As shown in the above diagram, a person starts from the centre O of a circular path AB, walks along the line indicated by arrows and returns to the same point. If the radius OA = OB = 100 metres, what is the total distance walked to the nearest metres?<\/p>\n
[amp_mcq option1=”703″ option2=”723″ option3=”743″ option4=”823″ correct=”option2″]<\/p>\n
The total distance walked would be the sum of the lengths of the segments OA, Arc AB, and BO.
\nDistance OA = radius = 100 m.
\nDistance BO = radius = 100 m.
\nDistance = 100 + Length of Arc AB + 100 = 200 + Length of Arc AB.<\/p>\n
The length of an arc AB is given by $L = r \\times \\theta$, where $r$ is the radius and $\\theta$ is the central angle AOB in radians. We need to determine the angle AOB from the context or options.
\nThe options for total distance are 703, 723, 743, 823 m.
\nThis implies the length of Arc AB is approximately:
\n703 – 200 = 503 m
\n723 – 200 = 523 m
\n743 – 200 = 543 m
\n823 – 200 = 623 m<\/p>\n
Let’s check if any common central angle results in an arc length close to these values when the radius is 100m.
\nIf the central angle is $\\theta$ radians, Arc length = $100\\theta$.
\nIf $\\theta = 503\/100 = 5.03$ rad $\\approx 288$ deg.
\nIf $\\theta = 523\/100 = 5.23$ rad $\\approx 300$ deg (using $\\pi \\approx 3.1416$, $5.23 \\times 180\/\\pi \\approx 299.57$ deg).
\nIf $\\theta = 543\/100 = 5.43$ rad $\\approx 311$ deg.
\nIf $\\theta = 623\/100 = 6.23$ rad $\\approx 357$ deg.<\/p>\n
A central angle of 300 degrees is a plausible value in geometry problems (e.g., a circle minus a 60-degree sector).
\nIf the central angle AOB is 300 degrees, which is $300 \\times \\frac{\\pi}{180} = \\frac{5\\pi}{3}$ radians.
\nArc length AB = $100 \\times \\frac{5\\pi}{3} = \\frac{500\\pi}{3}$ metres.
\nUsing $\\pi \\approx 3.14159$: Arc length $\\approx \\frac{500 \\times 3.14159}{3} \\approx \\frac{1570.795}{3} \\approx 523.598$ metres.
\nTotal distance = $100 + 523.598 + 100 = 723.598$ metres.
\nRounded to the nearest metre, this is 724 m.<\/p>\n
Let’s try a different approximation for $\\pi$, like $\\pi \\approx 22\/7$.
\nArc length $\\approx \\frac{500}{3} \\times \\frac{22}{7} = \\frac{11000}{21} \\approx 523.81$ metres.
\nTotal distance = $100 + 523.81 + 100 = 723.81$ metres. Rounded to 724m.<\/p>\n