A football player is attempting a field goal. His position on the field
is such that the ball is 7.3 m from the left upright of the goal post
and 10.2 m from the right upright of the goal post. The goal posts
are 4.2 m apart.
a. Find the angle marked θ.
b. If the ball is moved to the middle of the field (position P),
then the ball is equidistant from both uprights:
approximately 8.5 m each. Find the angle corresponding to
θ from this position.
To solve both parts of the problem, we can use the concept of trigonometry and specifically the tangent function.
a. To find the angle marked θ, we can use the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the distance between the ball and the left upright (7.3 m), and the adjacent side is half the distance between the uprights (2.1 m).
Therefore, the tangent of θ is given by:
tan(θ) = opposite/adjacent
tan(θ) = 7.3/2.1
tan(θ) ≈ 3.476
To find θ, we can take the inverse tangent (also known as arctan or tan^-1) of both sides of the equation:
θ = arctan(3.476)
θ ≈ 74.4 degrees
So the angle marked θ is approximately 74.4 degrees.
b. To find the angle corresponding to θ from the position in the middle of the field (P), we can use similar trigonometric reasoning. Now, the distance between the ball and each upright is 8.5 m, which means the opposite side is 8.5 m and the adjacent side is 2.1 m.
Using the same steps as before:
tan(θ) = opposite/adjacent
tan(θ) = 8.5/2.1
tan(θ) ≈ 4.048
θ = arctan(4.048)
θ ≈ 76.1 degrees
So the angle corresponding to θ from the position in the middle of the field is approximately 76.1 degrees.
To find the angle marked θ in both scenarios, we can use the properties of right-angled triangles.
a. Finding the angle θ with the original position:
In the original position, we have a right-angled triangle formed by the ball's position, the left upright, and the right upright. The length of the base (the distance between the left and right uprights) is given as 4.2 m.
Using trigonometry, we can find the angle θ by finding the vertical distance from the ball to a line passing through the midpoint of the base.
Let x be the vertical distance from the ball to the line passing through the midpoint of the base. Using the Pythagorean theorem, we have:
x^2 + (4.2/2)^2 = 7.3^2
x^2 + 2.1^2 = 7.3^2
x^2 = 7.3^2 - 2.1^2
x^2 = 53.29 - 4.41
x^2 = 48.88
x ≈ √48.88
x ≈ 6.98 m
Now, we can find the angle θ using the tangent ratio:
tan(θ) = (opposite/adjacent) = (x/(4.2/2)) = (6.98/2.1)
θ = tan^(-1)(6.98/2.1)
Using a calculator, θ ≈ 73.7°
b. Finding the angle θ with the new position (point P in the middle of the field):
In the new position, the ball is equidistant from both uprights, approximately 8.5 m from each. Here, we have a right-angled triangle formed by the ball's position, the left upright, and the right upright. The length of the base (the distance between the left and right uprights) is still given as 4.2 m.
Similar to the previous scenario, let x be the vertical distance from the ball to the line passing through the midpoint of the base. Using the Pythagorean theorem, we have:
x^2 + (4.2/2)^2 = 8.5^2
x^2 + 2.1^2 = 8.5^2
x^2 = 8.5^2 - 2.1^2
x^2 = 72.25 - 4.41
x^2 = 67.84
x ≈ √67.84
x ≈ 8.23 m
Using the same formula as before, we can find the angle θ:
tan(θ) = (opposite/adjacent) = (x/(4.2/2)) = (8.23/2.1)
θ = tan^(-1)(8.23/2.1)
Using a calculator, θ ≈ 76.7°