The graphs to the right illustrate a fountain shooting a jet of water with a laser beam illuminating

1
The graphs to the right illustrate a fountain
shooting a jet of water with a laser beam
illuminating the jet in two places, one of them
being (2 m ,3 m). Determine the height of the
other point illuminated by the laser.
Equation of the jet of water
x1 0.5 x1 2.5 (x,y) (2,3) y a(x x1)(x
x2) 3 a(2 0.5)( 2 2.5) 3 a(1.5)(-0.5) 3
-0.75a a -4
y -4(x 0.5)(x 2.5) y -4(x2 2.5x 0.5x
1.25) y -4x2 12x 5
Solution of the system
x 1 -4x2 12x 5 4x2 - 12x x 5 1
0 4x2 - 11x 6 0 4x2 - 3x - 8x 6 0 x(4x
3) 2(4x 3) 0 (4x 3)(x 2) 0 4x 3 0
OR x 2 0
Equation of the laser beam
The height of the other point illuminated by
water is 1.75 meters.
2
The proportion of men to women employed by a
company has varied over the years of operation.
The number of men was 428 in the year 2000 and
reached a maximum of 500 three years later
following a second degree curve. The number of
women in 2000 was 100 and increased at a rate of
25 per years after that.
If the trends continue, in what year will the
number of men and women be the same? How many
will be employed by the company at that time?
x number of years after 2000 y number of
employees
Equation for men
Vertex (3,500) Other point (0,428) y a(x
h)2 k 428 a(0 3)2 500 428 500 9a 9a
-72 a -8
Equation for women
y -8(x 3)2 500 y -8(x2 6x 9) 500 y
-8x2 48x - 72 500 y -8x2 48x 428
m 25 b 100 y 25x 100
Solution of the System
25x 100 -8x2 48x 428 8x2 - 48x 25x
428 100 0 8x2 - 23x 328 0 8x2 - 64x 41x
328 0 8x(x 8) 41(x 8) 0 (x 8)(8x
41) 0 x 8 0 OR 8x 41 0 x 8 OR x
-5.125
After 8 years, there were 300 men and 300 women
employed for a total of 600 employees
y 25x 100 y 25(8) 100 200 100 300
employees
3
A cold front sweeps into the town of Amos causing
the temperatures to drop steadily from 1C at 2h
to -4C at 21h and continue. As yet unaffected
by this cold front, on the same day Gaspe
experiences temperatures indicated in the table
to the right.
For how long is the temperature of Gaspe warmer
than that of Amos during that time?
Equation for Amos
x time y temperature
Solution of the System
Equation for Gaspe
Vertex (20,4) Other point (8, 0.4) y a(x
h)2 k 0.4 a(8 20)2 4 0.4 4 144a 144a
-3.6 a -0.025
y -0.025(x 20)2 4 y -0.025(x2 40x
400) 4 y -0.025x2 x - 10 4 y -0.025x2
x - 6
0.475x2 - 19x 5x 114 29 0 0.475x2 - 24x
143 0 a 0.475 b -24 c 143 ? b2 4ac
(-24)2 4(0.475)(143) 476 271.7 304.3
Gaspe was warmer than Amos between 6.91 hours and
43.6 hours 43.6 6.9 36.1 hours.
4
A display is constructed to take pictures of kids
with Santa Claus.
Two symmetrical braces extend up from the floor
to support the seat and attach to a piece of
board above the seat. The display is 2.25 meters
high and 1.5 meters wide at its base. The braces
are 42 cm apart at the floor and 58 cm apart at
the seat which is 48 cm above the floor. How long
are the braces? How far apart are the braces at
the top where they attach to the piece of board?
Equation of the brace
Solution of the System
Equation of the parabola
6x 1.26 -4x2 2.25 4x2 6x - 3.51 0 a
4 b 6 c -3.51 ? b2 4ac 62
4(4)(-3.51) 36 56.16 92.16
Vertex (0, 2.25) Other point (0.75, 0) y
a(x h)2 k 0 a(0.75 0)2 2.25 0 2.25
0.5625a 0.5625a -2.25 a -4
y -4x2 2.25
y 6x 1.26 6(0.45) 1.26 2.7 1.26
1.44
5
End points of right brace (0.21,0) (0.45,1.44)
Length of right brace (x1, y1) (0.21,0) (x2,
y2) (0.45,1.44)
End points of right brace (-0.45,1.44)
(0.45,1.44)
Length of right brace 0.45-(-0.45) 0.9 m
6
A new business is starting and the manager is
analyzing his expenses and revenues. The monthly
expenses start at 1950 and increase to a maximum
of 2550 after 5 months at which point the
expenses start to decline. The revenues increase
at a constant rate of 30 per month and reach
2256 after 12 months. At what point will there
be a zero deficit? ( i.e. revenue expenses)
Equation for revenue
Equation for expenses
m 30 ( x1 , y1) (12,2256)
Vertex (5,2550) Other point (0,1950) y a(x
h)2 k 1950 a(0 5)2 2550 1950 2550
25a 25a -600 a -24
y -24(x 5)2 2550 y -24(x2 - 10x 25)
2550 y -24x2 240x - 600 2550 y -24x2
240x 1950
Solution of the System
30x 1896 -24x2 240x 1950 24x2 - 240x
30x 1950 1896 0 24x2 - 210x 54 0 4x2 -
35x 9 0 4x2 1x - 36x 9 0 x(4x 1)
9(4x 1) 0 (4x 1)(x 9) 0 4x 1 0 OR
x - 9 0 x -0.25 OR x 9
There is a zero deficit after 9 months.
7
Plans for a highway indicate that it will cross a
river in 2 places. Use the coordinates from the
graph provided to determine the length of the
section of the highway to be built between the
bridges.
Equation for Highway
Equation for River
Vertex (5,1) Other point (9,4.2) y a(x
h)2 k 4.2 a(9 5)2 1 4.2 - 1 16a 16a
3.2 a 0.2
y 0.2(x 5)2 1 y 0.2(x2 10x 25) 1 y
0.2x2 2x 5 1 y 0.2x2 2x 6
Solution of the System
0.2x2 2x 6 0.4x 0.6 0.2x2 2x 0.4x
11 0.6 0 0.2x2 2.4x 5.4 0 x2 12x 27
0 (x 3)(x 9) 0 x 3 0 OR x 9 0 x
3 OR x 9
y 0.4(3) 0.6 1.2 0.6 1.8 (3, 1.8)
y 0.4(9) 0.6 3.6 0.6 4.2 (9, 4.2)