The pythagorean theorem of baseball is a formula for approximating a team's ratio of wins to games played. let R be the number of runs the team scores during the season, A be the number of run allowed to opponents, W be the # of wins, G be the total of games played. Then the formula W/G≈ R^2/R^2 + A^2 approximates the teams' ratio of wins to games played.
A. solve the formula for A in terms of w, g, and R
b. if the yankees scored 965 runs and won 111 of its 162 played games, approximately (to the nearest whole number) how many runs did the team allow their opponent to score?
I assume you meant
w/g = R^2/(R^2 + A^2)
R^2 + A^2 = R^2 * g/w
A^2 = R^2 (g/w - 1)
so A is the square root of that
Now plug in your numbers to find the desired A
To solve the formula for A in terms of W, G, and R, we need to rearrange the equation. Here's how you can do it:
1. Start with the formula: W/G ≈ R^2/(R^2 + A^2)
2. Multiply both sides of the equation by (R^2 + A^2) to get rid of the denominator on the right side: W/G * (R^2 + A^2) ≈ R^2
3. Distribute the W/G to both terms on the left side: (W/G) * R^2 + (W/G) * A^2 ≈ R^2
4. Move the term (W/G) * R^2 to the right side: (W/G) * A^2 ≈ R^2 - (W/G) * R^2
5. Factor out R^2 from the right side: (W/G) * A^2 ≈ R^2 * (1 - W/G)
6. Divide both sides by (W/G) to isolate A^2: A^2 ≈ R^2 * (1 - W/G) / (W/G)
7. Simplify the right side: A^2 ≈ R^2 * ((1 - W/G) / (W/G))
8. Divide numerator and denominator by (W/G): A^2 ≈ R^2 * ((1 - W/G) / (W/G)) = R^2 * ((1 - W/G) * (G/W))
9. Distribute (1 - W/G) and cancel out G and W in the denominator: A^2 ≈ R^2 * ((G - W) / W)
10. Take the square root of both sides to solve for A: A ≈ √(R^2 * ((G - W) / W))
Therefore, the formula to solve for A in terms of W, G, and R is:
A ≈ √(R^2 * ((G - W) / W))
Now let's move on to part B.
To find approximately how many runs the Yankees allowed their opponents to score, we can use the formula for A that we just derived.
Given:
R = 965 (runs scored by the team)
W = 111 (number of wins)
G = 162 (total games played)
Using the formula for A, we substitute the given values:
A ≈ √(R^2 * ((G - W) / W))
A ≈ √(965^2 * ((162 - 111) / 111))
Calculating this expression, we get:
A ≈ √(931,225 * (51 / 111))
A ≈ √(425,152.03)
Approximating to the nearest whole number, we get:
A ≈ 651
Therefore, the Yankees allowed their opponents to score approximately 651 runs.