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mark for setting up the equation: $\\pi \\times r^2 \\times 4 = 121\\pi$ M1\n- Award method mark for simplifying: $4r^2 = 121$ M1\n- Award method mark for finding $r^2$: $r^2 = 30.25$ M1\n- Final answer of 5.5 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":["5.5"],"generatedAt":"2026-05-06T14:02:54.721Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"use_cylinder_volume","label":"Use the cylinder volume formula","steps":[{"type":"text","order":0,"title":"Recall the formula","content":"For a cylinder, volume is the 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4=121\\pi}}$"},{"latex":"$$\\htmlData{anchor=without_pi}{\\color{@sky}{4r^2=121}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"with_pi"},{"color":"sky","style":"text-color","anchor":"without_pi"}],"connections":[{"to":"without_pi","from":"with_pi","color":"sky"}]},"order":2,"title":"Cancel pi","description":"Both sides contain the factor pi, so divide both sides by pi. This earns the second method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=before_divide}{\\color{@orange}{4r^2=121}}$"},{"latex":"$$\\htmlData{anchor=after_divide}{\\color{@orange}{r^2=30.25}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"before_divide"},{"color":"orange","style":"text-color","anchor":"after_divide"}],"connections":[{"to":"after_divide","from":"before_divide","color":"orange"}]},"order":3,"title":"Find the square","description":"Divide both sides by 4. Reaching the value of r squared earns the third method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=squared_radius}{\\color{@green}{r^2=30.25}}$"},{"latex":"$$\\htmlData{anchor=positive_root}{\\color{@green}{r=\\sqrt{30.25}}}$"},{"latex":"$$\\htmlData{anchor=final_radius}{\\color{@teal}{r=5.5\\text{ mm}}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"squared_radius"},{"color":"green","style":"text-color","anchor":"positive_root"},{"color":"teal","style":"text-color-with-background","anchor":"final_radius"}],"connections":[{"to":"positive_root","from":"squared_radius","color":"green","style":"solid"},{"to":"final_radius","from":"positive_root","color":"teal","style":"solid"}]},"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":"2026-07-19T18:14:14.911Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"2f2707d0-8b4d-4da9-8ec4-9c9e375e749e:algebraic","label":"Algebraic Approach","steps":[{"type":"text","order":0,"title":"Recall the volume formula","content":"The volume $V$ of a cylinder with radius $r$ and height $h$ is given by the formula:\n\n$$V = ̀\\pi r^2 h$$\n\nFrom the question, we are given:\n- The height of the cylinder is $h = 4\\text{ mm}$\n- The volume of the cylinder is $V = 121\\pi\\text{ mm}^3$","highlights":[{"text":"radius of $r$ mm and a height of 4 mm","location":"specification"},{"text":"volume of the metal pin is $121\\pi$ mm$^3$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$V = \\pi r^2 h$"},{"latex":"$$\\htmlData{anchor=vol-eq}{121\\pi} = \\pi r^2 \\times \\htmlData{anchor=h-eq}{4}$"}],"highlights":[{"color":"sky","style":"box","anchor":"vol-eq"},{"color":"amber","style":"box","anchor":"h-eq"}],"connections":[]},"order":1,"title":"Set up the equation","description":"Substitute the known values into the volume formula to form an equation in terms of $r$. This step earns the first method mark **M1**."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$121\\pi = 4\\pi r^2$"},{"latex":"$$\\htmlData{anchor=lh-simp}{121} = \\htmlData{anchor=rh-simp}{4r^2}$"}],"highlights":[{"color":"indigo","style":"underline","anchor":"lh-simp"},{"color":"indigo","style":"underline","anchor":"rh-simp"}],"connections":[{"to":"rh-simp","from":"lh-simp","color":"indigo","label":"divide by pi","style":"solid"}]},"order":2,"title":"Cancel pi and simplify","description":"Since $\\pi$ appears on both sides of the equation, we can divide both sides by $\\pi$ to cancel it out. This simplifies the equation and earns the second method mark **M1**."},{"type":"text","order":3,"title":"Solve for r^2","content":"To isolate $r^2$, divide both sides of the equation by $4$:\n\n$$\\begin{aligned} r^2 &= \\frac{121}{4} \\\\ r^2 &= 30.25 \\end{aligned}$$\n\nThis calculation earns the third method mark **M1**."},{"type":"text","order":4,"title":"Solve for r","content":"To find the value of $r$, take the positive square root of $30.25$ (since the radius must be a positive length):\n\n$$\\begin{aligned} r &= \\sqrt{30.25} \\\\ r &= 5.5 \\end{aligned}$$\n\nThis gives us our final answer and earns the accuracy mark **A1**.\n\n**Final Answer:**\n$$r = 5.5\\text{ mm}$$"}]},{"id":"2f2707d0-8b4d-4da9-8ec4-9c9e375e749e:calculator_solver","label":"Using Equation Solver","steps":[{"type":"text","order":0,"title":"Set up the equation","content":"The volume $V$ of a cylinder with radius $r$ and height $h$ is given by the formula:\n\n$$V = \\pi r^2 h$$\n\nGiven that the height is $4\\text{ mm}$ and the volume is $121\\pi\\text{ mm}^3$, we can substitute these values into the formula to form our equation:\n\n$$121\\pi = \\pi r^2 \\times 4$$\n\nWhich simplifies to:\n\n$$121\\pi = 4\\pi r^2$$","highlights":[{"text":"cylinder with a radius of $r$ mm and a height of 4 mm.","location":"specification"},{"text":"volume of the metal pin is $121\\pi$ mm$^3$.","location":"specification"}]},{"type":"text","order":1,"title":"Simplify the equation","content":"To make it easier to enter into the equation solver, we can first divide both sides of the equation by $\\pi$:\n\n$$121 = 4r^2$$"},{"type":"text","order":2,"title":"Solve using calculator","content":"We can now use a graphing calculator's numerical equation solver to find the positive value for $r$. We set up the equation as $4r^2 - 121 = 0$ (or $4x^2 - 121 = 0$).","calculator":[{"gdc":{"expression":"4*x^2 - 121 = 0","instructions":"Go to the Equation menu, select Solver, and enter the equation 4*x^2 - 121 = 0. Solve for x, selecting the positive result."},"ti84":{"expression":"4*X^2 = 121","instructions":"Press [MATH], scroll down to [Numeric Solver...]. Enter E1: 4*X^2 and E2: 121. Press [OK] (graph button), set a positive bound/guess (e.g., X=5), and press [SOLVE] (graph button)."},"desmos":{"expression":"4x^2 - 121 = 0","instructions":"Type the equation into the input bar or plot the function y = 4x^2 - 121 and find the positive x-intercept."},"description":"Use the numerical solver or graph solver to find the positive solution for $r$."}]},{"type":"text","order":3,"title":"State final answer","content":"The calculator yields two solutions: $r = -5.5$ and $r = 5.5$.\n\nSince a radius must be a positive length, we discard the negative value.\n\n$$r = 5.5$$"}]}],"generatedAt":"2026-06-12T09:55:53.769Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"9c326542-8bf8-4764-96b7-146a05aaf11e","revisionId":"7a8d908b-8acb-4682-a82e-74f6caee04c0","questionNumber":null,"questionType":"LA","specification":"A decorative glass paperweight is shaped like a semi-cylinder with a radius of $12$ cm and a length of $45$ cm.","difficultyBand":"hard","tier":null,"totalMarks":6,"sourceTitle":null,"calculatorUse":"calc","isCalculation":true,"parts":[{"id":"743fef30-0f15-440e-8861-fcbfb64a8b9c","label":"a","content":"Work out the volume of the semi-cylinder paperweight.\nGive your answer correct to 3 significant figures.","markscheme":"- Award method mark for area of semi-circle: $\\frac{\\pi \\times 12^2}{2} = 72\\pi$ M1\n- Award method mark for volume: $72\\pi \\times 45$ M1\n- Final answer of 10200 A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Calculation of the cross-sectional area of the semi-circle.","code":"M","acceptable":["72\\pi","226.2"]},{"id":"p2","marks":1,"description":"Multiplication of the cross-sectional area by the length to find volume.","code":"M","acceptable":["3240\\pi","10178.8"]},{"id":"p3","marks":1,"description":"Final answer correct to 3 significant figures.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:03:08.876Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]},{"id":"4e1bb35e-4b4b-4d4c-bb5f-99de1d4955d7","label":"b","content":"Work out the total surface area of the semi-cylinder paperweight.\nGive your answer correct to 3 significant figures.","markscheme":"- Award method mark for area of two semi-circles (one full circle): $\\pi \\times 12^2 = 144\\pi$ M1\n- Award method mark for curved surface area and rectangular base: $(\\pi \\times 12 \\times 45) + (24 \\times 45)$ M1\n- Final answer of 3230 A1","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Calculation of the area of both semi-circular ends.","code":"M","acceptable":["144\\pi","452.4"]},{"id":"p2","marks":1,"description":"Calculation of the curved surface area plus the area of the rectangular base.","code":"M","acceptable":["540\\pi + 1080","2776.5"]},{"id":"p3","marks":1,"description":"Final answer correct to 3 significant figures.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-06T14:03:08.876Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"split_the_surfaces","label":"Split the surfaces","steps":[{"type":"drawing","order":0,"title":"Identify every surface","width":900,"height":520,"shapes":[{"h":175,"w":485,"x":55,"y":40,"id":"curved_face","geo":"rectangle","fill":"semi","type":"geo","color":"blue","label":"curved rectangular face"},{"x":0,"y":0,"id":"curved_dimensions","gap":10,"size":"s","text":"45 cm by 12π cm","type":"text","color":"blue","anchorTo":"curved_face","orientation":"below"},{"h":105,"w":485,"x":55,"y":285,"id":"base","geo":"rectangle","fill":"semi","type":"geo","color":"light-blue","label":"rectangular base"},{"x":0,"y":0,"id":"base_dimensions","gap":10,"size":"s","text":"45 cm by 24 cm","type":"text","color":"blue","anchorTo":"base","orientation":"below"},{"id":"left_end","type":"plot","color":"green","segments":[[{"x":595,"y":175},{"x":600,"y":151},{"x":614,"y":130},{"x":635,"y":116},{"x":660,"y":111},{"x":685,"y":116},{"x":706,"y":130},{"x":720,"y":151},{"x":725,"y":175}]]},{"id":"left_diameter","type":"line","color":"green","points":[{"x":595,"y":175},{"x":725,"y":175}]},{"x":0,"y":0,"id":"left_end_label","gap":18,"size":"s","text":"semicircular end","type":"text","color":"green","anchorTo":"left_diameter","orientation":"below"},{"id":"left_radius","toX":660,"toY":111,"type":"arrow","color":"black","fromX":660,"fromY":175},{"x":0,"y":0,"id":"radius_label","gap":14,"size":"s","text":"12 cm","type":"text","color":"black","anchorTo":"left_radius","orientation":"left"},{"id":"right_end","type":"plot","color":"green","segments":[[{"x":730,"y":350},{"x":735,"y":326},{"x":749,"y":305},{"x":770,"y":291},{"x":795,"y":286},{"x":820,"y":291},{"x":841,"y":305},{"x":855,"y":326},{"x":860,"y":350}]]},{"id":"right_diameter","type":"line","color":"green","points":[{"x":730,"y":350},{"x":860,"y":350}]},{"x":0,"y":0,"id":"right_end_label","gap":18,"size":"s","text":"semicircular end","type":"text","color":"green","anchorTo":"right_diameter","orientation":"below"}],"caption":"An exploded net shows the four separate surfaces: one curved rectangular face, one rectangular base and two semicircular ends."},{"type":"text","order":1,"title":"Combine the ends","content":"The two semicircular ends together make one complete circle. Their total area is\n\n$$\\pi r^2=\\pi\\times 12^2=144\\pi\\text{ cm}^2.$$ \n\nWriting this earns the first **M1**.","highlights":[{"text":"radius of $12$ cm","location":"specification"}]},{"type":"text","order":2,"title":"Find the curved face","content":"A full cylinder's curved face has width equal to its circumference, $2\\pi r$. Because this is a semi-cylinder, the curved face uses half the circumference, so its width is $\\pi r$.\n\n$$\\text{curved area}=\\pi r\\times l=\\pi\\times12\\times45=540\\pi\\text{ cm}^2.$$","highlights":[{"text":"length of $45$ cm","location":"specification"}]},{"type":"text","order":3,"title":"Find the flat base","content":"The rectangular base has width equal to the diameter, $2\\times12=24$ cm, and length $45$ cm.\n\n$$\\text{base area}=24\\times45=1080\\text{ cm}^2.$$ \n\nIncluding both the curved area and this rectangular base earns the second **M1**.","highlights":[{"text":"semi-cylinder","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$A=\\color{@sky}{\\htmlData{anchor=pi_top}{144\\pi+540\\pi}}+\\color{@green}{\\htmlData{anchor=base_top}{1080}}$"},{"latex":"$$A=\\color{@sky}{\\htmlData{anchor=pi_bottom}{684\\pi}}+\\color{@green}{\\htmlData{anchor=base_bottom}{1080}}$"},{"latex":"$$A=3228.849375\\ldots\\ \\text{cm}^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"pi_top"},{"color":"sky","style":"text-color","anchor":"pi_bottom"},{"color":"green","style":"text-color","anchor":"base_top"},{"color":"green","style":"text-color","anchor":"base_bottom"}]},"order":4,"title":"Add every surface","description":"Add the two ends, the curved face and the rectangular base. Keep the calculator value unrounded until the final line."},{"type":"text","order":5,"title":"Round the answer","content":"To $3$ significant figures, the first three significant digits are $3$, $2$ and $2$. The next digit is $8$, so round the third digit up:\n\n$$\\boxed{3230\\text{ cm}^2}$$\n\nThe unit must be squared because this is an area. This correct rounded answer earns **A1**.","highlights":[{"text":"Give your answer correct to 3 significant figures.","location":"specification"}]}]}],"generatedAt":"2026-07-19T15:54:00.643Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"743fef30-0f15-440e-8861-fcbfb64a8b9c:cross_section_first","label":"(a) Using Cross-Section Area","steps":[{"type":"text","order":0,"title":"Identify the cross-section shape","content":"A semi-cylinder is a 3D prism-like shape with a semi-circular cross-section. \n\nTo find its volume using the cross-section method, we first need to calculate the area of this semi-circular cross-section, and then multiply it by the length of the paperweight.\n\nFrom the question, we are given:\n* Radius ($r$) = $12\\text{ cm}$\n* Length ($L$) = $45\\text{ cm}$","highlights":[{"text":"radius of 12 cm","location":"specification"},{"text":"length of 45 cm","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Area of a full circle} = \\pi r^2$"},{"latex":"$$\\text{Area of a semi-circle} = \\frac{1}{2}\\pi \\htmlData{anchor=r-val}{\\color{@sky}{r}}^2$"},{"latex":"$$\\phantom{\\text{Area of a semi-circle}} = \\frac{1}{2} \\times \\pi \\times \\htmlData{anchor=r-sub}{\\color{@sky}{12}}^2$"},{"latex":"$$\\phantom{\\text{Area of a semi-circle}} = 72\\pi \\approx 226.195\\text{ cm}^2$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"r-val"},{"color":"sky","style":"text-color","anchor":"r-sub"}],"connections":[{"to":"r-sub","from":"r-val","color":"sky","label":"substitute r","style":"solid"}]},"order":1,"title":"Calculate cross-section area","description":"We start by calculating the area of the semi-circular cross-section using the formula $A = \\frac{1}{2}\\pi r^2$. Awarding the first method mark (M1) is based on this calculation."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Volume} = \\htmlData{anchor=area-val}{\\color{@teal}{\\text{Cross-Section Area}}} \\times \\htmlData{anchor=len-val}{\\color{@orange}{\\text{Length}}}$"},{"latex":"$$\\phantom{\\text{Volume}} = \\htmlData{anchor=area-sub}{\\color{@teal}{72\\pi}} \\times \\htmlData{anchor=len-sub}{\\color{@orange}{45}}$"},{"latex":"$$\\phantom{\\text{Volume}} = 3240\\pi \\approx 10178.76\\text{ cm}^3$"}],"highlights":[{"color":"teal","style":"text-color","anchor":"area-val"},{"color":"teal","style":"text-color","anchor":"area-sub"},{"color":"orange","style":"text-color","anchor":"len-val"},{"color":"orange","style":"text-color","anchor":"len-sub"}],"connections":[{"to":"area-sub","from":"area-val","color":"teal","label":null,"style":"solid"},{"to":"len-sub","from":"len-val","color":"orange","label":null,"style":"solid"}]},"order":2,"title":"Calculate the volume","description":"Next, we multiply our cross-section area by the length of the paperweight ($45\\text{ cm}$) to find the total volume. Doing this earns the second method mark (M1)."},{"type":"text","order":3,"title":"Use calculator for precision","content":"To get the exact numerical value before rounding, perform the multiplication on a calculator using the $\\pi$ key to maintain accuracy.","calculator":[{"gdc":{"expression":"0.5 * π * 12^2 * 45","instructions":"Type `0.5 * π * 12^2 * 45` and press EXE."},"ti84":{"expression":"0.5 * π * 12^2 * 45","instructions":"Type `0.5 * π * 12^2 * 45` and press ENTER."},"desmos":{"expression":"0.5 * pi * 12^2 * 45","instructions":"Enter `0.5 * pi * 12^2 * 45` into the input bar."},"description":"Calculate the volume of the semi-cylinder on your graphing calculator."}]},{"type":"text","order":4,"title":"Round to significant figures","content":"Now we round our calculated value, $10178.76\\dots\\text{ cm}^3$, to $3$ significant figures:\n\n* The first three non-zero digits are $1$, $0$, and $1$ (yielding $10100$).\n* The fourth digit is $7$, which rounds the preceding $1$ up to $2$.\n\nTherefore, to 3 significant figures, the volume is:\n\n$$\\text{Volume} \\approx 10200\\text{ cm}^3$$\n\nThis final accuracy step awards the final mark (A1).","highlights":[{"text":"correct to 3 significant figures","location":"specification"}]}]},{"id":"743fef30-0f15-440e-8861-fcbfb64a8b9c:half_of_cylinder","label":"(a) Halving a Full Cylinder","steps":[{"type":"text","order":0,"title":"Identify given values","content":"First, let's identify the key measurements of the shape given in the question:\n- Radius of the circular face ($r$) = $12$ cm\n- Length (or height) of the cylinder ($h$) = $45$ cm\n\nWe need to find the volume of the semi-cylinder paperweight, which is exactly half of a full cylinder with these same dimensions.","highlights":[{"text":"radius of 12 cm","location":"specification"},{"text":"length of 45 cm","location":"specification"}]},{"type":"text","order":1,"title":"Find full cylinder volume","content":"We calculate the volume of a full cylinder using the standard formula:\n\n$$\\text{Volume of full cylinder} = \\pi r^2 h$$\n\nSubstituting our values in:\n\n$$\\begin{aligned} \\text{Volume of full cylinder} &= \\pi \\times 12^2 \\times 45 \\\\ &= \\pi \\times 144 \\times 45 \\\\ &= 6480\\pi \\text{ cm}^3 \\end{aligned}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Volume of semi-cylinder} = \\frac{\\htmlData{anchor=full-vol}{\\color{@sky}{6480\\pi}}}{2}$"},{"latex":"$$\\phantom{\\text{Volume of semi-cylinder}} = \\htmlData{anchor=half-vol}{\\color{@amber}{3240\\pi}} \\text{ cm}^3$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"full-vol"},{"color":"amber","style":"text-color","anchor":"half-vol"}],"connections":[{"to":"half-vol","from":"full-vol","color":"sky","label":"divide by 2","style":"solid"}]},"order":2,"title":"Halve the volume","description":"Since a semi-cylinder is exactly half of a full cylinder, we divide the full volume by 2."},{"type":"text","order":3,"title":"Calculate decimal value","content":"Now we evaluate $3240\\pi$ using a calculator:\n\n$$3240 \\times \\pi \\approx 10178.7602... \\text{ cm}^3$$","calculator":[{"gdc":{"expression":"3240 * π","instructions":"Type 3240 followed by the pi key (usually shift or option depending on your model) and press EXE."},"ti84":{"expression":"3240 * π","instructions":"Type 3240 followed by the [2nd] key and then [^] (which gives the π symbol), then press [ENTER]."},"desmos":{"expression":"3240\\pi","instructions":"Type 3240pi in the input cell."},"description":"To find the exact value of $3240\\pi$:"}]},{"type":"text","order":4,"title":"Round to 3 s.f.","content":"The question asks for the answer to be correct to **3 significant figures**.\n\nLooking at our value $10\\underline{1}78.76...$, the third significant figure is $1$ in the hundreds place. The digit to its right is $7$, which means we round up:\n\n$$10200 \\text{ cm}^3$$","highlights":[{"text":"correct to 3 significant figures","location":"specification"}]}]},{"id":"4e1bb35e-4b4b-4d4c-bb5f-99de1d4955d7:component_summation","label":"(b) Component Summation","steps":[{"type":"text","order":0,"title":"Identify the components","content":"To find the total surface area of the semi-cylinder paperweight using the **Component Summation** approach, we need to calculate the area of each individual surface:\n\n1. **Two semi-circular ends**: Together, these make one complete circle with a radius of $12\\text{ cm}$.\n2. **Curved surface**: This is half of the curved surface of a full cylinder.\n3. **Flat rectangular base**: This is a rectangle with dimensions equal to the diameter of the cylinder and the length of the paperweight.\n\nLet's calculate each of these components step-by-step.","highlights":[{"text":"semi-cylinder with a radius of 12 cm and a length of 45 cm","location":"specification"}]},{"type":"text","order":1,"title":"Two semi-circular ends","content":"The two semi-circular ends can be combined to form one complete circle with radius $r = 12\\text{ cm}$.\n\nThe area of a full circle is given by:\n$$\\text{Area} = \\pi r^2$$\n\nSubstituting the radius $r = 12$:\n$$\\text{Area of ends} = \\pi \\times 12^2 = 144\\pi \\approx 452.39\\text{ cm}^2$$\n\n*(This step earns the first method mark **M1**)*"},{"type":"text","order":2,"title":"Curved surface area","content":"The curved surface is half of a full cylinder's curved surface area ($2\\pi r l$).\n\nTherefore, the curved surface area is:\n$$\\text{Curved Area} = \\pi r l$$\n\nUsing radius $r = 12\\text{ cm}$ and length $l = 45\\text{ cm}$:\n$$\\text{Curved Area} = \\pi \\times 12 \\times 45 = 540\\pi \\approx 1696.46\\text{ cm}^2$$"},{"type":"text","order":3,"title":"Flat rectangular base","content":"The flat base is a rectangle. \n- One side is the diameter of the semi-cylinder: $d = 2 \\times 12 = 24\\text{ cm}$.\n- The other side is the length: $l = 45\\text{ cm}$.\n\n$$\\text{Base Area} = 24 \\times 45 = 1080\\text{ cm}^2$$\n\n*(Calculating the curved surface and the rectangular base earns the second method mark **M1**)*"},{"type":"text","order":4,"title":"Sum the components","content":"Now we add the areas of all components together to find the total surface area:\n\n$$\\text{Total Surface Area} = 144\\pi + 540\\pi + 1080$$\n$$\\text{Total Surface Area} = 684\\pi + 1080 \\approx 2148.85 + 1080 = 3228.85\\text{ cm}^2$$","calculator":[{"gdc":{"expression":"144*π + 540*π + 1080","instructions":"Enter the complete sum expression into the run menu:"},"ti84":{"expression":"144*π + 540*π + 1080","instructions":"Type the expression into the main screen and press ENTER:"},"desmos":{"expression":"144\\pi + 540\\pi + 1080","instructions":"Type the exact sum expression to obtain the decimal value:"},"description":"Calculate the total surface area on your calculator for precision."}]},{"type":"text","order":5,"title":"Round to 3 s.f.","content":"We need to round our final answer to **3 significant figures**:\n\n- The value is $3228.85...$\n- The third significant digit is the first $2$ in the tens place ($32\\underline{2}8.85$)\n- Since the following digit is $8$, we round up to get:\n\n$$\\text{Total Surface Area} \\approx 3230\\text{ cm}^2$$\n\n*(This correct value earns the final accuracy mark **A1**)*","highlights":[{"text":"Give your answer correct to 3 significant figures.","location":"specification"}]}]},{"id":"4e1bb35e-4b4b-4d4c-bb5f-99de1d4955d7:single_formula","label":"(b) Single Formula Approach","steps":[{"type":"text","order":0,"title":"Identify the formula","content":"To find the total surface area ($A$) of a semi-cylinder in one step, we use the consolidated formula:\n\n$$A = \\pi r^2 + \\pi r h + 2rh$$\n\nThis single formula combines all three parts of the surface area:\n- The two semi-circular ends: $2 \\times \\frac{1}{2}\\pi r^2 = \\pi r^2$\n- The curved surface: $\\frac{1}{2}(2\\pi r h) = \\pi r h$\n- The rectangular base: $2r \\times h = 2rh$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Radius } \\htmlData{anchor=r-val}{\\color{@sky}{r = 12}} \\text{ cm}, \\quad \\text{Length } \\htmlData{anchor=h-val}{\\color{@amber}{h = 45}} \\text{ cm}$"},{"latex":"$$A = \\pi \\htmlData{anchor=r-sub1}{\\color{@sky}{r}}^2 + \\pi \\htmlData{anchor=r-sub2}{\\color{@sky}{r}} \\htmlData{anchor=h-sub1}{\\color{@amber}{h}} + 2\\htmlData{anchor=r-sub3}{\\color{@sky}{r}} \\htmlData{anchor=h-sub2}{\\color{@amber}{h}}$"},{"latex":"$$A = \\pi (\\color{@sky}{12})^2 + \\pi (\\color{@sky}{12})(\\color{@amber}{45}) + 2(\\color{@sky}{12})(\\color{@amber}{45})$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"r-val"},{"color":"amber","style":"text-color","anchor":"h-val"}],"connections":[{"to":"r-sub1","from":"r-val","color":"sky","label":"r = 12","style":"solid"},{"to":"h-sub1","from":"h-val","color":"amber","label":"h = 45","style":"solid"}]},"order":1,"title":"Substitute the values","description":"We substitute the given radius $r = 12$ cm and length $h = 45$ cm directly into our formula."},{"type":"text","order":2,"title":"Calculate the area","content":"Now we simplify the calculation:\n\n$$\\begin{aligned} A &= 144\\pi + 540\\pi + 1080 \\\\ A &= 684\\pi + 1080 \\end{aligned}$$\n\nUsing a calculator to compute the exact value:\n\n$$A \\approx 2148.85 + 1080 = 3228.85...\\text{ cm}^2$$","calculator":[{"gdc":{"expression":"π * 12^2 + π * 12 * 45 + 2 * 12 * 45","instructions":"Type `π * 12^2 + π * 12 * 45 + 2 * 12 * 45` and press [EXE]."},"ti84":{"expression":"π * 12^2 + π * 12 * 45 + 2 * 12 * 45","instructions":"Type `π * 12^2 + π * 12 * 45 + 2 * 12 * 45` and press [ENTER]."},"desmos":{"expression":"\\pi \\cdot 12^2 + \\pi \\cdot 12 \\cdot 45 + 2 \\cdot 12 \\cdot 45","instructions":"Enter the expression into the input bar to get the decimal value."},"description":"Calculate the total surface area by entering the expression directly."}]},{"type":"text","order":3,"title":"Round the final answer","content":"Rounding $3228.85...$ to **3 significant figures**:\n\n$$A \\approx 3230 \\text{ cm}^2$$\n\n- The first three significant figures are $3$, $2$, and $2$.\n- The fourth figure is $8$, so we round up the third digit to $3$, giving $3230$."}]}],"generatedAt":"2026-06-12T23:56:41.342Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"gcse_maths__ccea","summaries":[{"id":"ad17fa2e-ebe8-41e9-b047-d53318d71589","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"cf7f70db-36a0-4554-89fe-5e8087edb5e9","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4a47ec50-257d-4ea9-918e-2a6072cf2c57","difficultyBand":"medium","calculatorUse":"calc","totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"54809ed6-85dc-453a-955d-c239bfd5d727","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5b3bebff-9b11-4351-9171-d0f1a89d56ae","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6a025a1f-fc62-4225-957e-24cc8dc9c9c4","difficultyBand":"medium","calculatorUse":"calc","totalMarks":6,"questionType":"LA","isCalculation":true},{"id":"73bf7a1a-3b29-4a77-ab98-562a56ac00e5","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"76083d0f-835a-4164-8b80-acd6399fa86e","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"78fe0fa5-8530-4882-bb17-37f311da5653","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"8363f95c-dfdb-4aa9-be39-9d170ad31873","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"96dd2974-7886-4136-b7ef-12fdf90e104d","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"9b41d421-d169-432e-822c-63a434f1f261","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"a331347a-62e4-4c5a-b528-a978dd75768d","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a40fed13-3b88-49bf-9771-195ab506d71f","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a5fc9d92-6dc6-4cf2-9c66-feee962b7da9","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a9760292-c939-46f3-affe-7f0cf462db6e","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bb35f599-afdd-4bfe-9b9e-41617bbea997","difficultyBand":"hard","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"be47c9c0-ca4b-4eaf-8820-c2b69f50571e","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d16ffeca-0740-47d7-a9f0-217f885e76fc","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d81e8b52-52e6-4953-b653-8c0b91bdbdf0","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"dc111005-2cc6-49ed-956c-c511db10af51","difficultyBand":"medium","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"ddb4e851-7719-4f5b-8595-9be2eab9b9d0","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"e4069437-3658-40b3-822f-73689baab712","difficultyBand":"hard","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f4d766e8-3a39-484b-ba7a-c3450b9dcf71","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f866d7ef-1c11-439f-a13c-716766e7e44f","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0383a797-1df6-48fa-a5bf-ec340aebab2f","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6970993f-e219-4fe2-bcba-32ba62bcff04","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"08240041-8ac0-47e4-877e-8e9f5e185b64","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"0e7dcdc9-6e48-490f-a305-a56d7586f19b","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1502390c-17f5-48dc-9ac1-0a72e58e7e2a","difficultyBand":"hard","calculatorUse":"calc","totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1c667eff-0f12-4829-832f-b5330fda2a2a","difficultyBand":"medium","calculatorUse":"calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1c972d7d-4879-4848-816a-58e78d74a072","difficultyBand":"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Cylinders

Question 6

Cylinders Questions

Cylinders

Question 6

Cylinders Questions