Math Question Problem

Question

 The area of a circle in relation to its radius is given on the table below:

Radius (cm) -> Area (cm^2
1	3.141593
2	12.56637
3	28.27433
4	50.26548
5	78.53982
6	113.0973
7	153.938
8	201.0619
9	254.469
10	314.1593
11	380.1327
12	452.3893

1. Graph the area as a function of radius.
2. Draw a secant on the graph corresponding to each time interval below. Then calculate the average rate of change.
3. 1 cm to 12 cm
4. 11 cm to 12 cm
5. 1 cm to 2 cm

Answer 1

Area = pi(r^2)

pi=about 3.141593...

A=pi(r^2) is an upward opening parabola with vertex at the origin, (0,0).

Plot A on the vertical y-axis and r on the horizontal x-axis

a secant line from r=1 to 2 is a line segment from (1,pi) to (2, 4pi)

a secant line from r=1 to 11 is a line segment from (1,pi) to (11, 121pi)

a secant line from r=11 to 12 is a line segment from (11, 121pi) to (12, 144pi)

rates of change are the slopes of the secant lines,

(4pi-pi)/(2-1) = 3pi,

(121pi-pi)/11-1) = 12pi,

(144pi-121pi)/(12-11) = 23pi

144pi = about 452.3893

121pi = about 380.1327

4pi = about 50.26548

3pi = about 9.424779

12pi = about 37.69916

23pi = about 72.256639