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?
a^2 + b^2 = c^2
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1. To find the distance from third base to first base on a baseball diamond, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, we have a right triangle with the two sides being the distance from third base to home plate and the distance from home plate to first base. Let's label the distance from third base to home plate as x and the distance from home plate to first base as y.
Now, we know that the length of each side of the square baseball diamond is 90 feet. Therefore, x + y = 90.
By the Pythagorean theorem, we have the equation x^2 + y^2 = (distance from third base to first base)^2.
To find the distance from third base to first base, we need to solve this system of equations.
2. a) The equation relating the distance, D, to the height, h, when viewing the horizon is given by:
D = sqrt(1.5 * h)
To solve this equation for h, we need to isolate h on one side of the equation.
First, square both sides of the equation to get rid of the square root:
D^2 = 1.5 * h
Now, divide both sides of the equation by 1.5 to isolate h:
h = D^2 / 1.5
Therefore, the equation solved for h is h = D^2 / 1.5.
b) To find how far you can see to the horizon from the top of Mammoth Mountain (which is 11,059 feet in elevation), plug the value of D (11,059) into the equation we found in part a:
h = (11,059)^2 / 1.5
Using a calculator to evaluate the expression on the right-hand side yields the answer.
Note: Make sure to use appropriate units throughout the calculations.