708-35\ 500=30\ 208\ text{m}^2`


♦ Mean mark (b) 44%.

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?

  1. 99
  2. 19.8
  3. 39.6
  4. 72
Show Answers Only

`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 2020 HSC 27

The shaded region on the diagram represents a garden. Each grid represents 5 m × 5 m.
 


 

  1. Use two applications of the trapezoidal rule to calculate the approximate area of the garden.   (3 marks)

  2. Should the answer to part (a) be more than, equal to or less than the actual area of the garden? Referring to the diagram above, briefly explain your answer.   (2 marks)

Show Answers Only

a.   `850 \ text{m}^2`

b.   `text{The estimate will be more than actual area}`

Show Worked Solution

a.     

`h = 4 xx 5 = 20 \ text{m}`

♦ Mean mark 47%.
`text{Area}` `= frac{h}{2} (x_1 + x_2) + frac{h}{2} (x_2 + x_3)`
   `= frac{20}{2} (25 + 20) + frac{20}{2} (20 + 20}`
  `= 450 + 400`
  `= 850 \ text{m}^2`

 

♦♦ Mean mark 23%.

b.     

`text{The trapezoidal rule captures the shaded area plus the}`

`text{the extra area highlighted above.}`

`therefore \ text{The estimate will be more than actual area}`

Measurement, STD2 M1 2008 HSC 28b*

A tunnel is excavated with a cross-section as shown.
 

 

  1. Find an expression for the area of the cross-section using the Trapezoidal rule.   (2 marks)

  2. The area of the cross-section must be 600 m2. The tunnel is 80 m wide. 

     

    If the value of `a` increases by 2 metres, by how much will `b` change?   (2 marks)

Show Answers Only

a.    `h(2a + b)`

b.    `b\ text(decreases by 4.)`

Show Worked Solution

a.    `\text{Strategy 1}`

`A` `~~h/2(0+a)+h/2(a+b)+h/2(b+a)+h/2(a+0)`  
  `~~ h/2(4a + 2b)`  
  `~~h(2a+b)`  

 
`\text{Strategy 2}`

`A~~ h/2[0 + 2(a + b + a) + 0]~~ h(2a + b)`
 

b.   `A = 600\ text(m)^2`

`text(If tunnel is 80 metres wide)`

`4h=80\ \ =>\ \ h=20`

 `text{Using part (a):}`

`600` `= 20(2a + b)`
`30` `= 2a+b`
`b` `= 30-2a`

 
`:.\ text(If)\ a\ text(increases by 2,)\ b\ text(must decrease by 4.)`

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?

  1. 105 m²
  2. 136 m²
  3. 210 m²
  4. 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.
 

2UG-2005-12MC

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

  1.  720 m²
  2.  880 m²
  3.  1140 m²
  4.  1440 m²
Show Answers Only

`A`

Show Worked Solution

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

  
`=> 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)

Show Answers Only

`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)

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)

Show Answers Only

`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.

2013 15a

The front of the tent has area `A\ text(m)^2`. 

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

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

Show Answers Only

a.    `3.96\ text(m)^2`

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.)`

Show Worked Solution

a.    `text(Solution 1)`

`A~~1.2/2(1.5+1.8)+1.2/2(1.8+1.5)~~3.96\ text(m)^2`

`text(Solution 2)`

`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(m)^2`

 

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.)`