According to Little League baseball official regulations, the diamond is a square 60 feet on a side. The pitching rubber is located 46 feet from home plate on a line joining home plate and second base.
(a)How far is it from the pitching rubber to first base?
(b)How far is it from the pitching rubber to second base?
(c)If a pitcher faces home plate, through what angle does he need to turn to face first base?
The correct answer and explanation is :
We have a Little League baseball diamond, which is a square where each side is 60 feet long. The pitching rubber is 46 feet from home plate along the line joining home plate and second base. We are tasked with determining the distance from the pitching rubber to first base, the distance from the pitching rubber to second base, and the angle through which the pitcher needs to turn to face first base.
(a) Distance from the pitching rubber to first base
To solve for the distance from the pitching rubber to first base, let’s visualize the situation:
- The square is formed by home plate (H), first base (F), second base (S), and third base (T).
- The pitching rubber (P) is located on the line joining home plate and second base, 46 feet from home plate.
Now, to find the distance from P to F (the first base), we use the Pythagorean theorem. The distance between the pitching rubber and first base forms a right triangle, where:
- One leg is the distance from P to home plate (46 feet),
- The other leg is the distance from home plate to first base (60 feet, since the diamond is a square).
Thus, the distance from P to F is the hypotenuse of this right triangle. Using the Pythagorean theorem:
[
\text{Distance from P to F} = \sqrt{(46)^2 + (60)^2} = \sqrt{2116 + 3600} = \sqrt{5716} \approx 75.6 \text{ feet}
]
(b) Distance from the pitching rubber to second base
The pitching rubber is 46 feet from home plate along the line joining home plate and second base. Therefore, the distance from the pitching rubber to second base is simply:
[
\text{Distance from P to S} = 60 – 46 = 14 \text{ feet}
]
(c) Angle to turn to face first base
To calculate the angle through which the pitcher must turn to face first base, we first note that the pitcher is facing home plate, and we need to find the angle between the line from the pitching rubber to home plate and the line from the pitching rubber to first base. This is an angle in a right triangle with sides:
- 46 feet (from P to H),
- 60 feet (from H to F).
The angle θ can be found using the tangent function:
[
\tan(\theta) = \frac{60}{46}
]
[
\theta = \tan^{-1}\left(\frac{60}{46}\right) \approx \tan^{-1}(1.304) \approx 52.4^\circ
]
Thus, the pitcher needs to turn approximately 52.4 degrees to face first base.
Summary
- (a) The distance from the pitching rubber to first base is approximately 75.6 feet.
- (b) The distance from the pitching rubber to second base is 14 feet.
- (c) The pitcher needs to turn approximately 52.4 degrees to face first base.
This solution involves basic geometry using the Pythagorean theorem and trigonometric functions, which are essential tools in solving such distance and angle problems on a baseball diamond.