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.)

Solution Summary

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

... the radius of this circle, and find the point-slope and slope-intercept forms of the equation of the line which is tangent to this circle at the point (5, -1 ...

... 9) Consider . a) Find an equation of the tangent line, T, at the point (2,8). b) Graph f and T on the same coordinate axes using a graphing calculator. ...

... the tangents, and use this along with the points given to find the equations of the tangents at those points. Then we use the slope of the tangent to calculate ...

... A tangent to a curve at a point is a ... a curve at the point without intersecting it (at the point). ... this notion more precise by considering tangents as limiting ...

... two segments tangent to a circle from a point outside the circle are congruent. That is, TP = TQ. Let's work out problems related to the chords and tangents to ...

... Hence the slope of the tangent at the point (0, 1) is sin0=0 which matches with our observation. ... At the point (0, 1) if the tangent is drawn, we see. ...

... What does it mean for the lines to be tangent to the circle? 1) they touch the circle at only one point 2) the derivative for the circle at that point has the ...

... Therefore, the equation of the tangent at the point (-5, 130) to the line y = -5 -5x2 is given as: (y-130) = y'(-5) {x-(-5)} Or, (y-130) = {(-10)(-5)}{x-(-5 ...