Q. 107 - Precalculus Enhanced with Graphing Utilities

Question

Baseball Refer to Problem 105. Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive x-axis lies in 

the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base.

(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.

(b) If the right fielder is located at (310, 15), how far is it from there to second base?

(c) If the center fielder is located at (300, 300), how far is it from there to third base?

Step-by-Step Solution

Answer

(a) First Base$\left(90,0\right)$ Second Base$\left(90,90\right)$ and Third Base $\left(0,90\right)$.

(b) Distance between right fielder and second base is $5\sqrt{2161}$

(c) The distance between centre fielder and third base is $30\sqrt{149}$

Step 1. Given Information

the positive x-axis lies in the direction from home plate to first base, and the positive

y-axis lies in the direction from home plate to third base.

First base lies 90 feet in the x direction and 0 feet in the y direction. So, points are $\left(90,0\right)$

Similarly, find the other bases.

Part (a). Step 1. Finding Base

The coordinate of the first, second and third base are $\left(90,0\right),\left(90,90\right) and \left(0,90\right)$ respectively.

Part (b). Step 1. Finding distance between right fielder and second base

$$\begin{gathered} d=\sqrt{{\left(310-90\right)}^2+{\left(5-90\right)}^2} \\ d=\sqrt{{220}^2+{\left(-75\right)}^2}=5\sqrt{2161} \end{gathered}$$

Part (c) Step 1. Finding distance between center fielder and third base

$$\begin{gathered} d=\sqrt{{\left(300-0\right)}^2+{\left(300-90\right)}^2} \\ d=\sqrt{{300}^2+{210}^2} \\ d=30\sqrt{149} \end{gathered}$$