154238 wrote:
leigimon wrote:
Getting a slightly different answer:
Prob(y<x) = \(\frac{Area(small Triangle)}{Area(large Triangle)}\)
Area(large Triangle) = 25
Area(small Triangle) = ?
The small triangle would be made up of 3 points:
1. the origin
2. (5,0),
3. a point on the hypotenuse where y=x
To figure out this point we build the equation for the hypotenuse, y = 10 - 2x, and calculate the intersection with y = x, solve the system to find that x = \(\frac{10}{3}\) = y
Area(small Triangle) = \(\frac{1}{2}(5 * \frac{10}{3} ) = \frac{50}{6}\)
\(Prob (y<x) = \frac{50}{6}/25 = \frac{1}{3}\)
Any chance 1/8 could've been 1/3?
Hi guys.. Can you please explain how did you find the equation of hypotenuse.. y=mx+c >> m=-2 , why c= 10 ?
When m and c are not 0, the line is not horizontal and will not pass through the origin. Then both the x and the y intercept will be non-zero.
The y intercept is the value of y for x = 0, which, for the equation y = mx + c, is c.
The x intercept is the value of x for y = 0, which is -c/m.
The given equation y = mx + c can be rewritten as -mx + y = c,
or \(\frac{x}{-c/m}+\frac{y}{c}=1\). You can see that the denominator of x is exactly the x intercept and the denominator of y is the y intercept.
Each line which doesn't go through the origin, has its equation as \(\frac{x}{x_i}+\frac{y}{y_i}=1\) , where \(x_i\) and \(y_i\) are the x and the y intercept, respectively.
In our case, we could have written directly the equation of the hypotenuse as \(\frac{x}{5}+\frac{y}{10}=1\) which we can rearrange and get \(y=-2x+10.\)
So, next time, if you have the two intercepts, for example you know that the line goes through the points (-3,0) and (0,4), you can immediately write the equation of the line as \(\frac{x}{-3}+\frac{y}{4}=1\) rearrange as you wish...
I mean you can save the time of finding the slope and write the standard equation of a line... Not that it is such a saving, but anyway, it is a nice mathematical property