Chapter 3.1, Problem 69PPS

(a)

To determine

Equations for winning times for men and women since 1964.

Answer to Problem 69PPS

For men:

  $y=-0.00705x+10$

For women:

  $y=-0.01409x+11.4$

Explanation of Solution

Given information:

Table for winning times (in seconds) for the 100 − meter dash at the Olympics between 1964 and 2008:

Years since 1964, x	Men’s Gold Medal Time	Women’s Gold Medal Time
0	10.0	11.4
4	9.90	11.0
8	10.14	11.07
12	10.06	11.08
16	10.25	11.06
20	9.99	10.97
24	9.92	10.54
28	9.96	10.82
32	9.84	10.94
36	9.87	10.75
40	9.85	10.93
44	9.69	10.78

Winning times for men:

Find the slope first using first and last winning points.

To find the slope, use the pair of points (0, 10.0) and (44, 9.69):

For slope:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Substitute using the pair of points:

  $m=\frac{9.69-10.0}{44-0}$

Perform operation inside parentheses:

  $m=\frac{-0.31}{44}$

Simplify the fraction:

  $m=-0.00705$

Now,

Equation for point slope form:

  $y-y_1=m\left(x-x_1\right)$

Let

  $x_1,y_1=\left(0,10.0\right)$

And

  $m=-0.00705$

Substitute the values:

  $y-10.0=-0.00705\left(x-0\right)$

Use distributive property:

  $y-10.0=-0.00705x+0$

Add 10.0 on each side:

  $y-10.0+10.0=-0.00705x+0+10.0$

Simplify:

  $y=-0.00705x+10.0$

Winning times for women:

Find the slope first using first and last winning points.

To find the slope, use the pair of points (0, 11.4) and (44, 10.78):

For slope:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Substitute using the pair of points:

  $m=\frac{10.78-11.4}{44-0}$

Perform operation inside parentheses:

  $m=\frac{-0.62}{44}$

Simplify the fraction:

  $m=-0.01409$

Now,

Equation for point slope form:

  $y-y_1=m\left(x-x_1\right)$

Let

  $x_1,y_1=\left(0,11.4\right)$

And

  $m=-0.01409$

Substitute the values:

  $y-11.4=-0.01409\left(x-0\right)$

Use distributive property:

  $y-11.4=-0.01409x+0$

Add 11.4 on each side:

  $y-11.4+11.4=-0.01409x+0+11.4$

Simplify:

  $y=-0.01409x+11.4$

(b)

To determine

Estimation of time at which women’s performance will catch up to men’s performance.

Graphically represent the equations for both men and women.

Answer to Problem 69PPS

Women’s performance will catch up to men’s performance approximately in the year 2162.

Explanation of Solution

Given information:

Table for winning times (in seconds) for the 100 − meter dash at the Olympics between 1964 and 2008:

Years since 1964, x	Men’s Gold Medal Time	Women’s Gold Medal Time
0	10.0	11.4
4	9.90	11.0
8	10.14	11.07
12	10.06	11.08
16	10.25	11.06
20	9.99	10.97
24	9.92	10.54
28	9.96	10.82
32	9.84	10.94
36	9.87	10.75
40	9.85	10.93
44	9.69	10.78

From Part (a),

We have

For men:

  $y=-0.00705x+10$

For women:

  $y=-0.01409x+11.4$

In the graph below,

Red line represents the equation for men.

And

Blue line represents the equation for women.

  

Graphing the two equations, the men’s and women’s winning points will meet approximately at (198, 8.6).

This implies

In the year 2162 (2164 is the actual Olympics),

Women will catch up to the performance of men.

However,

The prediction is not reasonable because the number of winning gold medal may vary from men’s and women’s division.