A Football Player Throws A Football. The Function H H H Given By H ( T ) = 6 + 75 T − 16 T 2 H(t) = 6 + 75t - 16t^2 H ( T ) = 6 + 75 T − 16 T 2 Describes The Football's Height In Feet T T T Seconds After It Is Thrown.Select All The Statements That Are True About This Situation.A.

Introduction

When a football player throws a football, it follows a parabolic trajectory under the influence of gravity. The height of the football at any given time can be described by a quadratic function. In this article, we will explore the function $h(t) = 6 + 75t - 16t^2$, which represents the height of the football in feet $t$ seconds after it is thrown. We will analyze the function to determine its key characteristics and identify the true statements about this situation.

Understanding the Function

The given function is a quadratic function in the form of $h(t) = at^2 + bt + c$, where $a = -16$, $b = 75$, and $c = 6$. The coefficient $a$ represents the rate of change of the height with respect to time, which is the acceleration due to gravity. The coefficient $b$ represents the initial velocity of the football, and the constant term $c$ represents the initial height of the football.

Vertex Form of the Parabola

To understand the behavior of the function, we can rewrite it in vertex form, which is given by $h(t) = a(t - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. To convert the function to vertex form, we need to complete the square.

Completing the Square

To complete the square, we start by factoring out the coefficient of the $t^2$ term, which is $-16$. We then add and subtract the square of half the coefficient of the $t$ term, which is $75/2 = 37.5$, inside the parentheses.

h(t) = -16(t^2 - 4.6875t) + 6

Next, we add and subtract the square of half the coefficient of the $t$ term, which is $(37.5)^2 = 1406.25$, outside the parentheses.

h(t) = -16(t^2 - 4.6875t + 1.90625) + 6 + 16(1.90625)

Simplifying the expression, we get:

h(t) = -16(t - 2.34375)^2 + 6 + 30.5

Vertex Form of the Parabola

The vertex form of the parabola is given by:

h(t) = -16(t - 2.34375)^2 + 36.5

The vertex of the parabola is at $(2.34375, 36.5)$, which represents the maximum height of the football.

Maximum Height and Time

The maximum height of the football is given by the $y$-coordinate of the vertex, which is $36.5$ feet. The time at which the football reaches its maximum height is given by the $x$-coordinate of the vertex, which is $2.34375$ seconds.

Initial Velocity and Acceleration

The initial velocity of the football is given by the coefficient of the $t$ term, which is $75$ feet per second. The acceleration due to gravity is given by the coefficient of the $t^2$ term, which is $-16$ feet per second squared.

True Statements

Based on the analysis of the function, the following statements are true:

• The maximum height of the football is $36.5$ feet.
• The time at which the football reaches its maximum height is $2.34375$ seconds.
• The initial velocity of the football is $75$ feet per second.
• The acceleration due to gravity is $-16$ feet per second squared.
• The function $h(t) = 6 + 75t - 16t^2$ represents the height of the football in feet $t$ seconds after it is thrown.

Conclusion

In conclusion, the function $h(t) = 6 + 75t - 16t^2$ represents the height of the football in feet $t$ seconds after it is thrown. The function has a maximum height of $36.5$ feet, which occurs at $2.34375$ seconds. The initial velocity of the football is $75$ feet per second, and the acceleration due to gravity is $-16$ feet per second squared.

Introduction

In our previous article, we explored the function $h(t) = 6 + 75t - 16t^2$, which represents the height of the football in feet $t$ seconds after it is thrown. We analyzed the function to determine its key characteristics and identified the true statements about this situation. In this article, we will answer some frequently asked questions about the function and the situation.

Q&A

Q: What is the maximum height of the football?

A: The maximum height of the football is $36.5$ feet.

Q: At what time does the football reach its maximum height?

A: The football reaches its maximum height at $2.34375$ seconds.

Q: What is the initial velocity of the football?

A: The initial velocity of the football is $75$ feet per second.

Q: What is the acceleration due to gravity?

A: The acceleration due to gravity is $-16$ feet per second squared.

Q: How can we determine the time at which the football hits the ground?

A: To determine the time at which the football hits the ground, we need to set the height function equal to zero and solve for $t$. This will give us the time at which the football hits the ground.

Q: What is the time at which the football hits the ground?

A: To determine the time at which the football hits the ground, we need to solve the equation $6 + 75t - 16t^2 = 0$. This is a quadratic equation, and we can solve it using the quadratic formula.

Q: How can we solve the quadratic equation $6 + 75t - 16t^2 = 0$?

A: We can solve the quadratic equation $6 + 75t - 16t^2 = 0$ using the quadratic formula. The quadratic formula is given by:

$$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = -16$, $b = 75$, and $c = 6$. Plugging these values into the quadratic formula, we get:

$$t = \frac{-75 \pm \sqrt{75^2 - 4(-16)(6)}}{2(-16)}$$

Simplifying the expression, we get:

$$t = \frac{-75 \pm \sqrt{5625 + 384}}{-32}$$

$$t = \frac{-75 \pm \sqrt{6009}}{-32}$$

$$t = \frac{-75 \pm 77.5}{-32}$$

Solving for $t$, we get two possible values:

$$t = \frac{-75 + 77.5}{-32} = \frac{2.5}{-32} = -0.078125$$

$$t = \frac{-75 - 77.5}{-32} = \frac{-152.5}{-32} = 4.765625$$

Since time cannot be negative, we discard the negative solution and keep the positive solution.

Q: What is the time at which the football hits the ground?

A: The time at which the football hits the ground is $4.765625$ seconds.

Conclusion

In conclusion, we have answered some frequently asked questions about the function $h(t) = 6 + 75t - 16t^2$ and the situation. We have determined the maximum height of the football, the time at which the football reaches its maximum height, the initial velocity of the football, the acceleration due to gravity, and the time at which the football hits the ground.