Question

3. The path of an underground stream is given by the function y = 4x² +17x -32. Two

new houses need wells to be dug. On the area plan, these houses lie on a line defined by

the equation y = –15x+10. Determine the coordinates where the two wells need to be

dug.

Answer 1

Answer:

The coordinate of the wells are

$(-4 -\sqrt[]{\frac{53}{2}}, 70+15\sqrt[]{\frac{53}{2}})$

$(-4 +\sqrt[]{\frac{53}{2}}, 70-15\sqrt[]{\frac{53}{2}})$

Step-by-step explanation:

The y coordinate of the stream is given by $y = 4x^2+17x-32$. Also, the y coordinate of the houses are determined by y=-15x+10. We will assume that the houses are goint to be built on the exact position where we build the wells. We want to build the wells at the exat position in which both functions cross each other, so we have the following equation

$4x^2+17x-32 = -15x+10$

or equivalently

$4x^2+32x-42=0$ (by summing 15x and substracting 10 on both sides)

Dividing by 2 on both sides, we get

$2x^2+16x-21=0$

Recall that given the equation of the form $ax^2+bx+c=0$ the solutions are

$x = \frac{-b \pm \sqrt[]{b^2-4ac}}{2a}$

Taking a =2, b = 16 and c = -21, we get the solutions

$x_1 = -4 -\sqrt[]{\frac{53}{2}}$

$x_2 = -4 +\sqrt[]{\frac{53}{2}}$

If we replace this values in any of the equations, we get

$y_1 = 70+15\sqrt[]{\frac{53}{2}}$

$y_2 = 70-15\sqrt[]{\frac{53}{2}}$