Grah. Headache brain drawing blank. thinking point of intersection of a line and circle but no idea how to do it. here’s question.
The longest crude oil pipeline in the world is the one from Edmonton, Alberta, to Buffalo, New York, a distance of 2858 km. In one region, the pipeline follows the path given by y=2x+20, where one unit on the grid represents 1km. A town in that region is centered at (50,5) and has radius 5km. New by laws require that the pipeline not be within 50km of an urban area.
A) was to find how far from city it was and I did this easily to get 46.something but
b) Does the pipeline need to be rerouted?(yes) If so, what is the mimimum length of the existing pipeline that must be rerouted? Show all your work. Explain and justify your reasoning.
Now I tried to use points to get a quadractic equation for the top half of the circle formed by the 50km radius around the city. points (50,60) and (-5,5). but this gave me a quadratic equation that didn’t make a semi circle. I was going to use this with poi formula to find 2 points I could use with the length equation to find the amount of pipeline but I’m stuck now.
There are no semicircles in this question, really, unless you make one up randomly. The arc that is formed when the pipeline crosses the 55km-radius circle doesn’t matter. What you need is the coordinates of those two intersecting points, and calculate their distance (note how the question says the minimum length of the <i>existing</i> pipeline that must be rerouted).
- Get the equation of the circle such that it is centered at (50,5) and has a radius of 5+50.
<i>(x - h)^2 + (y - k)^2 = r^2</i>
- Plug the equation y = 2x + 20 into the equation of the circle and get the points of intersection at which the line crosses the circle.
<i>y = 2x + 20, replace y in circle equation</i>
- Use those coordinates to calculate the distance between those two points of intersection
<i>Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2]</i>
- Done!
(your answer should be approximately 39.9872)
Yeah. We haven’t done much in the equation of circles so pretty much what I was thinking of doing. But how’d you get the equation of the circle?
The locus definition of a circle is a locus of points such that they are equidistant from a point (the center). Therefore, the pythagorean formula can be used (a^2 + b^2 = c^2). Since we’re working on the Cartesian plane, x and y are used instead of a and b, and r is used instead of c (r is the hypotenuse). Basically a circle centered on the origin would look like this:
<i>x^2 + y^2 = r^2</i>
However, what if the center is not (0,0)? Then we just use the distance formula sqrt[(x2 - x1)^2 + (y2 - y1)^2] = d to take into account that the origin may not be at (0,0).
So we get:
<i>(x - h)^2 + (y - k)^2 = r^2</i>
where (h,k) are the coordinates of the origin and r is the radius.
ooh. something I’ve never seen afore popped up.
((2x+20)-5)^2 do I have to go (2x+20)(2x+20)-5^2 or jsut drop the brackets or soemthing else?
[(2x + 20) - 5]^2
= [2x + 20 - 5]^2
Do the brackets first.
thankies and I got the same answer as you so THANK YOU VERY MUCH. why must our teacher make us figure out the equation of circles when origin is not 0,0 sheesh. THe fact I couldn’t remember what it was when y was 0,0 didn’t help much though.