Problem 50 - Algebra and Trigonometry

Question

A Little League baseball diamond has four bases forming a square whose sides measure 60 feet each. The pitcher's mound is 46 feet from home plate on a line joining home plate and second base. Find the distance from the pitcher's mound to third base. Round to the nearest tenth of a foot.

Step-by-Step Solution

Answer

The distance from the pitcher's mound to third base is approximately 75.6 feet.

Step 1: Identify the right triangle

Recognize that the diamond is divided into two right triangles by the line from home base to second base. Identify the right triangle formed by the home base, pitcher's mound and third base.

Step 2: Apply the Pythagorean theorem

Apply the Pythagorean theorem to this triangle. In a right triangle, the square of the length of the hypotenuse (C, the side opposite the right angle) is equal to the sum of the squares of the other two lengths (A and B). In this case, A is the distance from the mound to home base (46 feet), and B is the distance from home base to third base (60 feet). So, calculate C = sqrt(A^2 + B^2).

Step 3: Calculate the length of the hypotenuse

Plug in the given lengths into the formula: C = sqrt((46 feet)^2 + (60 feet)^2). Simplify inside the square root: C = sqrt(2116 square feet + 3600 square feet). Then, add the two numbers under the square root: C = sqrt(5716 square feet).

Step 4: Determine the final distance

Calculate the square root of 5716 to find the length of the hypotenuse. This comes out to approximately 75.6 feet. However, remember to round as instructed to the nearest tenth of a foot, giving you a final answer of 75.6 feet.