Distance between the goal posts, 19.5 feet, is the length of an arc on a circle of radius 50 yards. The kicker aims to kick the ball midway between the uprights. To score a field goal, what is the maximum number of degrees that the actual trajectory can deviate from the intended trajectory? What is the maximum number of degrees for a 20-yard field goal?
So I know that the length is 19.5 feet, and since 3 feet = 1 yard, that the 50 yard radius = 150 feet radius of circle.
I'm stuck in what to do next? Please help.
let's stick with yards. The distance between the goal posts is 6.5 yards.
So, since arc length s = rθ, we want θ such that
50θ = 6.5
θ = 0.13 radians = 7.45°
But, he can only deviate by 1/2 that, or 3.72°
Plug in r=20 for the other part.
Technically, the distance between the posts is the chord of the arc, but this is what the problem stated, and it is easier to deal with.
To find the maximum number of degrees that the actual trajectory can deviate from the intended trajectory, we can calculate the central angle corresponding to the arc length of 19.5 feet.
1. Convert the length of the arc from feet to yards:
19.5 feet / 3 feet/yard = 6.5 yards
2. Calculate the central angle using the formula:
Central angle = (Arc length / Radius) * (180 degrees / π)
Central angle = (6.5 yards / 50 yards) * (180 degrees / π)
Central angle ≈ 4.68 degrees
Therefore, the maximum number of degrees the actual trajectory can deviate from the intended trajectory is approximately 4.68 degrees.
To find the maximum number of degrees for a 20-yard field goal, we can follow the same steps:
1. Convert the length of the arc from yards to feet:
20 yards * 3 feet/yard = 60 feet
2. Calculate the central angle using the formula:
Central angle = (Arc length / Radius) * (180 degrees / π)
Central angle = (60 feet / 150 feet) * (180 degrees / π)
Central angle ≈ 68.8 degrees
Therefore, the maximum number of degrees the actual trajectory can deviate from the intended trajectory for a 20-yard field goal is approximately 68.8 degrees.
To find the maximum number of degrees that the actual trajectory can deviate from the intended trajectory, we need to determine the angle at the center of the circle that corresponds to the length of the arc between the goal posts.
First, let's convert the length of the arc from feet to yards. Since 3 feet = 1 yard, the length of the arc is 19.5 feet / 3 = 6.5 yards.
Now, we need to find the angle at the center of the circle that would produce a 6.5-yard arc on a circle with a 50-yard radius.
To do this, we can use the formula for the length of an arc: Length of Arc = Radius * Central Angle (in radians).
Since we have the length of the arc and the radius, we can rearrange the formula to solve for the central angle: Central Angle (in radians) = Length of Arc / Radius.
Plugging in the values, we get: Central Angle (in radians) = 6.5 yards / 50 yards.
Now, we need to convert the angle from radians to degrees. Since there are π (pi) radians in 180 degrees, we can use the conversion formula: Angle (in degrees) = Central Angle (in radians) * 180 / π.
Plugging in the values, we get: Angle (in degrees) = (6.5 yards / 50 yards) * 180 / π.
Now, we can calculate the maximum number of degrees that the actual trajectory can deviate from the intended trajectory for a 50-yard field goal.
For a 20-yard field goal, we repeat the same steps but with a different length of arc.
First, convert the length of the arc to yards: 20 yards.
Then, use the formula Central Angle (in radians) = Length of Arc / Radius to find the central angle.
Finally, convert the angle from radians to degrees using Angle (in degrees) = Central Angle (in radians) * 180 / π.
Now, you can plug in the values and calculate the maximum number of degrees for a 20-yard field goal.