 |
|
|
|
Solutions for Session 7, Part B
See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6 | B7 | B8 | B9 | B10
|
|
 |
Problem B2 | |
The length of the base is one-half the circle's circumference, since the entire circumference comprises the scalloped edges that run along the top and bottom of the figure, and exactly half of it appears on each side. The base length is C/2.
<< back to Problem B2
<< back to Problem B2 (non-interactive version)
|
|
|
|
|
 |
Problem B4 | |
As the number of wedges increases, each wedge becomes a nearly vertical piece. The base length becomes closer and closer to a straight line of length r (or half the circumference), while the height is equal to r. The area of such a rectangle is r r, or r2.
<< back to Problem B4
<< back to Problem B4 (non-interactive version)
|
|
|
|
 |
Problem B5 | |
Here is the completed table:
 |
 |
Circle |
 |
Radius of Circle |
 |
Area of Radius Square |
 |
Area of Circle |
 |
Number of Radius Squares Needed |
 |
 |
1 |
 |
6 |
 |
36 |
 |
36  |
 |
A little more than 3 |
2 |
4 |
16 |
16  |
A little more than 3 |
3 |
3 |
9 |
9  |
A little more than 3 |
|
 |
<< back to Problem B5
|
|
|
|
 |
Problem B6 | |
a. | In each case, it takes a little more than three radius squares to form the circle. If using approximations, it should always take around 3.14 of the squares to cover the circle. |
b. | The best estimate is somewhere between 3.1 and 3.2, which we know is roughly the value of . |
<< back to Problem B6
|
|
|
|
 |
Problem B7 | |
The formula for the area of a circle is A = r2. The activity helps one understand that a bit more than three times a radius square is needed to cover the circle. Namely, it illustrates why the formula is r2.
<< back to Problem B7
|
|
|
|
 |
Problem B8 | |
Think about a circle with a radius equal to 1 (r = 1). The circumference and the area of this circle are as follows:
C = 2 1 = 2
A = 12 = 
Now double the radius to 2 units (r = 2). The circumference and the area of the new circle are as follows:
C = 2 2 = 4 
A = 22 = 4 
The circumference of the new circle doubled, but the area is multiplied by a factor of 4 (the square of the scale factor). You can replace the 1 with any other number, or with a variable r, to see that this relationship will always hold.
<< back to Problem B8
|
|
|
|
|