Calculus Problem

Question

Of all the triangles that pass through the point (1,1) and have sides lying on the coordinate axes, one has the smallest area.    

 

what are the lengths of its sides?

Answer 1

PROBLEM

 

Calculus Problem
Of all the triangles that pass through the point (1,1) and have sides lying on the coordinate axes, one has the smallest area.

what are the lengths of its sides?

 

SOLUTION

 

Draw a diagram:

 

 

(0,y)

*

*    *

*         *  (1,1)

*              *

*                   *

*                        *

*    *   *   *    *   *  * (x,0)

 

Let's look at the slope of the hypotenuse...

 

 

Using the points (0,y), (1,1), and (x,0) as well as the formula for slope...

 

       y₂ - y₁

m = --------

       x₂ - x₁

 

         1 - y

m = -------- = 1 - y
         1 - 0

 

        0 - 1         -1
m = -------- = -------
        x - 1        x - 1

 

Set the slopes equal to each other

 

             -1

1 - y = ------

            x - 1

 

              1
y - 1 = -------
            x - 1

 

(x - 1)(y - 1) = 1

 

xy - x - y + 1 = 1

 

y(x - 1) - x = 0

 

         x
y = -------
       x - 1

 

Area of a triangle = bh/2

 

b = x, h = y

 

        x        x

A = --- * -------

        2      x - 1

 

A = x²/[2(x - 1)]

 

To find the minimum area, take first derivative and set it equal to zero...

 

dA/dx = 0 = (using quotient rule) [2(x-1)(2x) - x²(2)]/[2(x-1)]²

 

2x(x-2)/[4(x-1)²] = 0

 

x = 2

 

so, y = 2/(2-1) = 2