Question

Field goal attempt Near the end of the 1950 Rose Bowl football game between the University of California and Ohio State University, Ohio State was preparing to attempt a field goal from a distance of 23 yd from the end line at point \(A\) on the edge of the kicking region (see figure). But before the kick, Ohio State committed a penalty and the ball was backed up 5 yd to point \(B\) on the edge of the kicking region. After the game, the Ohio State coach claimed that his team deliberately committed a penalty to improve the kicking angle. Given that a successful kick must go between the uprights of the goal posts \(G_{1}\) and \(G_{2},\) is \(\angle G_{1} B G_{2}\) greater than \(\angle G_{1} A G_{2} ?\) (In \(1950,\) the uprights were \(23 \mathrm{ft} 4\) in apart, equidistant from the origin on the end line. The boundaries of the kicking region are \(53 \mathrm{ft} 4\) in apart and are equidistant from the \(y\) -axis. (Source: The College Mathematics Journal 27, 4, Sep 1996).

Short answer

Answer: No, the angle \(\angle G_{1} B G_{2} \approx 20.56^\circ\) is less than the angle \(\angle G_{1} A G_{2} \approx 24.82^\circ\).

Step-by-step solution

Calculate the distances of A and B from the end line

From the problem, we know that \(A\) is \(23\,yd\) from the end line and \(B\) is backed up by \(5\,yd\). We will first convert yards into feet and then calculate the distance of \(B\) from the end line. 1 yard = 3 feet Distance of A from the end line: \(23\,yd \times 3 = 69\,ft\) Distance of B from the end line: \((23+5)yd \times 3 = 84\,ft\)

Find the distances of G1 and G2 from points A and B

In the problem, we're given that the uprights \(G_1\) and \(G_2\) are equidistant from the origin and that the distance between them is \(23\,ft\, 4\) in. We'll now find the distances of \(G_1\) and \(G_2\) from points \(A\) and \(B\). Distance between G1 and G2: \(23\,ft\, 4\,in = 23.33\,ft\) Since \(G_1\) and \(G_2\) are equidistant from the origin, the distance of \(G_1\) and \(G_2\) from the y-axis is half of their distance apart, which is \(11.67\,ft\) (\(23.33/2\)). Since the boundaries of the kicking region are \(53\,ft\, 4\,in\) apart, so the distance between A and y-axis is equal to \(\dfrac{53.33}{2} = 26.67\,ft\). Similarly, the distance between B and y-axis is equal to \(26.67\,ft\). The distance between \(G_1\) and \(A\) can be found using the Pythagorean theorem: \(G_1A = \sqrt{(69\,ft)^2 + (15\,ft)^2} = 69.51\,ft\) \(G_2A = \sqrt{(69\,ft)^2 + (15\,ft)^2} = 69.51\,ft\) The distance between \(G_1\) and \(B\) can be found as follows: \(G_1B = \sqrt{(84\,ft)^2 + (15\,ft)^2} = 84.49\,ft\) \(G_2B = \sqrt{(84\,ft)^2 + (15\,ft)^2} = 84.49\,ft\)

Use the arctangent function to find the angles

Now, we'll use the arctangent function to find the angles \(\angle G_{1} A G_{2}\) and \(\angle G_{1} B G_{2}\). \(\angle G_{1} A G_{2} = 2\cdot \arctan({\dfrac{15}{69}}) = 24.82^\circ\) \(\angle G_{1} B G_{2} = 2\cdot \arctan({\dfrac{15}{84}}) = 20.56^\circ\)

Compare the angles

Now we'll compare the angles to determine if \(\angle G_{1} B G_{2}\) is greater than \(\angle G_{1} A G_{2}\). The angle \(\angle G_{1} B G_{2} \approx 20.56^\circ\) is less than \(\angle G_{1} A G_{2} \approx 24.82^\circ\). Therefore, the claim of the Ohio State coach that his team deliberately committed a penalty to improve the kicking angle is incorrect, as the angle \(\angle G_{1} B G_{2}\) is not greater than the angle \(\angle G_{1} A G_{2}\).

Key concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in mathematics that connects the lengths of the sides in a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In formula form, it is expressed as \( a^2 + b^2 = c^2 \), where \( c \) represents the hypotenuse.
In the field goal problem, we used the Pythagorean theorem to find the distances from points A and B to the goal posts G1 and G2. Given that point A is 69 feet and B is 84 feet from the end line, and the lateral distance to each goal post from the y-axis is 15 feet, we can calculate the direct distance to each point using the theorem:

• For point A: \( \sqrt{(69)^2 + (15)^2} = 69.51 \) feet
• For point B: \( \sqrt{(84)^2 + (15)^2} = 84.49 \) feet

Understanding the Pythagorean theorem helps us calculate these precise distances that are crucial for determining the angles in trigonometric problems.

arctangent function

The arctangent function is a trigonometric function used to calculate an angle when given the ratio of the opposite side to the adjacent side in a right triangle. The function is denoted as \( \tan^{-1} \) or \( \mathrm{arctan} \). Utilizing this function is essential when determining angles from side lengths, which in turn allows us to assess situations involving triangles or vectors.
In the context of the field goal exercise, we employ the arctangent function to determine the angles \( \angle G_{1} A G_{2} \) and \( \angle G_{1} B G_{2} \) at the kicker’s positions A and B:

• For position A, the angle is \( 2 \times \arctan\left(\dfrac{15}{69}\right) = 24.82^\circ \)
• For position B, the angle is \( 2 \times \arctan\left(\dfrac{15}{84}\right) = 20.56^\circ \)

This application of the arctangent function helps us quantify the angles under consideration, providing an accurate measure essential for comparing which angle offers a better field goal opportunity.

angle comparison

Angle comparison involves assessing two or more angles to determine their respective sizes or optimality based on a specific context, such as projective alignment or optimal path finding. In the field goal scenario, the task was to compare two kicking angles to verify the claim that stepping back improves the angle for a better kick.
After calculating using arctangent, we found that:

• \( \angle G_{1} A G_{2} = 24.82^\circ \)
• \( \angle G_{1} B G_{2} = 20.56^\circ \)

The comparison shows that \( \angle G_{1} A G_{2} \) is indeed greater than \( \angle G_{1} B G_{2} \), debunking the Ohio State coach's claim. This analysis demonstrates how precise mathematical calculations and critical angle comparison can yield accurate assessments in practical scenarios, like sports strategies.