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Answer :
We start with the height equation for the pebble:
[tex]$$
h = -16t^2 + 12t + 900.
$$[/tex]
When the pebble hits the ground, its height is zero. Therefore, we set:
[tex]$$
-16t^2 + 12t + 900 = 0.
$$[/tex]
This is a quadratic equation in the form
[tex]$$
at^2 + bt + c=0,
$$[/tex]
with
[tex]$$
a = -16, \quad b = 12, \quad c = 900.
$$[/tex]
We use the quadratic formula:
[tex]$$
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$[/tex]
The discriminant is computed as:
[tex]$$
\Delta = b^2 - 4ac = 57744.
$$[/tex]
Taking the square root gives:
[tex]$$
\sqrt{57744} \approx 240.29981273400944.
$$[/tex]
Substitute these values into the quadratic formula:
[tex]$$
t = \frac{-12 \pm 240.29981273400944}{2(-16)}.
$$[/tex]
This gives two solutions:
1.
[tex]$$
t_1 = \frac{-12 + 240.29981273400944}{-32} \approx -7.13437,
$$[/tex]
2.
[tex]$$
t_2 = \frac{-12 - 240.29981273400944}{-32} \approx 7.88437.
$$[/tex]
Since negative time is not physically meaningful, we reject [tex]$t_1$[/tex]. Thus, the time it takes for the pebble to hit the ground is approximately:
[tex]$$
t \approx 7.88437 \text{ seconds}.
$$[/tex]
Therefore, the pebble will hit the ground about [tex]$7.884$[/tex] seconds after it is thrown.
[tex]$$
h = -16t^2 + 12t + 900.
$$[/tex]
When the pebble hits the ground, its height is zero. Therefore, we set:
[tex]$$
-16t^2 + 12t + 900 = 0.
$$[/tex]
This is a quadratic equation in the form
[tex]$$
at^2 + bt + c=0,
$$[/tex]
with
[tex]$$
a = -16, \quad b = 12, \quad c = 900.
$$[/tex]
We use the quadratic formula:
[tex]$$
t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
$$[/tex]
The discriminant is computed as:
[tex]$$
\Delta = b^2 - 4ac = 57744.
$$[/tex]
Taking the square root gives:
[tex]$$
\sqrt{57744} \approx 240.29981273400944.
$$[/tex]
Substitute these values into the quadratic formula:
[tex]$$
t = \frac{-12 \pm 240.29981273400944}{2(-16)}.
$$[/tex]
This gives two solutions:
1.
[tex]$$
t_1 = \frac{-12 + 240.29981273400944}{-32} \approx -7.13437,
$$[/tex]
2.
[tex]$$
t_2 = \frac{-12 - 240.29981273400944}{-32} \approx 7.88437.
$$[/tex]
Since negative time is not physically meaningful, we reject [tex]$t_1$[/tex]. Thus, the time it takes for the pebble to hit the ground is approximately:
[tex]$$
t \approx 7.88437 \text{ seconds}.
$$[/tex]
Therefore, the pebble will hit the ground about [tex]$7.884$[/tex] seconds after it is thrown.
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