Ok, I got help [from outside MHF] for the last 2 problems but I still don't get the 1st one...
can anyone please help?
I see the graphs of these, using Geogebra, and I don't know how I can prove them..
1) Point A is @ (0,2) and point B is on the graph of y=x^2 + 1, what should be the approximate location of B so that the distance from A to B is as short as possible?
2) Point A is @ (3,1) and B is on a circle with center (1,2) and radius of 4. What is the closest point B should come to point A? Approximate to nearest thousandth.
And I'm having difficulty with this problem as well, but it doesn't use the same ideology of the above two.
A window is being built such that the bottom is a rectangle and the top is a semicircle. If there are 12 meter of framing material what should be the width of the window in order to let in the most light?
Any help is appreciated...
Let be the distance from (0,2) to the curve.
Now if you remember your distance formula, ( ), we have this relationship:
Now, we can rearrange the equation of the curve to get:
So our relationship becomes:
Differentiate both sides and solve for when the derivative is equal to 0 (as this will be the point where you have a minimum).
Two things to note:
(1) We are working in terms of . So whatever your answer will be, it will give you the y-coordinate in which the distance is shortest from (0,2) to the curve. Plug this in to your equation of the curve to get the x-coordinates.
(2) is a function of y. So you must implicitly differentiate.
Ok, so I solved for y as d was 0 on the left side
I got y = 3/2 + sq. root ( - 3/4), assuming I ignored the negative root, y equaled 2.36
and x is 1.16....
I looked at a graph and the shortest distance is supposed to be 0.87, and the coordinates for B are (0.61, 1.38) to (0.79, 1.62)
Are you sure you solved it correctly?
You should get: ..... i.e.,
Yes, you can use the fact that the shortest distance from the point to a curve is when the line segment is perpendicular. This involves solving:
The red represents the slope of the line segment from (0,2) to the a point on your curve. The blue represents the negative reciprical of the tangent at a given point. Whatever x values we get from solving this will be the values such that the line joining (0,2) to the curve is perpendicular to the curve. Note that x = 0 cannot be a solution to the equation but drawing a line segment from (0,2) to (0,1) fits the description above.
Much shorter actually.