College Algebra 9th txtbk

SECTION 3–4 Quadratic Functions 211
(C) When x 40, y 240 3(40) 120. Each pen is x by y2, or 40 feet by 60 feet.
The area of each pen is 40 feet 60 feet 2,400 square feet.
MATCHED PROBLEM 6
Repeat Example 6 with the owner constructing three identical adjacent pens instead of two.
The great sixteenth-century astronomer and physicist Galileo was the first to discover that
the distance an object falls is proportional to the square of the time it has been falling. This
makes quadratic functions a natural fit for modeling falling objects. Neglecting air resistance,
the quadratic function
h(t) h 0 16t 2
represents the height of an object t seconds after it is dropped from an initial height of h 0
feet. The constant 16 is related to the force of gravity and is dependent on the units used.
That is, 16 only works for distances measured in feet and time measured in seconds. If the
object is thrown either upward or downward, the quadratic model will also have a term involving
t. (See Problems 93 and 94 in Exercises 3-4.)
EXAMPLE 7 Projectile Motion
As a publicity stunt, a late-night talk show host drops a pumpkin from a rooftop that is 200
feet high. When will the pumpkin hit the ground? Round your answer to two decimal places.
SOLUTION
Because the initial height is 200 feet, the quadratic model for the height of the pumpkin is
Because h(t) 0 when the pumpkin hits the ground, we must solve this equation for t.
h(t) 200 16t 2 0
16t 2 200
t 2 200
16 12.5
t 112.5
h(t) 200 16t 2
3.54 seconds
Add 16t 2 to both sides.
Divide both sides by 16.
Take the square root of both sides.
Only the positive solution is relevant.
MATCHED PROBLEM 7
A watermelon is dropped from a rooftop that is 300 feet high. When will the melon hit the
ground? Round your answer to two decimal places.
Z Solving Quadratic Inequalities
Given a quadratic function f(x) ax 2 bx c, a 0, the zeros of f are the solutions of
the quadratic equation
ax 2 bx c 0
(see Section 1-5). If the equal sign in equation (1) is replaced with , , , or , the result
is a quadratic inequality in standard form. Just as was the case with linear inequalities
(see Section 1-2), the solution set for a quadratic inequality is the subset of the real number
line that makes the inequality a true statement. We can identify this subset by examining
the graph of a quadratic function. We begin with a specific example and then generalize the
results.
The graph of
f (x) x 2 2x 3 (x 3)(x 1)
(1)