":4,"difficultyBand":"medium","totalMarks":6,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"26ddb6cb-3526-4d50-947c-f80ca878e43e","label":null,"content":"Calculate the area of the region bounded by the arc $AB$ and the straight path $AB$ (the minor segment).\n\nGive your answer to three significant figures.","markscheme":"- Rearranges the circle equation by completing the square to find $r^2$: $(x - 7)^2 - 49 + (y + 4)^2 - 16 = 175$ M1\n- Identifies the radius squared as $r^2 = 175 + 49 + 16 = 240$ A1\n- Uses the formula for the area of a sector: $\\text{Area} = \\frac{1}{2}r^2\\theta = \\frac{1}{2} \\times 240 \\times 1.4 = 168$ M1\n- Uses the formula for the area of the triangle: $\\text{Area} = \\frac{1}{2}r^2\\sin\\theta = \\frac{1}{2} \\times 240 \\times \\sin(1.4) \\approx 118.25...$ M1\n- Subtracts the triangle area from the sector area to find the segment area: $168 - 118.25... = 49.746...$ M1\n- States the final answer as $49.7$ km$^2$ A1","marks":6,"order":0,"markingRubric":null,"explanation":{"methods":[{"id":"difference_of_areas","label":"Sector minus Triangle Area","steps":[{"type":"text","order":0,"title":"Analyze the problem","content":"We need to find the area of the minor segment bounded by the arc $AB$ and the chord $AB$.\n\nThe **Sector minus Triangle Area** method tells us that the area of a circular segment is found by subtracting the area of the triangle formed by the two radii and the chord from the area of the entire circular sector:\n\n$$\\text{Segment Area} = \\text{Sector Area} - \\text{Triangle Area}$$\n\nOur first step is to find the radius of the circle, $r$, or more specifically, the value of $r^2$. We can find this by rearranging the equation of the circle."},{"type":"text","order":1,"title":"Find radius squared","content":"The given equation of the circle is:\n\n$$x^2 + y^2 - 14x + 8y = 175$$\n\nTo find the radius, we complete the square for both the $x$ terms and the $y$ terms:\n\n$$\\begin{aligned} (x^2 - 14x) + (y^2 + 8y) &= 175 \\\\ \\left[(x - 7)^2 - 49\\right] + \\left[(y + 4)^2 - 16\\right] &= 175 \\\\ (x - 7)^2 + (y + 4)^2 - 65 &= 175 \\\\ (x - 7)^2 + (y + 4)^2 &= 175 + 65 \\\\ (x - 7)^2 + (y + 4)^2 &= 240 \\end{aligned}$$\n\nComparing this to the standard equation of a circle, $(x - h)^2 + (y - k)^2 = r^2$, we identify the radius squared as:\n\n$$r^2 = 240$$","highlights":[{"text":"x^2 + y^2 - 14x + 8y = 175","location":"specification"}]},{"type":"text","order":2,"title":"Calculate sector area","content":"The formula for the area of a sector when the angle $\\theta$ is in radians is:\n\n$$\\text{Sector Area} = \\frac{1}{2}r^2\\theta$$\n\nGiven that $\\theta = 1.4$ radians and $r^2 = 240$:\n\n$$\\text{Sector Area} = \\frac{1}{2} \\times 240 \\times 1.4 = 168 \\text{ km}^2$$","highlights":[{"text":"1.4","location":"specification"}]},{"type":"text","order":3,"title":"Calculate triangle area","content":"The formula for the area of the triangle formed by the two radii and the chord is:\n\n$$\\text{Triangle Area} = \\frac{1}{2}r^2\\sin\\theta$$\n\nMake sure your calculator is in **radian mode** before performing this calculation:\n\n$$\\text{Triangle Area} = \\frac{1}{2} \\times 240 \\times \\sin(1.4)$$\n$$\\text{Triangle Area} = 120 \\times \\sin(1.4) \\approx 118.254 \\text{ km}^2$$","calculator":[{"gdc":{"expression":"0.5 * 240 * sin(1.4)","instructions":"1. Set the calculator angle mode to Radians in Settings.\n2. In the scratchpad or calculation menu, type: 0.5 * 240 * sin(1.4)\n3. Press [ENTER]."},"ti84":{"expression":"0.5 * 240 * sin(1.4)","instructions":"1. Press [MODE] and ensure 'RADIAN' is highlighted, then press [ENTER].\n2. Press [2nd] [MODE] to return to the home screen.\n3. Type: 0.5 * 240 * sin(1.4) and press [ENTER]."},"desmos":{"expression":"0.5 * 240 * \\sin(1.4)","instructions":"1. Click the wrench icon in the top-right corner to open Graph Settings.\n2. Ensure 'Radians' is selected at the bottom.\n3. In an input cell, type: 0.5 * 240 * sin(1.4)"},"description":"Calcul

Radians

Question 12

Radians Questions

Radians

Question 12

Radians Questions