Hockey Shot Angle: 8m from 2m Net

Answer

To score, Lars must fire the puck within an angle of approximately $14^\circ$. This is calculated by modeling the geometry as two congruent right-angled triangles and applying the inverse tangent function.

Explanation

The image depicts a hockey puck positioned $8\text{ m}$ away from the center of a goal net. The net has a total width of $2\text{ m}$. We can observe that the distance from the puck to the center acts as an altitude, bisecting the total angle into two equal parts and creating two right-angled triangles.

1. Identify Known Quantities The total width of the net is $2\text{ m}$. Since the puck is shot toward the middle, we divide this by $2$ to find the "opposite" side of our right triangle.

   • Perpendicular distance to the net ($adj$): $8\text{ m}$
   • Half-width of the net ($opp$): $1\text{ m}$
   • Full angle to find: $\theta$

2. Establish the Trigonometric Relationship We define half of the desired angle as $\alpha$. In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. $\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}$ The tangent function relates the slope of the shot's trajectory to the dimensions of the field.

3. Calculate the Half-Angle Substitute the values into the formula and use the inverse tangent function ($\arctan$) to solve for $\alpha$. $\alpha = \tan^{-1}\left(\frac{1}{8}\right)$ This calculation determines the angle required to reach just one post of the net from the center line. $\alpha \approx 7.125^\circ$ ⚠️ This step is required on exams: ensure your calculator is set to Degree mode rather than Radians.

4. Determine the Total Shooting Angle The full angle $\theta$ within which the puck will enter the net is twice the value of $\alpha$, as the puck can go to either the left or right side of the center. $\theta = 2 \times \alpha$ Multiplying by two accounts for the entire $2\text{ m}$ aperture of the goal. $\theta = 2 \times 7.125^\circ = 14.25^\circ$

5. Final Rounding The question asks for the result to the nearest degree. $\theta \approx 14^\circ$

Final Answer

Lars must fire his shot within an angle of: $\boxed{14^\circ}$

Common Mistakes

• Using the full width: Students often use $2\text{ m}$ as the opposite side in a single right triangle calculation, which incorrectly assumes the puck is lined up with one of the posts rather than the middle.
• Calculator Mode: Failing to switch the calculator from Radians to Degrees will result in an answer of approximately $0.25$, which is incorrect for this context.