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volume formula."},{"text":"Double it to find the radius.","correct":false,"explanation":"Doubling confuses the relationship between radius and diameter. The diameter is twice the radius, so it must be halved to find the radius."},{"text":"Square it to find the radius.","correct":false,"explanation":"The radius is found before any squaring is performed. Halve the diameter first; the radius is then squared when using $V = \\pi r^2h$."},{"text":"Quarter it to find the radius.","correct":false,"explanation":"Dividing by 4 makes the measurement half the actual radius. Since the diameter equals two radii, it must be divided by 2."}]},{"id":"c96f7d28-91a9-ad29-ac56-4b8a19a2fb9d","cardType":"basic","points":1,"front":"State the formula for the volume of a cylinder.","back":"$$V = \\pi r^2 h$","mcqOptions":[{"text":"$$V = \\pi r^2 h$","correct":true,"explanation":"A cylinder’s volume is the area of its circular base, $\\pi r^2$, multiplied by its perpendicular height, $h$."},{"text":"$$V = 2\\pi rh$","correct":false,"explanation":"This calculates the curved surface area rather than volume. Volume requires the circular base area multiplied by the height, giving $V = \\pi r^2h$."},{"text":"$$V = \\pi d^2 h$","correct":false,"explanation":"Using the diameter in place of the radius makes the result four times too large. Halve the diameter first, then use $V = \\pi r^2h$."},{"text":"$$V = \\frac{1}{3}\\pi r^2 h$","correct":false,"explanation":"The factor $\\frac{1}{3}$ belongs to the volume formula for a cone. A cylinder has constant cross-section, so its volume is $V = \\pi r^2h$."}]},{"id":"97cd8e5b-4cb5-8a3a-087c-e0b5a643ac80","cardType":"cloze","points":null,"front":"The units for volume are [ ] units, such as $\\text{cm}^3$ or $\\text{m}^3$.","back":"The units for volume are cubic units, such as $\\text{cm}^3$ or $\\text{m}^3$.","mcqOptions":[{"text":"cubic","correct":true,"explanation":"Volume measures three-dimensional space, so its units are raised to the third power and are described as cubic units."},{"text":"square","correct":false,"explanation":"Square units measure two-dimensional area, not three-dimensional volume. Volume is measured in cubic units because three dimensions are multiplied together."},{"text":"linear","correct":false,"explanation":"Linear units measure one-dimensional lengths such as height or radius. Volume combines three dimensions, so the correct description is cubic units."},{"text":"fourth-power","correct":false,"explanation":"Volume involves three spatial dimensions, not four, so its units have an exponent of 3. The correct description is cubic units."}]},{"id":"2e6d0e50-7629-5dba-e992-99bb9cebf9b3","cardType":"cloze","points":null,"front":"The units for volume are cubic units, such as $\\text{[...]}$ or $\\text{m}^3$.","back":"The units for volume are cubic units, such as $\\boxed{\\text{cm}^3}$ or $\\text{m}^3$.","mcqOptions":[{"text":"$$\\text{cm}^3$","correct":true,"explanation":"Centimetres cubed are valid volume units because volume combines three length dimensions, giving an exponent of 3."},{"text":"$$\\text{cm}^2$","correct":false,"explanation":"Centimetres squared measure area, which has two dimensions. Volume has three dimensions, so the correct unit is $\\text{cm}^3$."},{"text":"$$\\text{cm}$","correct":false,"explanation":"Centimetres alone measure length rather than volume. A volume measured using centimetres must be written as $\\text{cm}^3$."},{"text":"$$\\text{cm}^4$","correct":false,"explanation":"An exponent of 4 does not represent ordinary three-dimensional volume. Multiplying three centimetre measurements produces the correct unit $\\text{cm}^3$."}]},{"id":"a1471e87-98d9-dc47-4c88-0d9cab7cca32","cardType":"basic","points":1,"front":"In the formula $V = \\pi r^2 h$, what does the term $\\pi r^2$ represent?","back":"The area of the circular base.","mcqOptions":[{"text":"The area of the circular base.","correct":true,"explanation":"The area of a circle is $\\pi r^2$, so this part of the formula gives the cylinder’s base area before it is multiplied by the height."},{"text":"The circumference of the circular base.","correct":false,"explanation":"The circumference of the base is $2\\pi r$, not $\\pi r^2$. The term $\\pi r^2$ represents the area of the circular base."},{"text":"The diameter of the circular base.","correct":false,"explanation":"The diameter is simply $2r$ and does not involve $\\pi$ or squaring. The term $\\pi r^2$ gives the area of the circular base."},{"text":"The volume of the circular base.","correct":false,"explanation":"A circular base is two-dimensional, so it has area rather than volume. Its area is $\\pi r^2$, which is then multiplied by the cylinder’s height."}]},{"id":"557dd27a-366d-1fbc-d5e5-8bb1b8b63fc5","cardType":"basic","points":1,"front":"What does it mean to give an answer \"in terms of $\\pi$\"?","back":"Do not calculate a decimal; leave the symbol $\\pi$ in the final result.","mcqOptions":[{"text":"Do not calculate a decimal; leave the symbol $\\pi$ in the final result.","correct":true,"explanation":"An exact answer in terms of $\\pi$ keeps the symbol rather than replacing it with a decimal approximation such as 3.14."},{"text":"Calculate a decimal; replace the symbol $\\pi$ with 3.14 in the final result.","correct":false,"explanation":"Replacing $\\pi$ with 3.14 produces an approximate decimal answer. In terms of $\\pi$ means keeping the symbol in the exact final result."},{"text":"Calculate a decimal; round the symbol $\\pi$ to 3 in the final result.","correct":false,"explanation":"Rounding $\\pi$ to 3 gives a particularly inaccurate approximation. An answer in terms of $\\pi$ must retain the symbol and avoid decimal evaluation."},{"text":"Do not calculate a decimal; remove the symbol $\\pi$ from the final result.","correct":false,"explanation":"Removing $\\pi$ changes the value rather than preserving it exactly. An answer in terms of $\\pi$ must leave the symbol in the final result."}]},{"id":"00c07f5f-15ec-7f30-e255-dec45089286b","cardType":"basic","points":1,"front":"State the formula for the total surface area (TSA) of a solid closed cylinder.","back":"$$\\text{TSA} = 2\\pi r^2 + 2\\pi rh$","mcqOptions":[{"text":"$$\\text{TSA} = 2\\pi r^2 + 2\\pi rh$","correct":true,"explanation":"A closed cylinder has two circular ends with total area $2\\pi r^2$ and a curved surface with area $2\\pi rh$, so these are added."},{"text":"$$\\text{TSA} = \\pi r^2 + 2\\pi rh$","correct":false,"explanation":"This includes only one circular end, as if the cylinder were open. A solid closed cylinder has two ends, giving $2\\pi r^2 + 2\\pi rh$."},{"text":"$$\\text{TSA} = 2\\pi r^2 + \\pi rh$","correct":false,"explanation":"The curved surface is circumference multiplied by height, so it is $2\\pi r \\times h = 2\\pi rh$. The missing factor of 2 halves that area."},{"text":"$$\\text{TSA} = 2\\pi r^2h$","correct":false,"explanation":"Multiplying the end areas by the height confuses surface area with volume and gives cubic dimensions. The correct areas are added: $2\\pi r^2 + 2\\pi rh$."}]},{"id":"dca420d4-6ef6-7e45-a2c4-f8c60c0c18b4","cardType":"basic","points":1,"front":"When a cylinder's curved surface is \"unwrapped\", what shape does it form?","back":"A rectangle.","mcqOptions":[{"text":"A rectangle.","correct":true,"explanation":"The unwrapped width is the circular base’s circumference, $2\\pi r$, and the other dimension is the cylinder’s height, forming a rectangle."},{"text":"A triangle.","correct":false,"explanation":"A triangle does not have the constant width of a cylinder’s curved surface. Unwrapping that surface produces a rectangle with width $2\\pi r$."},{"text":"A circle.","correct":false,"explanation":"The two ends of a cylinder are circles, but the unwrapped curved surface is a rectangle whose width equals the base’s circumference."},{"text":"A semicircle.","correct":false,"explanation":"A semicircle is not the net of the curved surface. Cutting and flattening the curved surface produces a rectangle with height $h$ and width $2\\pi r$."}]},{"id":"41d8cd8d-b04b-966e-2f2f-591b87ff9beb","cardType":"basic","points":2,"front":"What are the dimensions of the rectangle formed by unwrapping a cylinder's curved surface?","back":"

Width: circumference ($2\\pi r$)
Height: cylinder height ($h$)

","mcqOptions":[{"text":"

Width: circumference ($2\\pi r$)
Height: cylinder height ($h$)

","correct":true,"explanation":"The curved surface unwraps into a rectangle whose width is the circular base’s circumference, $2\\pi r$, and whose height equals the cylinder’s perpendicular height, $h$."},{"text":"

Width: diameter ($2r$)
Height: cylinder height ($h$)

","correct":false,"explanation":"Using the diameter for the width ignores the distance around the circular base. The width is the circumference, $2\\pi r$, while the rectangle’s height is $h$."},{"text":"

Width: circumference ($2\\pi r$)
Height: cylinder radius ($r$)

","correct":false,"explanation":"The rectangle’s second dimension comes from the cylinder’s height, not its radius. Its dimensions are circumference $2\\pi r$ by cylinder height $h$."},{"text":"

Width: radius ($r$)
Height: cylinder diameter ($2r$)

","correct":false,"explanation":"The radius and diameter describe the circular ends, not the unwrapped curved surface’s dimensions. The rectangle is $2\\pi r$ wide and $h$ high."}]},{"id":"d64a4cca-18b8-3f40-2d48-50ab59fbae3e","cardType":"cloze","points":null,"front":"The surface area of a [ ] cylinder includes two circular ends and one curved surface.","back":"The surface area of a closed cylinder includes two circular ends and one curved surface.","mcqOptions":[{"text":"closed","correct":true,"explanation":"A closed cylinder has a circular face at each end, so its surface area includes two circles as well as the curved surface."},{"text":"open","correct":false,"explanation":"An open cylinder is missing at least one circular end, so it does not necessarily include two circular ends. The required term is closed."},{"text":"hollow","correct":false,"explanation":"A hollow cylinder usually has inner and outer curved surfaces and may describe a tube rather than this standard solid. A closed cylinder has the stated three surfaces."},{"text":"semi-circular","correct":false,"explanation":"A semi-cylindrical shape has semicircular ends and only half the original curved surface. The cylinder with two complete circular ends is closed."}]},{"id":"a187080c-5a17-ae6d-cb55-64a0a0980b7c","cardType":"cloze","points":null,"front":"The surface area of a closed cylinder includes [ ] circular ends and one curved surface.","back":"The surface area of a closed cylinder includes two circular ends and one curved surface.","mcqOptions":[{"text":"two","correct":true,"explanation":"A closed cylinder has one circular end at the top and another at the bottom, giving two circular ends in total."},{"text":"one","correct":false,"explanation":"Counting only one end is the common mistake made when treating the cylinder as open. A closed cylinder has two circular ends."},{"text":"three","correct":false,"explanation":"Three circles may be confused with the three surface components, but the curved surface is not circular. There are exactly two circular ends."},{"text":"zero","correct":false,"explanation":"Having no circular ends would describe an open tube consisting only of a curved surface. A closed cylinder has two circular ends."}]},{"id":"efa67d74-ade9-7782-651c-abaa6f4cffda","cardType":"basic","points":1,"front":"How do you calculate the volume of a semi-cylinder?","back":"Calculate the volume of a full cylinder and halve it ($V = \\frac{1}{2}\\pi r^2 L$).","mcqOptions":[{"text":"Calculate the volume of a full cylinder and halve it ($V = \\frac{1}{2}\\pi r^2 L$).","correct":true,"explanation":"A semi-cylinder is half of a full cylinder cut lengthways, so its volume is half of $\\pi r^2L$, giving $V=\\frac{1}{2}\\pi r^2L$."},{"text":"Calculate the volume of a full cylinder and double it ($V = 2\\pi r^2 L$).","correct":false,"explanation":"Doubling produces the volume of two full cylinders rather than half a cylinder. The full-cylinder volume must instead be halved to obtain $\\frac{1}{2}\\pi r^2L$."},{"text":"Calculate the volume of a full cylinder and quarter it ($V = \\frac{1}{4}\\pi r^2 L$).","correct":false,"explanation":"Quartering halves the full-cylinder volume twice, making the result too small by a factor of two. A semi-cylinder requires one halving, giving $\\frac{1}{2}\\pi r^2L$."},{"text":"Calculate the volume of a full cylinder and keep it ($V = \\pi r^2 L$).","correct":false,"explanation":"Leaving the volume unchanged calculates a complete cylinder, not a semi-cylinder. Since the cross-section is halved, the full-cylinder volume must also be halved."}]},{"id":"f7c00df0-7242-cb6d-884a-b5510569659e","cardType":"basic","points":3,"front":"Which three distinct parts make up the total surface area of a solid semi-cylinder?","back":"

Two semi-circular ends (making one full circle)
Half the curved surface
The flat rectangular face

","mcqOptions":[{"text":"

Two semi-circular ends (making one full circle)
Half the curved surface
The flat rectangular face

","correct":true,"explanation":"The two semicircular ends combine to one circle, the original curved surface is halved, and the lengthways cut creates one flat rectangular face."},{"text":"

Two circular ends (making two full circles)
Half the curved surface
The flat rectangular face

","correct":false,"explanation":"A semi-cylinder has semicircular ends, not complete circular ends. The two semicircles together make one full circle, alongside half the curved surface and one rectangle."},{"text":"

Two semi-circular ends (making one full circle)
All the curved surface
The flat rectangular face

","correct":false,"explanation":"Cutting a cylinder lengthways halves its curved surface, so including the full curved surface overcounts the area. The remaining parts are two semicircular ends and one rectangle."},{"text":"

Two semi-circular ends (making one full circle)
Half the curved surface
Two flat rectangular faces

","correct":false,"explanation":"A single lengthways cut exposes one rectangular face, not two. The complete surface consists of that rectangle, two semicircular ends and half the original curved surface."}]},{"id":"fcb59cea-c933-85e5-55fd-44e74d2fc487","cardType":"basic","points":2,"front":"For a semi-cylinder with radius $r$ and length $L$, what are the dimensions of the flat rectangular face?","back":"Diameter ($2r$) by length ($L$).","mcqOptions":[{"text":"Diameter ($2r$) by length ($L$).","correct":true,"explanation":"The flat face spans straight across the semicircular cross-section, giving width $2r$, and extends along the entire cylinder length $L$."},{"text":"Radius ($r$) by length ($L$).","correct":false,"explanation":"Using $r$ gives only half the rectangle’s required width. The cut runs across the full diameter $2r$ and along the length $L$."},{"text":"Circumference ($2\\pi r$) by length ($L$).","correct":false,"explanation":"The circumference measures around the curved surface rather than across the flat cut. The rectangular face measures diameter $2r$ by length $L$."},{"text":"Diameter ($2r$) by radius ($r$).","correct":false,"explanation":"The radius is not the dimension extending along the cylinder. The face spans the diameter $2r$ across the end and the length $L$ along it."}]},{"id":"eee04dbf-5d47-f527-88fd-e3a2de4c9091","cardType":"basic","points":1,"front":"What is the final step when solving for a missing radius $r$ if the formula yields $r^2 = 49$?","back":"Take the square root (e.g. $\\sqrt{49} = 7$).","mcqOptions":[{"text":"Take the square root (e.g. $\\sqrt{49} = 7$).","correct":true,"explanation":"Taking the square root reverses the squaring operation. Although $r^2=49$ has numerical roots $\\pm7$, a physical radius must be positive, so $r=7$."},{"text":"Take the square (e.g. $49^2 = 2401$).","correct":false,"explanation":"Squaring again applies the operation twice rather than reversing it, producing an excessively large value. Taking the square root gives the positive radius $r=7$."},{"text":"Divide by two (e.g. $49 \\div 2 = 24.5$).","correct":false,"explanation":"Dividing by two does not undo squaring; it confuses a power with multiplication by two. Taking the square root of $49$ gives the radius $7$."},{"text":"Take the cube root (e.g. $\\sqrt[3]{49} \\approx 3.66$).","correct":false,"explanation":"A cube root reverses cubing, not squaring, so it cannot isolate $r$ from $r^2$. The required inverse operation is the square root, giving $7$."}]},{"id":"b4af85a5-b3f5-573b-9093-cff65a844ffa","cardType":"basic","points":1,"front":"How many $\\text{cm}^3$ are in 1 litre?","back":"1000 $\\text{cm}^3$","mcqOptions":[{"text":"1000 $\\text{cm}^3$","correct":true,"explanation":"One litre equals one cubic decimetre, and $1\\text{ dm}=10\\text{ cm}$. Cubing the conversion gives $10^3=1000$, so one litre is $1000\\text{ cm}^3$."},{"text":"100 $\\text{cm}^3$","correct":false,"explanation":"This uses a two-dimensional conversion factor, as though only two lengths were multiplied. Volume requires cubing the factor of $10$, so the answer is $1000\\text{ cm}^3$."},{"text":"10 $\\text{cm}^3$","correct":false,"explanation":"This converts only one length from decimetres to centimetres and ignores that volume has three dimensions. Cubing $10$ gives $1000\\text{ cm}^3$."},{"text":"10000 $\\text{cm}^3$","correct":false,"explanation":"This adds an unnecessary factor of ten to the cubic conversion. Since $1\\text{ dm}^3=(10\\text{ cm})^3$, one litre contains $1000\\text{ cm}^3$."}]},{"id":"013854c9-2af7-cdab-3729-b150d28ff3d9","cardType":"cloze","points":null,"front":"When converting area units, $1 \\text{ m}^2 = \\text{[...]} \\text{ cm}^2$.","back":"When converting area units, $1 \\text{ m}^2 = \\boxed{10,000} \\text{ cm}^2$.","mcqOptions":[{"text":"$$10,000$","correct":true,"explanation":"Area conversion factors must be squared. Since $1\\text{ m}=100\\text{ cm}$, it follows that $1\\text{ m}^2=100^2\\text{ cm}^2=10,000\\text{ cm}^2$."},{"text":"$$100$","correct":false,"explanation":"This uses the linear conversion factor without squaring it. Since an area has two dimensions, $100$ must be squared, giving $1\\text{ m}^2=10,000\\text{ cm}^2$."},{"text":"$$1,000$","correct":false,"explanation":"This incorrectly multiplies the linear conversion factor by $10$. Area conversion requires $100^2$, so $1\\text{ m}^2$ is equal to $10,000\\text{ cm}^2$."},{"text":"$$1,000,000$","correct":false,"explanation":"This applies the conversion factor three times, as if converting cubic units. Area units require $100^2$, so the correct conversion is $10,000\\text{ cm}^2$."}]},{"id":"2e92a597-8193-4d69-1282-68de78c7c02e","cardType":"basic","points":1,"front":"If a problem asks about \"filling\" a tank, which property should you calculate?","back":"Volume.","mcqOptions":[{"text":"Volume.","correct":true,"explanation":"Filling concerns the amount of three-dimensional space inside the tank, which is measured by its volume, usually in cubic units or capacity units."},{"text":"Surface area.","correct":false,"explanation":"Surface area measures the material covering the tank’s surfaces, not the space available inside it. Filling a tank therefore requires calculating its volume."},{"text":"Circumference.","correct":false,"explanation":"Circumference gives the distance around a circular cross-section and cannot measure how much the tank holds. The required property for filling is volume."},{"text":"Cross-sectional area.","correct":false,"explanation":"Cross-sectional area measures one flat slice of the tank. It must be combined with the tank’s length or height to obtain the required volume."}]},{"id":"6dafb532-84e2-0186-ffe6-a9d7b37fa32f","cardType":"basic","points":1,"front":"If a problem asks about \"painting\" a tank, which property should you calculate?","back":"Surface area.","mcqOptions":[{"text":"Surface area.","correct":true,"explanation":"Painting covers the exposed surfaces of the tank, so the amount of paint required depends on the total surface area rather than the space inside."},{"text":"Volume.","correct":false,"explanation":"Volume measures how much the tank can contain, which is relevant to filling it. Painting concerns the exposed surfaces, so surface area is required."},{"text":"Circumference.","correct":false,"explanation":"Circumference measures only the distance around a circular end. Painting the whole tank requires the areas of its surfaces, so calculate surface area."},{"text":"Cross-sectional area.","correct":false,"explanation":"Cross-sectional area describes one flat end or slice, not every surface being painted. The correct property is the tank’s total surface area."}]},{"id":"ec9963d2-39aa-4859-a5b1-754741d96dd2","cardType":"cloze","points":null,"front":"In real-life contexts like buying paint, if your calculation shows you need 3.2 pots, you must round up to [ ] pots.","back":"In real-life contexts like buying paint, if your calculation shows you need 3.2 pots, you must round up to 4 pots.","mcqOptions":[{"text":"4","correct":true,"explanation":"Paint pots must be bought as whole items, and three pots would not contain enough paint. Therefore, $3.2$ must be rounded up to $4$ pots."},{"text":"3","correct":false,"explanation":"Ordinary rounding gives $3$, but that would leave insufficient paint because the requirement exceeds three full pots. You must round up to $4$ pots."},{"text":"3.2","correct":false,"explanation":"This leaves the calculated amount unchanged, but shops normally sell only whole pots. To ensure enough paint is purchased, the answer must be $4$ pots."},{"text":"5","correct":false,"explanation":"Rounding up means choosing the next whole number, not adding an extra whole pot unnecessarily. The next integer above $3.2$ is $4$."}]},{"id":"cb33c1c6-ab96-d6a1-731c-5785bfbde6b6","cardType":"basic","points":1,"front":"What must be checked regarding units before substituting values into a cylinder formula?","back":"Ensure all measurements are in the same units.","mcqOptions":[{"text":"Ensure all measurements are in the same units.","correct":true,"explanation":"Cylinder formulas combine several measurements, so their units must be consistent before substitution. Otherwise, the calculated area or volume will have an invalid numerical value."},{"text":"Ensure all measurements are in the different units.","correct":false,"explanation":"Mixing units, such as centimetres and metres, makes direct substitution invalid. All measurements must first be converted so that they use the same units."},{"text":"Ensure all measurements are in the smallest units.","correct":false,"explanation":"Using the smallest available unit is unnecessary and may produce awkward numbers. The measurements may use any suitable unit, provided they all use the same units."},{"text":"Ensure all measurements are in the largest units.","correct":false,"explanation":"Using the largest available unit is not a mathematical requirement. Accuracy depends on converting every measurement into the same units before using the formula."}]},{"id":"348ccdc9-662c-fe7b-a19c-d7dc20da0bdc","cardType":"cloze","points":null,"front":"The distance all the way across a circle through the centre is the [ ], while the distance from the centre to the edge is the radius.","back":"The distance all the way across a circle through the centre is the diameter, while the distance from the centre to the edge is the radius.","mcqOptions":[{"text":"diameter","correct":true,"explanation":"The diameter is a straight line across the full width of a circle through its centre. It is twice the length of the radius."},{"text":"circumference","correct":false,"explanation":"The circumference is the curved distance around the outside of a circle, not a straight distance across it. The straight line through the centre is the diameter."},{"text":"radius","correct":false,"explanation":"The radius extends only from the centre to the circle’s edge, so it covers half the width. The full distance through the centre is the diameter."},{"text":"chord","correct":false,"explanation":"A chord joins two points on a circle but does not necessarily pass through the centre. The chord through the centre and across the full circle is the diameter."}]},{"id":"39e48b79-0bf4-fc56-6cd3-790c7ffa3b2b","cardType":"cloze","points":null,"front":"The distance all the way across a circle through the centre is the diameter, while the distance from the centre to the edge is the [ ].","back":"The distance all the way across a circle through the centre is the diameter, while the distance from the centre to the edge is the radius.","mcqOptions":[{"text":"radius","correct":true,"explanation":"The radius is the straight-line distance from the centre of a circle to its edge. Its length is half the circle’s diameter."},{"text":"diameter","correct":false,"explanation":"The diameter extends from one edge of the circle to the opposite edge through the centre. The distance from the centre to one edge is the radius."},{"text":"circumference","correct":false,"explanation":"The circumference is the curved distance around the outside of the circle. The straight distance from the centre to the edge is called the radius."},{"text":"chord","correct":false,"explanation":"A chord is a line segment joining two points on the circle’s edge. A segment joining the centre to one point on the edge is the radius."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Unit 1"},"heading":"Cylinders","title":"Cylinders Flashcards","topbar":{"backHref":"/gcse/maths/wjec/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"07ee8c90-ef36-4650-2983-88597a9d029b","walled":"$L3a"}],"$L3b","$L3c"]