Question
Max throws a ball into the air from a height of 3 meters with an initial velocity of 18 meters per second. What quadratic inequality represents the possible number of seconds, t, that the ball is at least 10 meters above the ground? Write the quadratic inequality that represents the description. (2 points)
Answers
GPT-4o mini
To form the quadratic inequality representing the scenario, we start with the equation that models the height of the ball over time \( t \). The height \( h(t) \) of the ball thrown into the air can be modeled using the following equation:
\[
h(t) = -4.9t^2 + vt + h_0
\]
where:
- \( v \) is the initial velocity (in meters per second),
- \( h_0 \) is the initial height (in meters),
- the coefficient \(-4.9\) comes from the acceleration due to gravity (approx. \(-9.8 \: \text{m/s}^2\) divided by \(2\)).
Given:
- Initial height, \( h_0 = 3 \) meters
- Initial velocity, \( v = 18 \) meters per second
Substituting the given values into the height equation:
\[
h(t) = -4.9t^2 + 18t + 3
\]
Next, we need to find when the height \( h(t) \) is at least 10 meters. This leads us to the inequality:
\[
-4.9t^2 + 18t + 3 \geq 10
\]
To simplify this inequality, we rearrange it:
\[
-4.9t^2 + 18t + 3 - 10 \geq 0
\]
\[
-4.9t^2 + 18t - 7 \geq 0
\]
To maintain a standard form where the leading coefficient is positive, we can multiply the entire inequality by \(-1\) (remember to flip the inequality sign):
\[
4.9t^2 - 18t + 7 \leq 0
\]
Thus, the quadratic inequality representing the possible number of seconds \( t \) that the ball is at least 10 meters above the ground is:
\[
4.9t^2 - 18t + 7 \leq 0
\]
\[
h(t) = -4.9t^2 + vt + h_0
\]
where:
- \( v \) is the initial velocity (in meters per second),
- \( h_0 \) is the initial height (in meters),
- the coefficient \(-4.9\) comes from the acceleration due to gravity (approx. \(-9.8 \: \text{m/s}^2\) divided by \(2\)).
Given:
- Initial height, \( h_0 = 3 \) meters
- Initial velocity, \( v = 18 \) meters per second
Substituting the given values into the height equation:
\[
h(t) = -4.9t^2 + 18t + 3
\]
Next, we need to find when the height \( h(t) \) is at least 10 meters. This leads us to the inequality:
\[
-4.9t^2 + 18t + 3 \geq 10
\]
To simplify this inequality, we rearrange it:
\[
-4.9t^2 + 18t + 3 - 10 \geq 0
\]
\[
-4.9t^2 + 18t - 7 \geq 0
\]
To maintain a standard form where the leading coefficient is positive, we can multiply the entire inequality by \(-1\) (remember to flip the inequality sign):
\[
4.9t^2 - 18t + 7 \leq 0
\]
Thus, the quadratic inequality representing the possible number of seconds \( t \) that the ball is at least 10 meters above the ground is:
\[
4.9t^2 - 18t + 7 \leq 0
\]