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The diagram shows the cross-section of a wall across a creek - HSC - SSCE Mathematics Standard - Question 24 - 2023 - Paper 1
Question 24
The diagram shows the cross-section of a wall across a creek. (a) Use two applications of the trapezoidal rule to estimate the area of the cross-section of the wall... show full transcript
Worked Solution & Example Answer:The diagram shows the cross-section of a wall across a creek - HSC - SSCE Mathematics Standard - Question 24 - 2023 - Paper 1
Step 1
Use two applications of the trapezoidal rule to estimate the area of the cross-section of the wall.
Answer
To estimate the area using the trapezoidal rule, we apply it twice:
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First application:
- The height of the first trapezoid is 1.9 m (left side) and 2.7 m (right side) with the base length of 4.0 m.
- Area (A1) = ( A_1 = \frac{1}{2} \times 4.0 \times (1.9 + 2.7) )
- Calculate: ( A_1 = \frac{1}{2} \times 4.0 \times 4.6 = 9.2 , \text{m}^2 )
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Second application:
- The height of the second trapezoid is 2.7 m (left side) and 1.7 m (right side) with the same base length of 4.0 m.
- Area (A2) = ( A_2 = \frac{1}{2} \times 4.0 \times (2.7 + 1.7) )
- Calculate: ( A_2 = \frac{1}{2} \times 4.0 \times 4.4 = 8.8 , \text{m}^2 )
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Total Area:
- Total area ( A = A_1 + A_2 )
- Total area = 9.2 m² + 8.8 m² = 18 m².
Step 2
How many tonnes of concrete are in the wall? Give the answer to two significant figures.
Answer
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Volume of the Wall:
- The wall has a uniform thickness of 0.80 m.
- Volume ( V = 18 imes 0.8 )
- Calculate: ( V = 14.4 , \text{m}^3 )
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Weight of Concrete:
- Given that 1 m³ of concrete is 3.52 tonnes,
- Total weight = ( 14.4 \times 3.52 )
- Calculate: ( 14.4 \times 3.52 = 50.688 , \text{tonnes} )
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Final Answer:
- Rounding to two significant figures: 51 tonnes.