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bg-background","children":["$","$L3a",null,{"backHref":"/a-level/maths/wjec/questions","basePath":"/a-level/maths/wjec/numerical-methods-5/questions","formulaUrl":"https://www.wjec.co.uk/media/4jfdytcz/ss-w27-18-e.pdf","initialIndex":0,"initialQuestionWindow":[{"id":"a3f4543a-299a-4a9e-91ff-cd3ea231da7f","revisionId":"c38bda77-746f-4a19-aee7-013a90d0e4d2","questionNumber":"1","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":"11.9 The Trapezium Rule","calculatorUse":null,"isCalculation":true,"parts":[{"id":"b908b330-a439-4831-8082-85388c37b036","label":"a","content":"Use the trapezium rule with 4 strips to estimate the value of\n\n$$\\int_1^3 x \\ln(x+1) \\, dx$$","markscheme":"- Calculate the step size $h = \\frac{3-1}{4} = 0.5$ M1\n- Evaluate the function at the required $x$ values: \n - $x=1, f(x)=\\ln 2$\n - $x=1.5, f(x)=1.5 \\ln 2.5$\n - $x=2, f(x)=2 \\ln 3$\n - $x=2.5, f(x)=2.5 \\ln 3.5$\n - $x=3, f(x)=3 \\ln 4$ M1\n- Apply the trapezium rule formula: $0.5 \\left( \\frac{\\ln 2}{2} + 1.5 \\ln 2.5 + 2 \\ln 3 + 2.5 \\ln 3.5 + \\frac{3 \\ln 4}{2} \\right)$ M1\n- Final answer $4.56$ (3sf) A1","marks":4,"order":0,"markingRubric":{"totalMarks":6,"points":[{"id":"p1","marks":1,"description":"Correctly identifies the interval width $h = 0.5$.","code":"M"},{"id":"p2","marks":2,"description":"Correctly evaluates the function at the boundaries and midpoints (at least 4 correct values).","code":"M"},{"id":"p3","marks":2,"description":"Correct substitution into the trapezium rule formula.","code":"M"},{"id":"p4","marks":1,"description":"Final answer of $4.56$ (3sf).","code":"A"}],"finalAnswers":["4.56"],"generatedAt":"2026-05-06T03:41:08.106Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"trapezium_rule","label":"Composite Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the strip width","content":"To begin, we need to find the step size (strip width), $h$, of our intervals. \n\nThe formula for $h$ is:\n\n$$h = \\frac{b - a}{n}$$\n\nwhere $a = 1$ is the lower limit of integration, $b = 3$ is the upper limit, and $n = 4$ is the number of strips.\n\n$$\\begin{aligned} h &= \\frac{3 - 1}{4} \\\\ &= \\frac{2}{4} \\\\ &= 0.5 \\end{aligned}$$\n\nThis earns the first mark **[M1]**."},{"type":"text","order":1,"title":"Tabulate the function values","content":"Now, we evaluate our function $f(x) = x \\ln(x+1)$ at each interval boundary $x_i$ starting from $x_0 = 1$ up to $x_4 = 3$, stepping by $h = 0.5$ each time.\n\n| $i$ | $x_i$ | $f(x_i) = y_i$ (Exact) |\n|---|---|---|\n| $0$ | $1$ | $1 \\ln(2) = \\ln 2$ |\n| $1$ | $1.5$ | $1.5 \\ln(2.5)$ |\n| $2$ | $2$ | $2 \\ln(3)$ |\n| $3$ | $2.5$ | $2.5 \\ln(3.5)$ |\n| $4$ | $3$ | $3 \\ln(4)$ |\n\nCorrectly identifying these values earns the second mark **[M1]**."},{"type":"text","order":2,"title":"Apply trapezium rule formula","content":"The composite trapezium rule is given by the formula:\n\n$$\\int_a^b f(x) \\, dx \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\n\nWe can also write this as:\n\n$$h \\left[ \\frac{y_0}{2} + y_1 + y_2 + y_3 + \\frac{y_4}{2} \\right]$$\n\nSubstituting our values into this formula, we get:\n\n$$0.5 \\left[ \\frac{\\ln 2}{2} + 1.5 \\ln 2.5 + 2 \\ln 3 + 2.5 \\ln 3.5 + \\frac{3 \\ln 4}{2} \\right]$$\n\nSubstituting the values into the formula correctly earns the third mark **[M1]**."},{"type":"text","order":3,"title":"Calculate the final estimate","content":"Let's compute the value of each term inside the brackets:\n\n- $\\frac{\\ln 2}{2} \\approx 0.34657$\n- $1.5 \\ln 2.5 \\approx 1.37444$\n- $2 \\ln 3 \\approx 2.19722$\n- $2.5 \\ln 3.5 \\approx 3.13191$\n- $\\frac{3 \\ln 4}{2} \\approx 2.07944$\n\nSumming these terms:\n\n$$\\text{Sum} = 0.34657 + 1.37444 + 2.19722 + 3.13191 + 2.07944 \\approx 9.12958$$\n\nNow, multiply by $h = 0.5$:\n\n$$\\text{Estimate} = 0.5 \\times 9.12958 \\approx 4.56479$$\n\nTo $3$ significant figures, our estimate is:\n\n$$\\text{Estimate} \\approx 4.56$$\n\nThis final accuracy earns the final mark **[A1]**.","calculator":[{"gdc":{"expression":"0.5*(ln(2)/2 + 1.5*ln(2.5) + 2*ln(3) + 2.5*ln(3.5) + 3*ln(4)/2)","instructions":"Enter the composite trapezium rule formula into your main run-matrix calculation screen."},"ti84":{"expression":"0.5*(ln(2)/2 + 1.5*ln(2.5) + 2*ln(3) + 2.5*ln(3.5) + 3*ln(4)/2)","instructions":"Type the expression directly into the main screen:\n1. Press [0] [.] [5] [parenthesis]\n2. Press [LN] [2] [parenthesis] [/] [2] [+] [1] [.] [5] [LN] [2] [.] [5] [parenthesis]\n3. Press [+] [2] [LN] [3] [parenthesis] [+] [2] [.] [5] [LN] [3] [.] [5] [parenthesis]\n4. Press [+] [3] [LN] [4] [parenthesis] [/] [2] [parenthesis]\n5. Press [ENTER] to get the result."},"desmos":{"expression":"0.5 * ( (ln(2)/2) + 1.5*ln(2.5) + 2*ln(3) + 2.5*ln(3.5) + (3*ln(4)/2) )","instructions":"Type the arithmetic expression into a calculation line to get the approximate decimal value."},"description":"Calculate the complete trapezium rule expression using a calculator."}]}]}],"generatedAt":"2026-06-09T08:33:02.868Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"b908b330-a439-4831-8082-85388c37b036:trapezium_rule","label":"Composite Trapezium Rule","steps":[{"type":"text","order":0,"title":"Calculate the strip width","content":"To begin, we need to find the step size (strip width), $h$, of our intervals. \n\nThe formula for $h$ is:\n\n$$h = \\frac{b - a}{n}$$\n\nwhere $a = 1$ is the lower limit of integration, $b = 3$ is the upper limit, and $n = 4$ is the number of strips.\n\n$$\\begin{aligned} h &= \\frac{3 - 1}{4} \\\\ &= \\frac{2}{4} \\\\ &= 0.5 \\end{aligned}$$\n\nThis earns the first mark **[M1]**."},{"type":"text","order":1,"title":"Tabulate the function values","content":"Now, we evaluate our function $f(x) = x \\ln(x+1)$ at each interval boundary $x_i$ starting from $x_0 = 1$ up to $x_4 = 3$, stepping by $h = 0.5$ each time.\n\n| $i$ | $x_i$ | $f(x_i) = y_i$ (Exact) |\n|---|---|---|\n| $0$ | $1$ | $1 \\ln(2) = \\ln 2$ |\n| $1$ | $1.5$ | $1.5 \\ln(2.5)$ |\n| $2$ | $2$ | $2 \\ln(3)$ |\n| $3$ | $2.5$ | $2.5 \\ln(3.5)$ |\n| $4$ | $3$ | $3 \\ln(4)$ |\n\nCorrectly identifying these values earns the second mark **[M1]**."},{"type":"text","order":2,"title":"Apply trapezium rule formula","content":"The composite trapezium rule is given by the formula:\n\n$$\\int_a^b f(x) \\, dx \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\n\nWe can also write this as:\n\n$$h \\left[ \\frac{y_0}{2} + y_1 + y_2 + y_3 + \\frac{y_4}{2} \\right]$$\n\nSubstituting our values into this formula, we get:\n\n$$0.5 \\left[ \\frac{\\ln 2}{2} + 1.5 \\ln 2.5 + 2 \\ln 3 + 2.5 \\ln 3.5 + \\frac{3 \\ln 4}{2} \\right]$$\n\nSubstituting the values into the formula correctly earns the third mark **[M1]**."},{"type":"text","order":3,"title":"Calculate the final estimate","content":"Let's compute the value of each term inside the brackets:\n\n- $\\frac{\\ln 2}{2} \\approx 0.34657$\n- $1.5 \\ln 2.5 \\approx 1.37444$\n- $2 \\ln 3 \\approx 2.19722$\n- $2.5 \\ln 3.5 \\approx 3.13191$\n- $\\frac{3 \\ln 4}{2} \\approx 2.07944$\n\nSumming these terms:\n\n$$\\text{Sum} = 0.34657 + 1.37444 + 2.19722 + 3.13191 + 2.07944 \\approx 9.12958$$\n\nNow, multiply by $h = 0.5$:\n\n$$\\text{Estimate} = 0.5 \\times 9.12958 \\approx 4.56479$$\n\nTo $3$ significant figures, our estimate is:\n\n$$\\text{Estimate} \\approx 4.56$$\n\nThis final accuracy earns the final mark **[A1]**.","calculator":[{"gdc":{"expression":"0.5*(ln(2)/2 + 1.5*ln(2.5) + 2*ln(3) + 2.5*ln(3.5) + 3*ln(4)/2)","instructions":"Enter the composite trapezium rule formula into your main run-matrix calculation screen."},"ti84":{"expression":"0.5*(ln(2)/2 + 1.5*ln(2.5) + 2*ln(3) + 2.5*ln(3.5) + 3*ln(4)/2)","instructions":"Type the expression directly into the main screen:\n1. Press [0] [.] [5] [parenthesis]\n2. Press [LN] [2] [parenthesis] [/] [2] [+] [1] [.] [5] [LN] [2] [.] [5] [parenthesis]\n3. Press [+] [2] [LN] [3] [parenthesis] [+] [2] [.] [5] [LN] [3] [.] [5] [parenthesis]\n4. Press [+] [3] [LN] [4] [parenthesis] [/] [2] [parenthesis]\n5. Press [ENTER] to get the result."},"desmos":{"expression":"0.5 * ( (ln(2)/2) + 1.5*ln(2.5) + 2*ln(3) + 2.5*ln(3.5) + (3*ln(4)/2) )","instructions":"Type the arithmetic expression into a calculation line to get the approximate decimal value."},"description":"Calculate the complete trapezium rule expression using a calculator."}]}]}],"generatedAt":"2026-06-09T08:33:34.797Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":21},"walkthroughVideo":null},{"id":"244102ec-b9cd-4011-8acb-e970dfc01abf","revisionId":"03ffcf07-f136-43f6-af98-c6553855842a","questionNumber":null,"questionType":"LA","specification":"A marine biologist is monitoring the rate of oxygen consumption, \$R\$ (measured in mg/L/h), by a specific coral species in a research tank. The rate can be modelled by the equation\n\n\\[R = 1.2 + 2^{t-4} + 2^{4-t} - 0.5(t-4)^2, \\quad 1 \\le t \\le 7\\]\n\nwhere \$t\$ is the time in hours after the start of the observation.\n\nA table of values for \$t\$ and \$R\$ is shown below, with values of \$R\$ given to 4 significant figures where appropriate.\n\n| Time, \$t\$ (h) | 2 | 2.5 | 3 | 3.5 | 4 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| Rate, \$R\$ (mg/L/h) | | 3.257 | 3.2 | | 3.2 |","difficulty":3,"difficultyBand":"easy","tier":null,"totalMarks":5,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"b857618d-073b-41a3-a934-491e0795ecd3","label":"a","content":"Use the given equation to complete the table, giving the values of \$R\$ to 4 significant figures where appropriate.","markscheme":"- Substitutes $t = 2$ into the equation: $R = 1.2 + 2^{2-4} + 2^{4-2} - 0.5(2-4)^2 = 1.2 + 0.25 + 4 - 2 = 3.45$ B1\n- Substitutes $t = 3.5$ into the equation: $R = 1.2 + 2^{-0.5} + 2^{0.5} - 0.5(3.5-4)^2 \\approx 3.196$ B1","marks":2,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_substitution","label":"Direct Algebraic Substitution","steps":[{"type":"text","order":0,"title":"Identify the task","content":"We need to find the rate of oxygen consumption $R$ for the missing times in the table: $t = 2$ and $t = 3.5$.\n\nWe will do this by substituting these values directly into the model equation:\n\n$$R = 1.2 + 2^{t-4} + 2^{4-t} - 0.5(t-4)^2$$","highlights":[{"text":"t = 2","location":"specification"},{"text":"t = 3.5","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$R = 1.2 + 2^{\\htmlData{anchor=t2-exp1}{\\color{@sky}{2}}-4} + 2^{4-\\htmlData{anchor=t2-exp2}{\\color{@sky}{2}}} - 0.5(\\htmlData{anchor=t2-exp3}{\\color{@sky}{2}}-4)^2$"},{"latex":"$$R = 1.2 + 2^{-2} + 2^{2} - 0.5(-2)^2$"},{"latex":"$$R = 1.2 + 0.25 + 4 - 0.5(4)$"},{"latex":"$$R = 1.2 + 0.25 + 4 - 2 = \\htmlData{anchor=t2-ans}{\\color{@sky}{3.45}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t2-exp1"},{"color":"sky","style":"text-color","anchor":"t2-exp2"},{"color":"sky","style":"text-color","anchor":"t2-exp3"},{"color":"sky","style":"box","anchor":"t2-ans"}],"connections":[]},"order":1,"title":"Substitute t = 2","description":"Substitute $t = 2$ into the model equation. Simplify the exponents and the squared term step-by-step."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$R = 1.2 + 2^{\\htmlData{anchor=t35-exp1}{\\color{@purple}{3.5}}-4} + 2^{4-\\htmlData{anchor=t35-exp2}{\\color{@purple}{3.5}}} - 0.5(\\htmlData{anchor=t35-exp3}{\\color{@purple}{3.5}}-4)^2$"},{"latex":"$$R = 1.2 + 2^{-0.5} + 2^{0.5} - 0.5(-0.5)^2$"},{"latex":"$$R = 1.2 + \\frac{1}{\\sqrt{2}} + \\sqrt{2} - 0.5(0.25)$"},{"latex":"$$R = 1.2 + 0.70711 + 1.41421 - 0.125 \\approx \\htmlData{anchor=t35-ans}{\\color{@purple}{3.196}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"t35-exp1"},{"color":"purple","style":"text-color","anchor":"t35-exp2"},{"color":"purple","style":"text-color","anchor":"t35-exp3"},{"color":"purple","style":"box","anchor":"t35-ans"}],"connections":[]},"order":2,"title":"Substitute t = 3.5","description":"Now substitute $t = 3.5$ into the model equation. Evaluate the fractional powers and the squared term."},{"type":"text","order":3,"title":"Calculate using GDC","content":"You can perform these calculations efficiently on your graphing calculator or scientific calculator by evaluating the expressions directly.","calculator":[{"gdc":{"expression":"1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2","instructions":"For t = 2:\nType `1.2 + 2^(2-4) + 2^(4-2) - 0.5*(2-4)^2` and press [EXE].\n\nFor t = 3.5:\nType `1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2` and press [EXE]."},"ti84":{"expression":"1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2","instructions":"For t = 2:\nType `1.2 + 2^(2-4) + 2^(4-2) - 0.5*(2-4)^2` and press [ENTER].\n\nFor t = 3.5:\nType `1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2` and press [ENTER]."},"desmos":{"expression":"1.2 + 2^{3.5-4} + 2^{4-3.5} - 0.5(3.5-4)^2","instructions":"Type the formula with substituted values:\nFor t = 2: `1.2 + 2^{2-4} + 2^{4-2} - 0.5(2-4)^2` \nFor t = 3.5: `1.2 + 2^{3.5-4} + 2^{4-3.5} - 0.5(3.5-4)^2`"},"description":"To calculate both rates, type the expressions directly into the main screen of your calculator."}]},{"type":"text","order":4,"title":"State final table values","content":"Rounding the values to 4 significant figures where appropriate:\n\n* For $t = 2$, we get exactly $R = 3.45$ (which has 3 significant figures).\n* For $t = 3.5$, we get $R \\approx 3.196$ (rounded to 4 significant figures).\n\nHere is the completed row of the table:\n\n| Time, $t$ (h) | 2 | 2.5 | 3 | 3.5 | 4 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| Rate, $R$ (mg/L/h) | **3.45** | 3.257 | 3.2 | **3.196** | 3.2 |"},{"type":"text","order":5,"title":"Apply the trapezium rule","content":"To find an estimate for the total oxygen consumption between $t = 2$ and $t = 4$, we use the trapezium rule with a step size of $h = 0.5$:\n\n$$\\text{Area} \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + y_3) \\right]$$\n\nSubstituting our values into the formula:\n\n$$\\text{Area} \\approx \\frac{0.5}{2} \\left[ 3.45 + 3.2 + 2(3.257 + 3.2 + 3.196) \\right]$$\n\n$$\\text{Area} \\approx 0.25 \\left[ 6.65 + 2(9.653) \\right] = 0.25 \\left[ 6.65 + 19.306 \\right] = 6.489$$\n\nRounding to 2 decimal places, we get **6.49** mg/L."}]},{"id":"gdc_table","label":"Using Graphing Calculator Table/Function","steps":[{"type":"text","order":0,"title":"Define the function","content":"To find the missing values of $R$ at $t = 2$ and $t = 3.5$ quickly and accurately, we can define our rate function $R(t)$ in a graphing calculator. Let's rewrite the equation in terms of the variable $x$ for the calculator:\n\n$$Y_1 = 1.2 + 2^{x-4} + 2^{4-x} - 0.5(x-4)^2$$"},{"type":"text","order":1,"title":"Evaluate at t = 2","content":"Using the table or function evaluation tool on your calculator, find the value when $x = 2$.\n\nWhen we evaluate $Y_1(2)$, we get:\n\n$$R = 3.45$$","calculator":[{"gdc":{"expression":"Y1(2)","instructions":"Navigate to the Table view or use the Calculate tool on the graph at $x = 2$."},"ti84":{"expression":"Y_1(2)","instructions":"Press [2nd] [TABLE]. Scroll to where $X = 2$ to read the corresponding $Y_1$ value. Alternatively, from the home screen, press [ALPHA] [TRACE] [ENTER] to select $Y_1$, type '(2)', and press [ENTER]."},"desmos":{"expression":"f(2)","instructions":"In a new cell, type f(2) to output the value."},"description":"Find the value of the function at x = 2."}]},{"type":"text","order":2,"title":"Evaluate at t = 3.5","content":"Now, evaluate the function at $x = 3.5$ using your calculator.\n\nEvaluating $Y_1(3.5)$ yields:\n\n$$R \\approx 3.19635...$$","calculator":[{"gdc":{"expression":"Y1(3.5)","instructions":"Evaluate the function for an input value of 3.5."},"ti84":{"expression":"Y_1(3.5)","instructions":"If your table step size is 1, you can change the independent variable to 'Ask' mode by pressing [2nd] [TBLSET] and changing 'Indpnt' to 'Ask'. Then press [2nd] [TABLE] and type '3.5'. Alternatively, evaluate $Y_1(3.5)$ on the home screen."},"desmos":{"expression":"f(3.5)","instructions":"Type f(3.5) to find the decimal value."},"description":"Find the value of the function at x = 3.5."}]},{"type":"text","order":3,"title":"Record the final values","content":"Round the values to $4$ significant figures as requested by the question specification:\n\n- At $t = 2$, we have $R = 3.45$ (which has $3$ significant figures and is exact).\n- At $t = 3.5$, we round $3.19635...$ to $4$ significant figures, giving $R \\approx 3.196$.\n\nThe completed table is:\n\n| Time, $t$ (h) | 2 | 2.5 | 3 | 3.5 | 4 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| Rate, $R$ (mg/L/h) | **3.45** | 3.257 | 3.2 | **3.196** | 3.2 |"},{"type":"text","order":4,"title":"Perform trapezium integration","content":"Using the completed table, we estimate the area under the curve between $t = 2$ and $t = 4$. With $h = 0.5$:\n\n$$\\text{Area} \\approx \\frac{0.5}{2} [y_{\\text{first}} + y_{\\text{last}} + 2(\\text{sum of middle values})]$$\n\n$$\\text{Area} \\approx 0.25 [3.45 + 3.2 + 2(3.257 + 3.2 + 3.196)]$$\n\n$$\\text{Area} \\approx 0.25 [6.65 + 19.306] = 6.489$$\n\nRounding to 2 decimal places, the total estimated oxygen consumption is **6.49** mg/L."}]}],"generatedAt":"2026-06-09T03:55:34.038Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"161a1671-1cde-4dad-9c78-281722622410","label":"b","content":"The total amount of oxygen consumed by the coral during the interval between \$t = 2\$ and \$t = 4\$ can be estimated by calculating the area under the curve. Use the trapezium rule, with all the values of \$R\$ in the completed table, to find an estimate for this total oxygen consumption. Give your answer to 2 decimal places.","markscheme":"- States $h = 0.5$ (or uses $h = 0.5$ in the formula) B1\n- Substitutes their values into the trapezium rule formula: $\\text{Area} \\approx \\frac{0.5}{2} [3.45 + 3.2 + 2(3.257 + 3.2 + 3.196)]$ M1\n- Calculates total sum $0.25 [6.65 + 19.306] = 6.489$ and rounds to $6.49$ A1","marks":3,"order":1,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"direct_trapezium","label":"Trapezium Rule Application","steps":[{"type":"text","order":0,"title":"Calculate missing rate values","content":"To apply the trapezium rule, we first need to find the missing rate values, $R$, at $t = 2$ and $t = 3.5$ using the model equation:\n\n$$R = 1.2 + 2^{t-4} + 2^{4-t} - 0.5(t-4)^2$$\n\n**For $t = 2$:**\n$$\\begin{aligned} R(2) &= 1.2 + 2^{2-4} + 2^{4-2} - 0.5(2-4)^2 \\\\ &= 1.2 + 2^{-2} + 2^{2} - 0.5(-2)^2 \\\\ &= 1.2 + 0.25 + 4 - 0.5(4) \\\\ &= 1.2 + 0.25 + 4 - 2 \\\\ &= 3.45 \\end{aligned}$$\n\n**For $t = 3.5$:**\n$$\\begin{aligned} R(3.5) &= 1.2 + 2^{3.5-4} + 2^{4-3.5} - 0.5(3.5-4)^2 \\\\ &= 1.2 + 2^{-0.5} + 2^{0.5} - 0.5(-0.5)^2 \\\\ &= 1.2 + \\frac{1}{\\sqrt{2}} + \\sqrt{2} - 0.5(0.25) \\\\ &\\approx 1.2 + 0.7071 + 1.4142 - 0.125 \\\\ &\\approx 3.196 \\text{ (to 4 s.f.)} \\end{aligned}$$\n\nNow, the completed table of values is:\n\n| $t$ (h) | 2 | 2.5 | 3 | 3.5 | 4 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| $R$ (mg/L/h) | **3.45** | 3.257 | 3.2 | **3.196** | 3.2 |","calculator":[{"gdc":{"expression":"f(x) = 1.2 + 2^(x-4) + 2^(4-x) - 0.5*(x-4)^2","instructions":"Define the function $f(x) = 1.2 + 2^{x-4} + 2^{4-x} - 0.5(x-4)^2$. Evaluate $f(2)$ and $f(3.5)$."},"ti84":{"expression":"1.2 + 2^(X-4) + 2^(4-X) - 0.5*(X-4)^2","instructions":"Enter the formula into the main screen to calculate each value:\nTo find $R(2)$: press [1][.][2] [+] [2] [^] [(] [2] [-] [4] [)] [+] [2] [^] [(] [4] [-] [2] [)] [-] [0][.][5] [^] [(] [2] [-] [4] [)] [x²] [ENTER]\nTo find $R(3.5)$: press [1][.][2] [+] [2] [^] [(] [3][.][5] [-] [4] [)] [+] [2] [^] [(] [4] [-] [3][.][5] [)] [-] [0][.][5] [^] [(] [3][.][5] [-] [4] [)] [x²] [ENTER]"},"desmos":{"expression":"R(t) = 1.2 + 2^{t-4} + 2^{4-t} - 0.5(t-4)^2","instructions":"Type the formula for $R(t)$ into the first input bar, then evaluate $R(2)$ and $R(3.5)$ directly in the subsequent bars."},"description":"Calculate $R(2)$ and $R(3.5)$ using your graphing calculator."}]},{"type":"text","order":1,"title":"Identify step size","content":"Next, we find the width of each strip ($h$) from our table. The time values are spaced at equal intervals:\n\n$$h = 2.5 - 2 = 0.5$$\n\nIdentifying that $h = 0.5$ earns the first mark **(B1)**."},{"type":"text","order":2,"title":"Set up trapezium rule","content":"Recall the standard formula for the trapezium rule:\n\n$$\\text{Area} \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\n\nHere, our boundary ordinates are:\n* Endpoints: $y_0 = 3.45$ (at $t=2$) and $y_n = 3.2$ (at $t=4$)\n* Interior points: $3.257$, $3.2$, and $3.196$\n\nSubstituting our values into the formula gives:\n\n$$\\text{Area} \\approx \\frac{0.5}{2} \\left[ 3.45 + 3.2 + 2(3.257 + 3.2 + 3.196) \\right]$$\n\nSetting up this substitution correctly earns the method mark **(M1)**."},{"type":"text","order":3,"title":"Calculate the estimated area","content":"Now, we compute the sum inside the brackets first:\n\n$$\\text{Sum of interior terms} = 3.257 + 3.2 + 3.196 = 9.653$$\n\nMultiply this by $2$:\n\n$$2 \\times 9.653 = 19.306$$\n\nAdd the sum of the endpoints:\n\n$$\\text{Total bracketed sum} = 3.45 + 3.2 + 19.306 = 25.956$$\n\nFinally, multiply by $\\frac{h}{2} = 0.25$:\n\n$$\\text{Area} \\approx 0.25 \\times 25.956 = 6.489$$\n\nRounding to **2 decimal places** as requested in the question, we obtain:\n\n$$\\text{Total oxygen consumption} \\approx 6.49 \\text{ mg/L}$$\n\nCalculating this final rounded value correctly awards the final accuracy mark **(A1)**.","calculator":[{"gdc":{"expression":"0.25*(3.45 + 3.2 + 2*(3.257 + 3.2 + 3.196))","instructions":"Evaluate the full expression: $0.25 \\times (3.45 + 3.2 + 2 \\times (3.257 + 3.2 + 3.196))$"},"ti84":{"expression":"0.25*(3.45 + 3.2 + 2*(3.257 + 3.2 + 3.196))","instructions":"Enter the following on your main screen:\npress [0][.][2][5] [ ( ] [3][.][4][5] [+] [3][.][2] [+] [2] [ ( ] [3][.][2][5][7] [+] [3][.][2] [+] [3][.][1][9][6] [ ) ] [ ) ] [ENTER]"},"desmos":{"expression":"0.25 * (3.45 + 3.2 + 2 * (3.257 + 3.2 + 3.196))","instructions":"Enter the complete formula to calculate the final decimal answer."},"description":"Compute the full trapezium rule expression in one calculation step to avoid rounding errors."}]}]}],"generatedAt":"2026-06-09T03:56:15.360Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"b857618d-073b-41a3-a934-491e0795ecd3:direct_substitution","label":"(a) Direct Algebraic Substitution","steps":[{"type":"text","order":0,"title":"Identify the task","content":"We need to find the rate of oxygen consumption $R$ for the missing times in the table: $t = 2$ and $t = 3.5$.\n\nWe will do this by substituting these values directly into the model equation:\n\n$$R = 1.2 + 2^{t-4} + 2^{4-t} - 0.5(t-4)^2$$","highlights":[{"text":"t = 2","location":"specification"},{"text":"t = 3.5","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$R = 1.2 + 2^{\\htmlData{anchor=t2-exp1}{\\color{@sky}{2}}-4} + 2^{4-\\htmlData{anchor=t2-exp2}{\\color{@sky}{2}}} - 0.5(\\htmlData{anchor=t2-exp3}{\\color{@sky}{2}}-4)^2$"},{"latex":"$$R = 1.2 + 2^{-2} + 2^{2} - 0.5(-2)^2$"},{"latex":"$$R = 1.2 + 0.25 + 4 - 0.5(4)$"},{"latex":"$$R = 1.2 + 0.25 + 4 - 2 = \\htmlData{anchor=t2-ans}{\\color{@sky}{3.45}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t2-exp1"},{"color":"sky","style":"text-color","anchor":"t2-exp2"},{"color":"sky","style":"text-color","anchor":"t2-exp3"},{"color":"sky","style":"box","anchor":"t2-ans"}],"connections":[]},"order":1,"title":"Substitute t = 2","description":"Substitute $t = 2$ into the model equation. Simplify the exponents and the squared term step-by-step."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$R = 1.2 + 2^{\\htmlData{anchor=t35-exp1}{\\color{@purple}{3.5}}-4} + 2^{4-\\htmlData{anchor=t35-exp2}{\\color{@purple}{3.5}}} - 0.5(\\htmlData{anchor=t35-exp3}{\\color{@purple}{3.5}}-4)^2$"},{"latex":"$$R = 1.2 + 2^{-0.5} + 2^{0.5} - 0.5(-0.5)^2$"},{"latex":"$$R = 1.2 + \\frac{1}{\\sqrt{2}} + \\sqrt{2} - 0.5(0.25)$"},{"latex":"$$R = 1.2 + 0.70711 + 1.41421 - 0.125 \\approx \\htmlData{anchor=t35-ans}{\\color{@purple}{3.196}}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"t35-exp1"},{"color":"purple","style":"text-color","anchor":"t35-exp2"},{"color":"purple","style":"text-color","anchor":"t35-exp3"},{"color":"purple","style":"box","anchor":"t35-ans"}],"connections":[]},"order":2,"title":"Substitute t = 3.5","description":"Now substitute $t = 3.5$ into the model equation. Evaluate the fractional powers and the squared term."},{"type":"text","order":3,"title":"Calculate using GDC","content":"You can perform these calculations efficiently on your graphing calculator or scientific calculator by evaluating the expressions directly.","calculator":[{"gdc":{"expression":"1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2","instructions":"For t = 2:\nType `1.2 + 2^(2-4) + 2^(4-2) - 0.5*(2-4)^2` and press [EXE].\n\nFor t = 3.5:\nType `1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2` and press [EXE]."},"ti84":{"expression":"1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2","instructions":"For t = 2:\nType `1.2 + 2^(2-4) + 2^(4-2) - 0.5*(2-4)^2` and press [ENTER].\n\nFor t = 3.5:\nType `1.2 + 2^(3.5-4) + 2^(4-3.5) - 0.5*(3.5-4)^2` and press [ENTER]."},"desmos":{"expression":"1.2 + 2^{3.5-4} + 2^{4-3.5} - 0.5(3.5-4)^2","instructions":"Type the formula with substituted values:\nFor t = 2: `1.2 + 2^{2-4} + 2^{4-2} - 0.5(2-4)^2` \nFor t = 3.5: `1.2 + 2^{3.5-4} + 2^{4-3.5} - 0.5(3.5-4)^2`"},"description":"To calculate both rates, type the expressions directly into the main screen of your calculator."}]},{"type":"text","order":4,"title":"State final table values","content":"Rounding the values to 4 significant figures where appropriate:\n\n* For $t = 2$, we get exactly $R = 3.45$ (which has 3 significant figures).\n* For $t = 3.5$, we get $R \\approx 3.196$ (rounded to 4 significant figures).\n\nHere is the completed row of the table:\n\n| Time, $t$ (h) | 2 | 2.5 | 3 | 3.5 | 4 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| Rate, $R$ (mg/L/h) | **3.45** | 3.257 | 3.2 | **3.196** | 3.2 |"},{"type":"text","order":5,"title":"Apply the trapezium rule","content":"To find an estimate for the total oxygen consumption between $t = 2$ and $t = 4$, we use the trapezium rule with a step size of $h = 0.5$:\n\n$$\\text{Area} \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + y_3) \\right]$$\n\nSubstituting our values into the formula:\n\n$$\\text{Area} \\approx \\frac{0.5}{2} \\left[ 3.45 + 3.2 + 2(3.257 + 3.2 + 3.196) \\right]$$\n\n$$\\text{Area} \\approx 0.25 \\left[ 6.65 + 2(9.653) \\right] = 0.25 \\left[ 6.65 + 19.306 \\right] = 6.489$$\n\nRounding to 2 decimal places, we get **6.49** mg/L."}]},{"id":"b857618d-073b-41a3-a934-491e0795ecd3:gdc_table","label":"(a) Using Graphing Calculator Table/Function","steps":[{"type":"text","order":0,"title":"Define the function","content":"To find the missing values of $R$ at $t = 2$ and $t = 3.5$ quickly and accurately, we can define our rate function $R(t)$ in a graphing calculator. Let's rewrite the equation in terms of the variable $x$ for the calculator:\n\n$$Y_1 = 1.2 + 2^{x-4} + 2^{4-x} - 0.5(x-4)^2$$"},{"type":"text","order":1,"title":"Evaluate at t = 2","content":"Using the table or function evaluation tool on your calculator, find the value when $x = 2$.\n\nWhen we evaluate $Y_1(2)$, we get:\n\n$$R = 3.45$$","calculator":[{"gdc":{"expression":"Y1(2)","instructions":"Navigate to the Table view or use the Calculate tool on the graph at $x = 2$."},"ti84":{"expression":"Y_1(2)","instructions":"Press [2nd] [TABLE]. Scroll to where $X = 2$ to read the corresponding $Y_1$ value. Alternatively, from the home screen, press [ALPHA] [TRACE] [ENTER] to select $Y_1$, type '(2)', and press [ENTER]."},"desmos":{"expression":"f(2)","instructions":"In a new cell, type f(2) to output the value."},"description":"Find the value of the function at x = 2."}]},{"type":"text","order":2,"title":"Evaluate at t = 3.5","content":"Now, evaluate the function at $x = 3.5$ using your calculator.\n\nEvaluating $Y_1(3.5)$ yields:\n\n$$R \\approx 3.19632034...$$","calculator":[{"gdc":{"expression":"Y1(3.5)","instructions":"Evaluate the function for an input value of 3.5."},"ti84":{"expression":"Y_1(3.5)","instructions":"If your table step size is 1, you can change the independent variable to 'Ask' mode by pressing [2nd] [TBLSET] and changing 'Indpnt' to 'Ask'. Then press [2nd] [TABLE] and type '3.5'. Alternatively, evaluate $Y_1(3.5)$ on the home screen."},"desmos":{"expression":"f(3.5)","instructions":"Type f(3.5) to find the decimal value."},"description":"Find the value of the function at x = 3.5."}]},{"type":"text","order":3,"title":"Record the final values","content":"Round the values to $4$ significant figures as requested by the question specification:\n\n- At $t = 2$, we have $R = 3.45$ (which has $3$ significant figures and is exact).\n- At $t = 3.5$, we round $3.19635...$ to $4$ significant figures, giving $R \\approx 3.196$.\n\nThe completed table is:\n\n| Time, $t$ (h) | 2 | 2.5 | 3 | 3.5 | 4 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| Rate, $R$ (mg/L/h) | **3.45** | 3.257 | 3.2 | **3.196** | 3.2 |"},{"type":"text","order":4,"title":"Perform trapezium integration","content":"Using the completed table, we estimate the area under the curve between $t = 2$ and $t = 4$. With $h = 0.5$:\n\n$$\\text{Area} \\approx \\frac{0.5}{2} [y_{\\text{first}} + y_{\\text{last}} + 2(\\text{sum of middle values})]$$\n\n$$\\text{Area} \\approx 0.25 [3.45 + 3.2 + 2(3.257 + 3.2 + 3.196)]$$\n\n$$\\text{Area} \\approx 0.25 [6.65 + 19.306] = 6.489$$\n\nRounding to 2 decimal places, the total estimated oxygen consumption is **6.49** mg/L."}]},{"id":"161a1671-1cde-4dad-9c78-281722622410:direct_trapezium","label":"(b) Trapezium Rule Application","steps":[{"type":"text","order":0,"title":"Calculate missing rate values","content":"To apply the trapezium rule, we first need to find the missing rate values, $R$, at $t = 2$ and $t = 3.5$ using the model equation:\n\n$$R = 1.2 + 2^{t-4} + 2^{4-t} - 0.5(t-4)^2$$\n\n**For $t = 2$:**\n$$\\begin{aligned} R(2) &= 1.2 + 2^{2-4} + 2^{4-2} - 0.5(2-4)^2 \\\\ &= 1.2 + 2^{-2} + 2^{2} - 0.5(-2)^2 \\\\ &= 1.2 + 0.25 + 4 - 0.5(4) \\\\ &= 1.2 + 0.25 + 4 - 2 \\\\ &= 3.45 \\end{aligned}$$\n\n**For $t = 3.5$:**\n$$\\begin{aligned} R(3.5) &= 1.2 + 2^{3.5-4} + 2^{4-3.5} - 0.5(3.5-4)^2 \\\\ &= 1.2 + 2^{-0.5} + 2^{0.5} - 0.5(-0.5)^2 \\\\ &= 1.2 + \\frac{1}{\\sqrt{2}} + \\sqrt{2} - 0.5(0.25) \\\\ &\\approx 1.2 + 0.7071 + 1.4142 - 0.125 \\\\ &\\approx 3.196 \\text{ (to 4 s.f.)} \\end{aligned}$$\n\nNow, the completed table of values is:\n\n| $t$ (h) | 2 | 2.5 | 3 | 3.5 | 4 |\n| :--- | :--- | :--- | :--- | :--- | :--- |\n| $R$ (mg/L/h) | **3.45** | 3.257 | 3.2 | **3.196** | 3.2 |","calculator":[{"gdc":{"expression":"f(x) = 1.2 + 2^(x-4) + 2^(4-x) - 0.5*(x-4)^2","instructions":"Define the function $f(x) = 1.2 + 2^{x-4} + 2^{4-x} - 0.5(x-4)^2$. Evaluate $f(2)$ and $f(3.5)$."},"ti84":{"expression":"1.2 + 2^(X-4) + 2^(4-X) - 0.5*(X-4)^2","instructions":"Enter the formula into the main screen to calculate each value:\nTo find $R(2)$: press [1][.][2] [+] [2] [^] [(] [2] [-] [4] [)] [+] [2] [^] [(] [4] [-] [2] [)] [-] [0][.][5] [^] [(] [2] [-] [4] [)] [x²] [ENTER]\nTo find $R(3.5)$: press [1][.][2] [+] [2] [^] [(] [3][.][5] [-] [4] [)] [+] [2] [^] [(] [4] [-] [3][.][5] [)] [-] [0][.][5] [^] [(] [3][.][5] [-] [4] [)] [x²] [ENTER]"},"desmos":{"expression":"R(t) = 1.2 + 2^{t-4} + 2^{4-t} - 0.5(t-4)^2","instructions":"Type the formula for $R(t)$ into the first input bar, then evaluate $R(2)$ and $R(3.5)$ directly in the subsequent bars."},"description":"Calculate $R(2)$ and $R(3.5)$ using your graphing calculator."}]},{"type":"text","order":1,"title":"Identify step size","content":"Next, we find the width of each strip ($h$) from our table. The time values are spaced at equal intervals:\n\n$$h = 2.5 - 2 = 0.5$$\n\nIdentifying that $h = 0.5$ earns the first mark **(B1)**."},{"type":"text","order":2,"title":"Set up trapezium rule","content":"Recall the standard formula for the trapezium rule:\n\n$$\\text{Area} \\approx \\frac{h}{2} \\left[ y_0 + y_n + 2(y_1 + y_2 + \\dots + y_{n-1}) \\right]$$\n\nHere, our boundary ordinates are:\n* Endpoints: $y_0 = 3.45$ (at $t=2$) and $y_n = 3.2$ (at $t=4$)\n* Interior points: $3.257$, $3.2$, and $3.196$\n\nSubstituting our values into the formula gives:\n\n$$\\text{Area} \\approx \\frac{0.5}{2} \\left[ 3.45 + 3.2 + 2(3.257 + 3.2 + 3.196) \\right]$$\n\nSetting up this substitution correctly earns the method mark **(M1)**."},{"type":"text","order":3,"title":"Calculate the estimated area","content":"Now, we compute the sum inside the brackets first:\n\n$$\\text{Sum of interior terms} = 3.257 + 3.2 + 3.196 = 9.653$$\n\nMultiply this by $2$:\n\n$$2 \\times 9.653 = 19.306$$\n\nAdd the sum of the endpoints:\n\n$$\\text{Total bracketed sum} = 3.45 + 3.2 + 19.306 = 25.956$$\n\nFinally, multiply by $\\frac{h}{2} = 0.25$:\n\n$$\\text{Area} \\approx 0.25 \\times 25.956 = 6.489$$\n\nRounding to **2 decimal places** as requested in the question, we obtain:\n\n$$\\text{Total oxygen consumption} \\approx 6.49 \\text{ mg/L}$$\n\nCalculating this final rounded value correctly awards the final accuracy mark **(A1)**.","calculator":[{"gdc":{"expression":"0.25*(3.45 + 3.2 + 2*(3.257 + 3.2 + 3.196))","instructions":"Evaluate the full expression: $0.25 \\times (3.45 + 3.2 + 2 \\times (3.257 + 3.2 + 3.196))$"},"ti84":{"expression":"0.25*(3.45 + 3.2 + 2*(3.257 + 3.2 + 3.196))","instructions":"Enter the following on your main screen:\npress [0][.][2][5] [ ( ] [3][.][4][5] [+] [3][.][2] [+] [2] [ ( ] [3][.][2][5][7] [+] [3][.][2] [+] [3][.][1][9][6] [ ) ] [ ) ] [ENTER]"},"desmos":{"expression":"0.25 * (3.45 + 3.2 + 2 * (3.257 + 3.2 + 3.196))","instructions":"Enter the complete formula to calculate the final decimal answer."},"description":"Compute the full trapezium rule expression in one calculation step to avoid rounding 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