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rectangular base and a half-cylinder lid.](https://assets.mathsgenie.co.uk/question-image-redraws/raster-v1/1502390c-17f5-48dc-9ac1-0a72e58e7e2a/image-1-c854c0f798a3d103f8d796d5.svg)\n\nWork out the volume of the chest.\nGive your answer correct to 3 significant figures.","markscheme":"- Award method mark for volume of rectangular base: $60 \\times (80 - 30) \\times 120 = 360000$ M1\n- Award method mark for volume of half-cylinder lid: $\\frac{1}{2} \\times \\pi \\times 30^2 \\times 120 = 54000\\pi$ M1\n- Award method mark for adding volumes: $360000 + 54000\\pi$ M1\n- Final answer of 530000 A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Calculate the volume of the rectangular base.","code":"M"},{"id":"p2","marks":1,"description":"Calculate the volume of the half-cylinder lid.","code":"M"},{"id":"p3","marks":1,"description":"Sum the base and lid volumes.","code":"M"},{"id":"p4","marks":1,"description":"Final answer rounded to 3 significant figures.","code":"A"}],"finalAnswers":["530000"],"generatedAt":"2026-05-06T14:02:59.186Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"split_the_solid","label":"Split into two solids","steps":[{"type":"text","order":0,"title":"Find the radius","content":"The lid has a semicircular cross-section. Its diameter is the chest's width, so\n\n$$r=60\\div2=30\\text{ cm}$$\n\nThe rectangular base is therefore\n\n$$80-30=50\\text{ cm}$$\n\nhigh.","highlights":[{"text":"width of 60 cm","location":"specification"},{"text":"total height of 80 cm","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=base_sub}{V_{\\mathrm{base}}=60\\times(80-30)\\times120}"},{"latex":"\\htmlData{anchor=base_value}{V_{\\mathrm{base}}=360000\\ \\mathrm{cm}^3}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"base_sub"}],"connections":[{"to":"base_value","from":"base_sub","color":"sky","label":"calculate"}]},"order":1,"title":"Calculate the base","description":"Use the cuboid formula: width times height times length. This correct substitution earns the first M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=lid_sub}{V_{\\mathrm{lid}}=\\frac{1}{2}\\times\\pi\\times30^2\\times120}"},{"latex":"\\htmlData{anchor=lid_value}{V_{\\mathrm{lid}}=54000\\pi\\ \\mathrm{cm}^3}"}],"highlights":[{"color":"amber","style":"text-color-with-background","anchor":"lid_sub"}],"connections":[{"to":"lid_value","from":"lid_sub","color":"amber","label":"simplify"}]},"order":2,"title":"Calculate the lid","description":"A cylinder has volume pi times radius squared times length. The lid is half a cylinder, so include one half. This substitution earns the second M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=total_sum}{V_{\\mathrm{chest}}=360000+54000\\pi}"},{"latex":"\\htmlData{anchor=total_value}{V_{\\mathrm{chest}}=529646.003\\ldots\\ \\mathrm{cm}^3}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"total_sum"}],"connections":[{"to":"total_value","from":"total_sum","color":"green","label":"evaluate"}]},"order":3,"title":"Add both volumes","description":"The chest contains both solids, so add their volumes. Writing this sum earns the third M1."},{"type":"text","order":4,"title":"Round to three figures","content":"The first three significant figures are $5$, $2$ and $9$. The next digit is $6$, so round the $9$ up:\n\n$$\\boxed{530000\\text{ cm}^3}$$\n\nThis correctly rounded answer earns A1.","highlights":[{"text":"Give your answer correct to 3 significant figures.","location":"specification"}]}]}],"generatedAt":"2026-07-20T10:38:57.186Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"8696e696-6b59-4b93-818e-56c71f520057:separate_volumes","label":"Summing 3D Volumes","steps":[{"type":"text","order":0,"title":"Analyze the shapes","content":"To find the total volume of the storage chest, we can split it into two distinct 3D shapes:\n1. A **rectangular prism** (the base)\n2. A **semi-cylinder** (the half-cylinder lid)\n\nWe will calculate the volume of each part separately and then add them together to find the total volume.","highlights":[{"text":"rectangular box with a semi-cylindrical lid","location":"specification"}]},{"type":"text","order":1,"title":"Find dimensions of lid","content":"The lid is a semi-cylinder (half of a cylinder) lying on its side along the length of the chest.\n- The width of the chest is $60\\text{ cm}$, which represents the **diameter** of the circular cross-section of the lid.\n- Therefore, the **radius** ($r$) of the semi-cylinder is:\n $$r = \\frac{60}{2} = 30\\text{ cm}$$\n\nSince the lid is a semi-cylinder, its maximum height is equal to its radius ($30\\text{ cm}$).","highlights":[{"text":"width of 60 cm","location":"specification"}]},{"type":"text","order":2,"title":"Find height of base","content":"The total height of the chest is $80\\text{ cm}$. Since the semi-cylinder lid has a height equal to its radius of $30\\text{ cm}$, we can calculate the height of the rectangular base:\n\n$$\\text{Height of base} = 80 - 30 = 50\\text{ cm}$$","highlights":[{"text":"total height of 80 cm","location":"specification"}]},{"type":"text","order":3,"title":"Calculate rectangular base volume","content":"Now we calculate the volume of the rectangular prism base using the formula:\n$$\\text{Volume} = \\text{width} \\times \\text{height} \\times \\text{length}$$\n\nSubstitute our values in:\n$$\\text{Volume}_{\\text{base}} = 60 \\times 50 \\times 120$$\n$$\\text{Volume}_{\\text{base}} = 360,000\\text{ cm}^3$$","highlights":[{"text":"length of 120 cm","location":"specification"}]},{"type":"text","order":4,"title":"Calculate semi-cylinder lid volume","content":"The volume of a full cylinder is $\\pi r^2 h$. Since the lid is a semi-cylinder (half of a cylinder), its volume formula is:\n$$\\text{Volume}_{\\text{lid}} = \\frac{1}{2} \\pi r^2 L$$\nwhere $r = 30\\text{ cm}$ and $L = 120\\text{ cm}$ (the length of the chest).\n\nSubstitute the values:\n$$\\text{Volume}_{\\text{lid}} = \\frac{1}{2} \\times \\pi \\times 30^2 \\times 120$$\n$$\\text{Volume}_{\\text{lid}} = 0.5 \\times \\pi \\times 900 \\times 120 = 54,000\\pi\\text{ cm}^3$$\n$$\\text{Volume}_{\\text{lid}} \\approx 169,646\\text{ cm}^3$$"},{"type":"text","order":5,"title":"Sum the volumes","content":"Add the volumes of the rectangular base and the semi-cylindrical lid to get the total volume:\n\n$$\\text{Total Volume} = 360,000 + 54,000\\pi \\approx 529,646\\text{ cm}^3$$","calculator":[{"gdc":{"expression":"360000 + 54000 * π","instructions":"Type `360000 + 54000 * π` and press ENTER."},"ti84":{"expression":"360000 + 54000 * π","instructions":"Type `360000 + 54000 * π` and press ENTER."},"desmos":{"expression":"360000 + 54000\\pi","instructions":"Type `360000 + 54000\\pi` into the expression bar."},"description":"Enter the sum into your calculator to find the decimal value."}]},{"type":"text","order":6,"title":"Round to 3 s.f.","content":"We need to round our final answer to **3 significant figures**:\n\n$$529,646 \\rightarrow 530,000\\text{ cm}^3$$\n\nThus, the volume of the chest is **$530,000\\text{ cm}^3$**.","highlights":[{"text":"Give your answer correct to 3 significant figures.","location":"specification"}]}]},{"id":"8696e696-6b59-4b93-818e-56c71f520057:cross_sectional_area","label":"Using Cross-Sectional Area","steps":[{"type":"text","order":0,"title":"Analyze the cross-section","content":"The storage chest is a prism where the front face represents its uniform cross-sectional area. This cross-section is a composite shape made up of:\n1. A **rectangle** at the bottom.\n2. A **semicircle** (the lid) on the top.\n\nTo find the volume of the chest, we will calculate the total area of this front cross-section and then multiply it by the length of the chest ($120\\text{ cm}$).\n\n$$\\text{Volume} = \\text{Cross-Sectional Area} \\times \\text{Length}$$","highlights":[{"text":"width of 60 cm","location":"specification"},{"text":"total height of 80 cm","location":"specification"},{"text":"length of 120 cm","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Width of the chest} = \\htmlData{anchor=width}{\\color{@sky}{60}}\\text{ cm}$"},{"latex":"$$\\text{Radius of the semicircle } (r) = \\frac{\\htmlData{anchor=diam}{\\color{@sky}{60}}}{2} = \\htmlData{anchor=rad}{\\color{@purple}{30}}\\text{ cm}$"},{"latex":"$$\\text{Height of the rectangular part} = \\htmlData{anchor=tot-ht}{\\color{@green}{80}} - \\htmlData{anchor=rad-sub}{\\color{@purple}{30}} = \\htmlData{anchor=rect-ht}{\\color{@amber}{50}}\\text{ cm}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"width"},{"color":"purple","style":"text-color","anchor":"rad"},{"color":"amber","style":"text-color","anchor":"rect-ht"}],"connections":[{"to":"diam","from":"width","color":"sky","label":"diameter","style":"solid"},{"to":"rad-sub","from":"rad","color":"purple","label":"lid height","style":"solid"}]},"order":1,"title":"Find the dimensions","description":"Since the width of the chest is $60\\text{ cm}$, the diameter of the semicircle is also $60\\text{ cm}$. This means the radius (which is also the height of the lid) is $30\\text{ cm}$. Subtracting this from the total height of $80\\text{ cm}$ gives the height of the rectangle."},{"type":"text","order":2,"title":"Calculate the area","content":"Now, we calculate the area of the two parts of the cross-section:\n\n* **Area of the rectangular bottom:**\n$$\\text{Area}_{\\text{rectangle}} = \\text{width} \\times \\text{height} = 60 \\times 50 = 3000\\text{ cm}^2$$\n\n* **Area of the semicircular lid:**\n$$\\text{Area}_{\\text{semicircle}} = \\frac{1}{2} \\pi r^2 = \\frac{1}{2} \\times \\pi \\times 30^2 = 450\\pi\\text{ cm}^2$$\n\nAdding these together gives the total cross-sectional area:\n$$\\text{Total Area} = 3000 + 450\\pi\\text{ cm}^2$$"},{"type":"text","order":3,"title":"Calculate the volume","content":"Multiply the total cross-sectional area by the length of the chest ($120\\text{ cm}$):\n\n$$\\begin{aligned} \\text{Volume} &= \\text{Total Area} \\times \\text{Length} \\\\ &= (3000 + 450\\pi) \\times 120 \\\\ &= (3000 \\times 120) + (450\\pi \\times 120) \\\\ &= 360000 + 54000\\pi \\end{aligned}$$\n\nThis matches the sum of the volume of the rectangular prism ($360000\\text{ cm}^3$) and the volume of the half-cylinder ($54000\\pi\\text{ cm}^3$) specified in the markscheme."},{"type":"text","order":4,"title":"Round the final answer","content":"Now, let's calculate the numerical value and round it to 3 significant figures.\n\n$$\\text{Volume} = 360000 + 54000\\pi \\approx 360000 + 169646 = 529646\\text{ cm}^3$$\n\nRounding $529646$ to 3 significant figures gives:\n\n$$\\text{Volume} \\approx 530000\\text{ cm}^3$$","calculator":[{"gdc":{"expression":"360000 + 54000 * π","instructions":"Type 360000 + 54000 * π and press EXE."},"ti84":{"expression":"360000 + 54000 * π","instructions":"Type 360000 + 54000 * π and press ENTER."},"desmos":{"expression":"360000 + 54000\\pi","instructions":"Type 360000 + 54000 * pi into the prompt to see the decimal result."},"description":"Calculate the final volume using $\\pi$ and round to 3 significant figures."}],"highlights":[{"text":"correct to 3 significant figures","location":"specification"}]}]}],"generatedAt":"2026-06-12T10:28:05.713Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"bf2e2c33-a200-4b05-be44-cf9383d0c8c1","revisionId":"dd8df6b7-eff8-48e6-ac85-3aba04f5e241","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"hard","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"5410279e-7fc2-487c-bbdd-2d590fa9e5e5","label":"a","content":"A cylindrical storage drum has a radius of $r$ m and a height of 8 m.\n\nThe volume of the drum is $450\\pi$ m$^3$.\n\nFind the value of $r$.","markscheme":"- Award method mark for setting up the equation: $\\pi \\times r^2 \\times 8 = 450\\pi$ M1\n- Award method mark for simplifying: $8r^2 = 450$ M1\n- Award method mark for finding $r^2$: $r^2 = 56.25$ M1\n- Final answer of 7.5 A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Sets up the volume equation $\\pi r^2 h = V$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies the equation by dividing both sides by $\\pi$.","code":"M"},{"id":"p3","marks":1,"description":"Solves for $r^2$ by dividing the constant by the height.","code":"M"},{"id":"p4","marks":1,"description":"Calculates the final value of $r$ correctly.","code":"A"}],"finalAnswers":["7.5"],"generatedAt":"2026-05-06T14:02:52.896Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"use_cylinder_volume","label":"Use the cylinder formula","steps":[{"type":"text","order":0,"title":"Recall the formula","content":"For a cylinder, volume is the area of the circular base multiplied by the height:\n\n$$V=\\pi r^2h$$\n\nHere, the height is $8\\text{ m}$ and the volume is $450\\pi\\text{ m}^3$.","highlights":[{"text":"a radius of $r$ m and a height of 8 m","location":"specification"},{"text":"The volume of the drum is $450\\pi$ m$^3$.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=formula}{\\color{@sky}{V=\\pi r^2h}}$"},{"latex":"$$\\htmlData{anchor=substitution}{\\color{@sky}{\\pi r^2\\times 8=450\\pi}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substitution"}],"connections":[{"to":"substitution","from":"formula","color":"sky"}]},"order":1,"title":"Substitute the values","description":"Put the given height and volume directly into the cylinder formula. Writing this equation earns the first method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=with_pi}{\\color{@red}{\\pi r^2\\times 8=450\\pi}}$"},{"latex":"$$\\htmlData{anchor=without_pi}{\\color{@red}{8r^2=450}}$"}],"highlights":[{"color":"red","style":"text-color","anchor":"without_pi"}],"connections":[{"to":"without_pi","from":"with_pi","color":"red"}]},"order":2,"title":"Cancel pi","description":"Divide both sides by pi. This simplification earns the second method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=before_division}{\\color{@orange}{8r^2=450}}$"},{"latex":"$$\\htmlData{anchor=after_division}{\\color{@orange}{r^2=\\dfrac{450}{8}}}$"},{"latex":"$$r^2=56.25$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"after_division"}],"connections":[{"to":"after_division","from":"before_division","color":"orange"}]},"order":3,"title":"Find the square","description":"Divide both sides by 8. Reaching r squared equals 56.25 earns the third method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=squared_radius}{\\color{@green}{r^2=56.25}}$"},{"latex":"$$\\htmlData{anchor=positive_root}{\\color{@green}{r=\\sqrt{56.25}}}$"},{"latex":"$$r=7.5\\text{ m}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"positive_root"}],"connections":[{"to":"positive_root","from":"squared_radius","color":"green"}]},"order":4,"title":"Take the square root","description":"A radius is a length, so use the positive square root. The correct final value earns A1."}]}],"generatedAt":"2026-07-20T11:32:09.728Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"5410279e-7fc2-487c-bbdd-2d590fa9e5e5:algebraic","label":"Algebraic Solution","steps":[{"type":"text","order":0,"title":"Recall the volume formula","content":"To solve this problem, we need the formula for the volume of a cylinder:\n\n$$V = \n\\pi r^2 h$$\n\nwhere:\n* $V$ is the volume\n* $r$ is the radius\n* $h$ is the height"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$V = \\pi r^2 \\htmlData{anchor=h-val}{\\color{@amber}{h}}$"},{"latex":"$$\\htmlData{anchor=vol-val}{\\color{@sky}{450\\pi}} = \\pi r^2 (\\htmlData{anchor=h-sub}{\\color{@amber}{8}})$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"h-val"},{"color":"sky","style":"text-color","anchor":"vol-val"}],"connections":[{"to":"h-sub","from":"h-val","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Substitute the given values","description":"We are given that the height is $8$ m and the volume is $450\\pi$ m$^3$. Substituting these into our formula gives us an equation in terms of $r$."},{"type":"text","order":2,"title":"Simplify and cancel pi","content":"Next, we divide both sides of the equation by $\\pi$ to simplify it:\n\n$$\\begin{aligned}\n450\\pi &= 8\\pi r^2 \\\\\n450 &= 8r^2\n\\end{aligned}$$\n\n*This step earns a method mark M1 for setting up the equation, and another mark M1 for simplifying it to $8r^2 = 450$.*"},{"type":"text","order":3,"title":"Solve for r^2","content":"Now, isolate $r^2$ by dividing both sides by $8$:\n\n$$\\begin{aligned}\nr^2 &= \\frac{450}{8} \\\\\nr^2 &= 56.25\n\\end{aligned}$$\n\n*This earns the third method mark M1.*"},{"type":"text","order":4,"title":"Find the radius r","content":"Finally, find $r$ by taking the square root of both sides of the equation:\n\n$$r = \\sqrt{56.25} = 7.5$$\n\n*(Note: We only consider the positive square root because a radius must be a positive length.)*\n\nTherefore, the value of $r$ is **$7.5$**. \n\n*This final step completes the solution and earns the accuracy mark A1.*"}]},{"id":"5410279e-7fc2-487c-bbdd-2d590fa9e5e5:calculator_solver","label":"Calculator Solver","steps":[{"type":"text","order":0,"title":"Set up the equation","content":"To find the radius $r$, we first need to recall the formula for the volume of a cylinder. From our formula sheets, the volume $V$ of a cylinder is given by:\n\n$$V = \\pi r^2 h$$\n\nWe are given:\n- The height of the cylinder, $h = 8$ m\n- The volume of the cylinder, $V = 450\\pi$ m$^3$\n\nSubstituting these values into the volume formula gives us the equation:\n\n$$\\pi \\times r^2 \\times 8 = 450\\pi$$","highlights":[{"text":"radius of $r$ m","location":"specification"},{"text":"height of 8 m","location":"specification"},{"text":"volume of the drum is $450\\pi$ m$^3$","location":"specification"}]},{"type":"text","order":1,"title":"Simplify the equation","content":"We can simplify this equation by dividing both sides by $\\pi$ to eliminate it from the equation:\n\n$$8r^2 = 450$$\n\nNow, we have a simple quadratic equation that we can easily solve using our graphing calculator's solver or graphing features."},{"type":"text","order":2,"title":"Solve using a calculator","content":"To solve $8r^2 = 450$, we can rewrite it as $8r^2 - 450 = 0$ (or $8x^2 - 450 = 0$ on your calculator) and use the built-in solver or find the $x$-intercepts of the graph.","calculator":[{"gdc":{"expression":"8*x^2 - 450 = 0","instructions":"Method 1: Equation Solver\n1. Go to the 'Equation' or 'Solver' menu.\n2. Choose 'Polynomial' solver (Degree 2) or 'Solver'.\n3. Input the equation 8*x^2 - 450 = 0.\n4. Solve to get the roots.\n\nMethod 2: Graphing\n1. In the Graph menu, plot Y1 = 8*x^2 - 450.\n2. Use the 'G-Solv' (Graph Solve) tool and select 'ROOT' to find the positive x-intercept."},"ti84":{"expression":"8*x^2 = 450","instructions":"Method 1: Numerical Solver\n1. Press [MATH] and scroll down to select 'Numeric Solver...'.\n2. Enter 'E1: 8*x^2' and 'E2: 450', then press [OK] (graph button).\n3. Enter a positive guess for x (e.g., 5) and press [SOLVE] (graph button).\n\nMethod 2: Graphing\n1. Press [Y=] and enter Y1 = 8*X^2 - 450.\n2. Press [GRAPH] (adjust window to see the x-intercept if needed, e.g., Xmax = 10, Ymin = -50).\n3. Press [2nd] [CALC] and select '2: zero'. Move the cursor to find the positive zero."},"desmos":{"expression":"y = 8x^2 - 450","instructions":"1. In the first expression bar, type: y = 8x^2 - 450\n2. Locate the positive x-intercept on the graph where the curve crosses the horizontal axis.\n3. Click on the gray point on the x-axis to reveal its coordinates."},"description":"Solve the equation using the numerical solver or graphing features."}]},{"type":"text","order":3,"title":"State the final answer","content":"Using our calculator, we find the two solutions to the quadratic equation:\n\n$$r = \\pm 7.5$$\n\nSince the radius of a cylinder must be a positive length, we discard the negative solution.\n\nTherefore, the value of the radius is:\n\n$$r = 7.5\\text{ m}$$","highlights":[{"text":"Find the value of $r$.","location":"specification"}]}]}],"generatedAt":"2026-06-13T03:28:53.229Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/review/17afbfa7-933c-403d-93aa-0bacc9681bce.mp4","durationSeconds":84,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/review/17afbfa7-933c-403d-93aa-0bacc9681bce.jpg"}},{"id":"f6db3504-5c0b-4e01-9c3a-79ff0d6f7d80","revisionId":"9fd94061-3535-4c24-9de2-d314360c9c37","questionNumber":null,"questionType":"LA","specification":"A large cylindrical water tank has a diameter of $12\\text{ cm}$ and a height of $7\\text{ cm}$.\n\n![A diagram of a cylinder with diameter 12 cm and height 7 cm.](https://assets.mathsgenie.co.uk/question-image-redraws/raster-v1/f6db3504-5c0b-4e01-9c3a-79ff0d6f7d80/image-1-34c13e7062a28130edf0a9bd.svg)","difficultyBand":"hard","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":"calc","isCalculation":true,"parts":[{"id":"c88f1223-1f44-4e3a-a933-ac7fdb683006","label":"a","content":"Work out the volume of the cylinder.\nGive your answer correct to 1 decimal place.","markscheme":"- Finds radius: $6$ B1\n- Applies volume formula: $\\pi \\times 6^2 \\times 7$ M1\n- Final answer of $791.7$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Identifies the radius as 6 cm.","code":"B"},{"id":"p2","marks":1,"description":"Applies formula $\\pi \\times 6^2 \\times 7$.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of 791.7.","code":"A"}],"finalAnswers":["791.7"],"generatedAt":"2026-05-06T14:02:59.951Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"base_area_times_height","label":"Base area times height","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$d=12\\text{ cm}$"},{"latex":"$$r=\\color{@green}{\\htmlData{anchor=half_expression}{\\dfrac{12\\text{ cm}}{2}}}$"},{"latex":"$$r=\\color{@green}{\\htmlData{anchor=radius_result}{6\\text{ cm}}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"half_expression"},{"color":"green","style":"text-color-with-background","anchor":"radius_result"}],"connections":[{"to":"radius_result","from":"half_expression","color":"green","label":"divide by 2"}]},"order":0,"title":"Find the radius","description":"The formula needs the radius, but the diagram gives the diameter. The radius is half the diameter; this earns B1."},{"type":"text","order":1,"title":"Choose the formula","content":"A cylinder is a circular prism, so its volume is the area of the circular base multiplied by its height:\n\n$$V=\\pi r^2h$$\n\nHere, use $r=6\\text{ cm}$ and $h=7\\text{ cm}$.","highlights":[{"text":"volume of the cylinder","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$V=\\pi r^2h$"},{"latex":"$$V=\\color{@sky}{\\htmlData{anchor=substitution}{\\pi\\times6^2\\times7}}$"},{"latex":"$$V=\\color{@purple}{\\htmlData{anchor=calculator_value}{791.6813487046279\\ldots\\text{ cm}^3}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substitution"},{"color":"purple","style":"text-color","anchor":"calculator_value"}],"connections":[{"to":"calculator_value","from":"substitution","color":"purple","label":"calculate"}]},"order":2,"title":"Substitute the values","description":"Substituting the correct radius and height into the cylinder formula earns M1. Keep the full calculator value for the rounding step."},{"type":"text","order":3,"title":"Round the answer","content":"The calculator value is $791.6813487046279\\ldots$, so to $1$ decimal place the answer is\n\n$$\\boxed{791.7\\text{ cm}^3}$$\n\nThe final answer of $791.7$ earns A1.","highlights":[{"text":"Give your answer correct to 1 decimal place.","location":"specification"}]}]}],"generatedAt":"2026-07-19T06:49:08.063Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"db9eccbb-f6ec-4a2e-9850-9cfba7d14db1","label":"b","content":"The volume of another cylinder is $4500\\text{ cm}^3$. James says that $4500\\text{ cm}^3$ is the same as $4.5\\text{ m}^3$. Is James correct? You must give a reason for your answer.","markscheme":"- States 'No' with valid reason C1\n- Reason: $1\\text{ m}^3 = 100^3\\text{ cm}^3 = 1,000,000\\text{ cm}^3$.","marks":1,"order":1,"markingRubric":{"totalMarks":1,"points":[{"id":"p1","marks":1,"description":"States 'No' and explains that conversion requires a factor of $1,000,000$.","code":"C"}],"finalAnswers":["No"],"generatedAt":"2026-05-06T14:02:59.951Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"cube_the_conversion","label":"Cube the conversion factor","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$1\\text{ m}=\\htmlData{anchor=linear}{\\color{@sky}{100\\text{ cm}}}$"},{"latex":"$$1\\text{ m}^3=\\htmlData{anchor=cubed}{\\color{@sky}{100^3\\text{ cm}^3}}$"},{"latex":"$$1\\text{ m}^3=\\htmlData{anchor=result}{\\color{@green}{1{,}000{,}000\\text{ cm}^3}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"linear"},{"color":"sky","style":"text-color","anchor":"cubed"},{"color":"green","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"cubed","from":"linear","color":"sky","label":"cube for volume"}]},"order":1,"title":"Convert the cubic unit","description":"Cube both the number and the centimetre unit."},{"type":"text","order":3,"title":"Reject James's claim","content":"No, James is not correct. $1\\text{ m}^3=100^3\\text{ cm}^3=1{,}000{,}000\\text{ cm}^3$, so $4500\\text{ cm}^3$ cannot equal $4.5\\text{ m}^3$. This valid reason earns **C1**.","highlights":[{"text":"James says that $4500\\text{ cm}^3$ is the same as $4.5\\text{ m}^3$. Is James correct? You must give a reason for your 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