Page 780 - Algebra 1

Example
4
Application: Avalanches
A ski patrol fires an explosive arrow to trigger a controlled avalanche. The
x
path of the arrow is modeled by the equation y = -_2 + 2x and the
_3x
distance and y is the vertical distance. At what altitude will the arrow strike
1600
shape of the mountainside is modeled by y = 4 where x is the horizontal
the mountain? (Assume all dimensions are in feet.)
SOLUTION
Understand The path of the arrow is modeled by a parabola. The mountainside is modeled by a straight line.
_2 y=-x +2x 1600
_ y = 3x
path of arrow
Math Reasoning
Write Describe the meaning of the x-coordinate in the solution (2000, 1500).
mountain
cannon
4
Plan The equation of the arrow’s path and the equation of the shape of the mountain form a system of equations.
Solving this system will determine the points at which the two graphs intersect.
y
x
Solve One way of solving the system is by graphing the two equations. The cannon is located at the base of the mountain, so both graphs pass through (0, 0). The non-origin solution to the system is (2000, 1500). The altitude at which the arrow will strike the side of the mountain is 1500 feet.
Check
1500 1000 500 0
1000
2000
_ _2
y=3x y=-x +2x
4 1600
3(2000) 20002 __
1500 4 1500 - 1600 + 2(2000) __
Horizontal distance (ft)
1500 6000 1500 - 4,000,000 + 4000 4 1600
1500 = 1500 ✓ 1500 -2500 + 4000 1500 = 1500 ✓
Lesson Practice
Solve each system of equations by graphing.
(Ex 1)
a. y=x2 b. y=x2
y = 16 y = 6x - 9
c. y=x2
y = -2x + 3
Lesson 112 765
Vertical distance (ft)