math question help (1 Viewer)

[poptarts12345]

1. The point A lies on the positive half of the x-axis, and the point B lies on the positive half of the y-axis, and the interval AB passes through the point P(5,3). Find the coordinates of A and B so that triangle AOB has minimum area.

2. A man in a rowing boat is presently 6km from the nearest point A on the shore. He wants to reach as soon as possible a point B that is a further 20km down from the shore from A. If he can row at 8km/hr and run at 10km/hr , how far from A should he land.

 

[ultra908]

1. Let A be a variable point on the x axis, (x,0). Notice for any A, there can only be one B for which AB passes through (5,3). Thus we don't really have to worry about B for now. Then find an equation for area of triangle AOB, and its min using calculus. Using ur min value of x, find A and B!

2. Draw a diagram (assuming the coast is a straight line). Let x be the distance from A he lands. Find an equation for the time of the trip wrt to x,. Then find its min using calculus and thus the distance x.

 

[Hannah0605]

Send your question on Airstudy and see whether someone can help. They have a request teacher's function.

 

[CM_Tutor]

1. The point A lies on the positive half of the x-axis, and the point B lies on the positive half of the y-axis, and the interval AB passes through the point P(5,3). Find the coordinates of A and B so that triangle AOB has minimum area.

Let A be at and B be at where a and b are positive. Since the line AB passes through , we actually know that and .
(We know this as (i) if then AP is vertical and so does not pass through B and (i) if then AP has a positive slope and so intersects the negative half of the y-axis, violating the restriction on the position of B. The same reasoning applies to restricting that .)

Now, the line AB has equation as the coordinates for A and B satisfy this linear equation. Further, the triangle AOB is right-angled at O and so has area given by . So, we need only re-write this expression in two variables and apply calculus or other methods to find the required minimum. Since P lies on AB, we know that:



So, we have stationary points at and and the derivative is undefined at . Fortunately, we know the domain for valid answers is and so we need only investigate .

When a is near to but below 10,
And, when a is near to but above 10,

It follows that in the vicinity of , A is first decreasing, then stationary, then increasing, meaning that the stationary point there is a local minimum, where and .

Thus, the minimum area triangle AOB has A at (10, 0), B at (0, 6), and has area of 30 square units.

If you have TEX with tikzpicture installed, here is the picture I would draw to accompany this solution:
\begin{tikzpicture}
\fill[blue!20] (0, 0) -- (8, 0) -- (0, 8);
\draw[latex-latex] (-1.5, 0) -- (10, 0) node {$x$};
\draw[latex-latex] (0, -1.5) -- (0, 9) node {$y$};
\draw[thick,blue] (8, 0) -- (0, 8);
\node at (4.5, 5) {$\color{blue} \cfrac{x}{a} + \cfrac{y}{b} = 1 \color{black}$};
\node at (-0.3, -0.3) {$O$};
\node at (8.5, 0.3) {$A(a, 0)$};
\node at (0.8, 8) {$B(0, b)$};
\node at (5.8, 3) {$P(5, 3)$};
\fill[black] (5, 3) circle (0.1);
\fill[black] (8, 0) circle (0.075);
\fill[black] (0, 8) circle (0.075);
\draw[dashed] (0, 3) -- (5, 3);
\draw[dashed] (5, 0) -- (5, 3);
\draw[dotted] (-1, 8) -- (0, 8);
\draw[dotted] (8, -1) -- (8, 0);
\draw[<-] (0, -0.75) -- (3.7, -0.75);
\draw[<-] (8, -0.75) -- (4.3, -0.75);
\node at (4.05, -0.75) {$a$};
\draw[<-] (-0.75, 0) -- (-0.75, 3.7);
\draw[<-] (-0.75, 8) -- (-0.75, 4.3);
\node at (-0.75, 3.95) {$b$};
\node at (5, -0.3) {$5$};
\node at (-0.3, 3) {$3$};
\node at (2.5, 1.5) {Area$_{\Delta AOB} = \cfrac{ab}{2}$};
\end{tikzpicture}

 

[CM_Tutor]

2. A man in a rowing boat is presently 6km from the nearest point A on the shore. He wants to reach as soon as possible a point B that is a further 20km down from the shore from A. If he can row at 8km/hr and run at 10km/hr , how far from A should he land.

Firstly, let's look at the answer that is sought:

Let S be the starting point for the rowing boat, which is 6 km from point A on the shore, which is the closest point on the shore to S.. We know that B is 20 km from A along the shore, and let point P be located a distance of x km from A, so that the distance PB is (20 - x) km.

The attached file contains this information shown in a diagram. We seek the position of P (and thus the value of x) so as to minimise the travel time from S to B. Let T be the travel time (in hours) from S to B, made up of the travel time from S to P (t_SP) and the travel time from P to B (t_PB). In otherwise, we need to minmise:



Defining the distances SP and PB (in km) as d_SP and d_PB, respectively, and remembering that distance equals speed times times, we can see that:



Now, applying Pythagoras' Theorem to triangle SAP, we can see that SA² + AP² = SP²:



To find stationary points, we need to set this derivative to zero:









So, there is a stationary point at (x = 8 km, T = 2 h 27 min)

Following the path SAB (corresponding to x = 0), the rower rows 6 km taking 45 min and then runs 20 km taking 2 h, so that T = 2 h 45 min.
Following path SB (rowing directly from S to B, corresponding to x = 20), the distance rowed is (6² + 20²)^0.5 = 20.8806... km taking a little over 2 h 36 min. It follows that the stationary point is a minimum, and thus the rower should row to point P, 8 km from A towards B.

Of course, that x = 8 corresponds to a minimum can also be established by the usual methods of examining the behaviour of dT/dx in the vicinity of x = 8 or by showing that the second derivative is positive.

Note, however, that we have assumed that the shore line from A to B is a straight line with AB perpendicular to AS. I believe that this is what is intended, but the question does not actually specify this and if the shoreline is different from this then the answer may change.