Question

A longhorn kicker recently kicked a 54-yard field goal during practice. The football started at a height of 0 ft, and the football was 12 ft off the ground when it made it a horizontal distance of 54 yards. Assume that the football hits the ground 10 yards after it passes the goal post. What was the maximum height of the football? Round to the nearest whole number in feet (1 yard = 3 feet).

Answer 1

To solve this problem, we can model the flight of the football as a projectile motion problem. The trajectory of the football can be represented by a quadratic equation of the form
`\( y = ax^2 + bx + c \)`, where
`\( y \)` is the height of the football and
`\( x \)` is the horizontal distance it has traveled.

From the information given:

- The football starts at a height of 0 feet, so
`\( c = 0 \)`.

- The football is 12 feet off the ground at a horizontal distance of 54 yards (which is
`\( 54 * 3 = 162 \) feet)`, giving us the point
`\( (162, 12) \)`.

- The football hits the ground again 10 yards (which is
`\( 10 * 3 = 30 \) feet)` after passing the goal post, giving us the point
`\( (162 + 30, 0) \) or \( (192, 0) \)`.

We can now use these points to set up equations to solve for the coefficients
`\( a \) and \( b \)` in the quadratic equation
`\( y = ax^2 + bx \) (since \( c = 0 \))`.

1. Substituting the point
`\( (162, 12) \)`:

`\[ 12 = a(162)^2 + b(162) \]`

2. Substituting the point
`\( (192, 0) \)`:

`\[ 0 = a(192)^2 + b(192) \]`

We can solve this system of equations to find the values of
`\( a \) and \( b \)`, and then we can find the maximum height of the football by finding the vertex of the parabola, since the maximum height will be at the vertex.

Let's perform these calculations.

The maximum height of the football is approximately
`\( 22.76 \)` feet.

Rounded to the nearest whole number, the maximum height is
`\( 23 \)`feet.

Answer 2

Without the initial speed and angle of projection, we cannot calculate the exact maximum height mathematically. However, based on the provided information and the symmetry of projectile motion, we estimate the maximum height to be slightly higher than 12 feet, around 13-15 feet, rounded to the nearest whole foot.

Step-by-step explanation:

To determine the maximum height of the football, we start by understanding that we're dealing with a projectile motion problem. The football followed a parabolic trajectory after being kicked. Given that the football started at a height of 0 ft and was at 12 ft when it passed the goal post at a horizontal distance of 54 yards (162 ft), and considering that it landed 10 yards (30 ft) beyond the goal post, its trajectory can be modeled accordingly.

The football's greatest height would occur at the vertex of the parabolic path. Since the ball landed 10 yards beyond the goal post and considering the symmetrical properties of a projectile's path, we can determine that the maximum height was reached 5 yards (15 feet) before the football passed the goal post.

To find the maximum height, we would need information such as the initial speed and the angle of the projection, but as this information is not provided in your question, we are unable to compute an exact answer mathematically. However, typically in these types of problems, the maximum height occurs at the midpoint of the horizontal range, so we can make an educated guess that it would be slightly higher than 12 ft at that midpoint.

Since we're lacking the specific details needed for a precise calculation, and given that we are asked to round to the nearest whole number, we can estimate the ball's maximum height to be slightly higher than 12 ft. Therefore, a reasonable estimate for the maximum height of the football might be around 13-15 ft, rounded to the nearest whole foot.