Question 118365
A fountain in the town square sprays water in a parabolic arc. The water spray starts at 1/2 metre above the ground and reaches a maximum height of 5 metres after 3 seconds. Determine the quadratic function that models the path followed by the water in the fountain and use it to determine the height of the water at 1&3/4 seconds . Round your answer to the nearest tenth of a metre.
:
Using the form: y = ax^2 + bx + c
:
y = .5 when x = 0; c = .5
:
Using the vertex, x = 3; y = 5
9a + 3b + .5 = 5
9a + 3b = 5 - .5
9a + 3b = 4.5
:
Using the symmetry of a parabola we know that 3 sec after the vertex, y = .5 again
x = 6; y = .5
36a + 6b + .5 = .5
36a + 6b = .5 - .5
36a + 6b = 0
:
Multiply the vertex equation by 2 and subtract from the above equation:
36a + 6b = 0
18a + 6b = 9
-------------subtracting eliminates b, find a
18a + 0b = -9
a = -9/18
a = -.5
:
Find b using 9a + 3b = 4.5
9(-.5) + 3b = 4.5
-4.5 + 3b = 4.5
3b = 4.5 + 4.5
3b = 9
b = 3
:
Our equation: y = -.5x^2 + 3x + .5
:
Illustrated by the graph:
{{{ graph( 300, 200, -2, 10, -2, 6, -.5x^2 + 3x + .5) }}}
:
Find the height after 1.75 seconds:
h = -.5(1.75^2) + 3(1.75) -.5 
h = -1.53125 + 5.25 + .5
h = 4.2 meters after 1.75 seconds