1. A baseball diamond is actually a square, 90 feet on each side. What is the distance from third base to first base?
2. The equation gives the distance, D, in miles that a person can see to the horizon from a height, h, in feet.
a) Solve this equation for h
b) Mammoth Mountain is 11,059 feet in elevation. How far can you see to the horizon from the top of Mammoth Mountain?
1) Use pythagoras theorem:
a right triangle with 90 as the two sides.
1. To find the distance from third base to first base in a baseball diamond, you can use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse (the longest side) of a right triangle is equal to the sum of the squares of the other two sides. In this case, we can imagine a right triangle with the distance from third base to home plate as one side (90 feet) and the distance from first base to home plate as another side (also 90 feet). The missing side, which represents the distance from third base to first base, can be found by using the Pythagorean theorem.
Using the formula c^2 = a^2 + b^2, where c is the hypotenuse and a and b are the other two sides, we can plug in the values:
c^2 = 90^2 + 90^2
c^2 = 8100 + 8100
c^2 = 16200
c ≈ √16200
c ≈ 127.28 feet
Therefore, the distance from third base to first base in a baseball diamond is approximately 127.28 feet.
2a. The equation D = √(2Rh) gives the distance, D, in miles that a person can see to the horizon from a height, h, in feet.
To solve the equation for h, we need to isolate h on one side of the equation.
Start by squaring both sides of the equation:
D^2 = 2Rh
Now, divide both sides of the equation by 2R:
D^2 / (2R) = h
Therefore, the equation can be solved for h by dividing D^2 by 2R.
2b. If Mammoth Mountain is 11,059 feet in elevation, we can use the equation from part 2a to calculate how far you can see to the horizon from the top of Mammoth Mountain.
Let's assume R represents the radius of Earth, which is approximately 3,959 miles.
Using the equation D = √(2Rh), we can substitute the values into the equation:
D = √(2 * 3959 * 11059)
D ≈ √(87903522)
D ≈ 9369.68
Therefore, from the top of Mammoth Mountain, you can see approximately 9,369.68 miles to the horizon.