5.8 Trapezoidal ruleIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The surface of a lake is surveyed. Starting at one end, its width is measured every 20 m along its length. The six widths are 12 m, 30 m, 44 m, 50 m, 36 m and 8 m.

   (a)

   How many trapezoids are used when the trapezoidal rule is applied to all the measurements?

   [1 mark]

   • A6
   • B4
   • C5
   • D20

   (b)

   Use the trapezoidal rule to estimate the area of the surface of the lake.

   [1 mark]

   • A$1800$ m²
   • B$3400$ m²
   • C$6800$ m²
   • D$3440$ m²

   (c)

   The average depth of the lake is 2.5 m. Estimate the volume of water in the lake in m³.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Water flows into a tank at a rate of $R(t)=t^2+1$ litres per minute for $0\le t\le 4$, where $t$ is in minutes. The total volume of water that enters is the area under the graph of $R$.

   (a)

   Which list gives the values of $R(t)$ at $t=0,1,2,3,4$?

   [1 mark]

   • A$1,2,3,4,5$
   • B$0,1,4,9,16$
   • C$2,3,6,11,18$
   • D$1,2,5,10,17$

   (b)

   Use the trapezoidal rule with 4 intervals to estimate the total volume of water.

   [1 mark]

   • A$26$ litres
   • B$52$ litres
   • C$35$ litres
   • D$34$ litres

   (c)

   Use your GDC to find $\int_0^4 (t^2+1)\,dt$. Hence state whether the trapezoidal estimate is an over-estimate or an under-estimate, giving a reason.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A flower bed is bounded by the $x$-axis, the lines $x=1$ and $x=5$, and the curve $y=\sqrt{x}$, where $x$ and $y$ are in metres.

   (a)

   Use the trapezoidal rule with 4 intervals of equal width to estimate the area of the flower bed.

   [3 marks]

   (b)

   (i) Use your GDC to find $\int_1^5\sqrt{x}\,dx$.
   (ii) Find the percentage error in the estimate from part (a).
   (iii) Explain why the trapezoidal estimate is less than the exact area.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   During a storm the rate of rainfall, $r$ mm per hour, is measured every half hour. The readings at $t=0,\,0.5,\,1,\,1.5,\,2,\,2.5,\,3$ hours are 0.4, 4.2, 9.6, 12.4, 8.8, 3.6 and 0.8 mm per hour. Each reading is rounded to 1 decimal place.

   (a)

   (i) Write down the width $h$ of each interval.
   (ii) Use the trapezoidal rule to estimate the area under the graph of $r$ against $t$ for $0\le t\le3$.
   (iii) Interpret this area in context, and explain why it is only an estimate.

   [6 marks]

   (b)

   (i) Find the lower and upper bounds for the trapezoidal estimate in part (a), using the bounds of each reading.
   (ii) A second student uses a more accurate method and finds a total of 19.9 mm. Comment on whether the difference from the estimate in part (a) can be explained by rounding of the readings.

   [6 marks]

   Total for question 4: 12 marks

End of questions