Answer :
Answer:
Step-by-step explanation:
Part a: Find the values of
x when
ℎ
=
0
h=0.
The height
ℎ
h of the golf ball is given by the equation:
ℎ
=
−
2
+
100
h=−x
2
+100x
To find the values of
x when
ℎ
=
0
h=0, we set
ℎ
=
0
h=0 in the equation:
0
=
−
2
+
100
0=−x
2
+100x
Factor out
x:
(
−
+
100
)
=
0
x(−x+100)=0
This equation will be true if either
=
0
x=0 or
−
+
100
=
0
−x+100=0.
=
0
x=0
−
+
100
=
0
−x+100=0
−
=
−
100
−x=−100
=
100
x=100
So, the values of
x when
ℎ
=
0
h=0 are
=
0
x=0 and
=
100
x=100.
Part b: Interpret your answer from part a.
From part a, we found that
=
0
x=0 and
=
100
x=100 are the values of
x when
ℎ
=
0
h=0.
=
0
x=0 represents the horizontal distance from where the ball was hit when the height
ℎ
h is 0. This means the ball is on the ground at the point where it was initially hit.
=
100
x=100 represents the horizontal distance from where the ball was hit when the height
ℎ
h is 0 again. This indicates that the ball has traveled 100 meters horizontally and has returned to the ground level.
Part c: Find how far the ball has traveled horizontally when the height is 196 m.
We are given that
ℎ
=
196
h=196 m. We need to find
x, the horizontal distance traveled, when
ℎ
=
196
h=196.
Use the equation for
ℎ
h:
−
2
+
100
=
196
−x
2
+100x=196
Subtract 196 from both sides to set the equation to 0:
−
2
+
100
−
196
=
0
−x
2
+100x−196=0
Now, solve this quadratic equation using the quadratic formula
=
−
±
2
−
4
2
x=
2a
−b±
b
2
−4ac
, where
=
−
1
a=−1,
=
100
b=100, and
=
−
196
c=−196.
Calculate the discriminant:
Δ
=
2
−
4
=
10
0
2
−
4
(
−
1
)
(
−
196
)
Δ=b
2
−4ac=100
2
−4(−1)(−196)
Δ
=
10000
−
784
Δ=10000−784
Δ
=
9216
Δ=9216
Now, find
x:
=
−
100
±
9216
2
(
−
1
)
x=
2(−1)
−100±
9216
=
−
100
±
96
−
2
x=
−2
−100±96
Calculate the two possible values of
x:
1
=
−
100
+
96
−
2
=
−
4
−
2
=
2
x
1
=
−2
−100+96
=
−2
−4
=2
2
=
−
100
−
96
−
2
=
−
196
−
2
=
98
x
2
=
−2
−100−96
=
−2
−196
=98
Therefore, the ball has traveled horizontally
=
2
x=2 meters and
=
98
x=98 meters when the height is 196 m.
So, the ball has traveled 2 meters and 98 meters horizontally when the height is 196 m.
