Question

A dog trainer is trying to find a triangular area behind his house that encloses 1800 square feet for his dog to run. He has the first two fence posts at (6, 0) and at (0, 25). The final fence post is on the property line at x = 30. Find the point where the trainer can place the final fence post.

Answer 1

`C = (30,500)`

Explanation:

Given

`Area = 1800ft^2`

`A = (6,0)`

`B = (0,25)`

`C(x,y) = (30,y)`

Required

Determine the coordinates of C

The area of triangle is calculated as thus:

`Area = (|A_x(B_y - C_y) + B_x(C_y - A_y) + C_x(A_y - B_y)|)/(2)`

This gives:

`1800= (|6(25 - y) + 0(y - 0) + 30(0 - 25)|)/(2)`

`1800= (|6(25 - y) + 30(0- 25)|)/(2)`

`1800= (|6(25 - y) + 30(-25)|)/(2)`

`1800= (|150 - 6y -750 |)/(2)`

`1800= (|150 -750- 6y |)/(2)`

`1800= (|-600- 6y |)/(2)`

Multiply through by 2

`2 * 1800= |-600- 6y|`

`3600= |-600- 6y|`

`3600= |-(600+ 6y)|`

`3600= 600+ 6y`

Solve for 6y

`6y = 3600 - 600`

`6y = 3000`

Solve for y

`y = 3000/6`

`y = 500`

Hence:

The point is:

`C = (30,500)`