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Tangent through a point

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A portion of a river has the shape of the equation y=1-x^2/4, where distances are measured in tens of kilometres, and the positive y-axis represents due north. the town of Coopers Crossing is situated on the river at its most northerly point. The town of Black Stump is 10 kilometres due south of Coopers Crossing. the town of Andrewsville is 25 kilometres east of Black Stump. It is planned to build another town, Macquarie Flats, due north of Coopers Crossing and a road will be built connecting Macquarie Flats to Andrewsville. It has been stipulated that this road must be in a straight line and must not cross the river. What is the shortest length of road (to the nearest kilometre)?

(HINT- Find the co-ordinates of the points representing Coopers Crossing, Black Stump and Andrewsville and draw a diagram. Remember that 1 unit represents 10 kilometres. The solution has something to do with finding a tangent through a certain point.)

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This shows how to find shortest length of road by computing tangent through a point.

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A portion of a river has the shape of the equation y=1-x^2/4, where distances are measured in tens of kilometres, and the positive y-axis represents due north. the town of Coopers Crossing is situated on the river at its most northerly point. The town of Black Stump is 10 kilometres due south of Coopers Crossing. the town of Andrewsville is 25 kilometres east of Black Stump. It is planned to build another town, Macquarie Flats, due north of Coopers Crossing and a road will be built connecting Macquarie Flats to ...

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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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