The Conics
By: Diana Brown
Day
Three:
Some
Circle Investigations
What happens to the circumference of a circle if you double the
diameter? If you triple the diameter? If you half the diameter? As the diameter
increases (or decreases) in measure, how does the circumference change? Why
does this change occur?
First we will construct a circle in Geometer’s Sketchpad
(GSP):
Here are some measurements of the above circle:
Radius: 4.42
Diameter: 8.84
Circumference: 27.76
Now let’s create another circle that has a diameter half
the size of the above circle and find the same measurements:
Radius: 2.21
Diameter: 4.42
Circumference: 13.88
Notice that each of the measures (even the Circumference)
is exactly one half of the first circle. So as the diameter is halved the
circumference is halved.
Why is this so?
Well, what do we know about the circumference of a
circle?
The circumference of a circle can be found by multiplying
pi (which is equal to 3.14) by the diameter of the circle.
C = πD,
Examples:
Original Diameter = 10
C = π (10) = 31.4
C = π (5) = 15.7
31.4 / 15.7 = 2
Notice that the formula for the Circumference of a circle
is a direct variation formula, with Pi being the constant and the diameter and
the circumference being the variables.
What is direct variation?
When two variable quantities have a constant
(unchanged) ratio, their relationship is called a direct
variation.
It is said that one variable "varies directly" as the
other.
The constant ratio is called the constant of variation.
We can show this
by circumscribing a square around a circle.
The circumference C
of a circle varies directly as the perimeter P of the circumscribed
square.
C = k1P
As the boundary of the square
changes, the boundary of the circle changes proportionally.
But the perimeter varies directly
as the side. Hence, the circumference varies directly as the side --
because varies directly is a transitive
relation:
If
a varies directly as b, and b varies directly as c,
then a varies directly as c.
If a = k1b, and b = k2c, then a = k1k2c.
Therefore,
the circumference C varies directly as the side s.
But
s is equal to the diameter D!
C = kD
Suppose a 6-foot tall man is walking around the world.
How much further does his head travel than his feet?
What is the circumference of the
earth?
The circumference of the earth at
the equator is 24,901.55 miles (40,075.16 kilometers).
But, if you measure the earth through the poles the
circumference is a bit shorter - 24,859.82 miles (40,008 km). This earth is a
tad wider than it is tall, giving it a slight bulge at the equator. This shape
is known as an ellipsoid or more properly, geoid (earth-like).
Information obtained from: http://geography.about.com
Lets use a circumference of is 24,901.55 miles. If this is so then we can find the diameter
by substituting the circumference into the formula:
C = πD,
24,901.55 = πD
Dividing both sides by π gives a diameter of:
D = 7,926.41 miles
To convert this to feet we
multiply 7,926.41 miles by 5280 ft/mi = 41,851,442 feet.
Look at the following diagram:
Notice that if FG is the diameter of the earth and the
man is the length of EF (F being the point of his feet and E being the point of
his head) which is 6 feet tall then we need to add 12 feet to the diameter of
the earth to get the circumference that the man’s head is traveling. Here is how the math goes to determine how
much further his head travels than his feet.
will be the
circumference of the earth. = 24,901.55 x 5280 =
131,480,184 feet. Which is also the
distance the man’s feet travels.
will be the
circumference that the man’s head travels.
= π (41,851,454)
= 131,480,222
To determine how much further his head travels than his
feet we will subtract from :
131,480,222- 131,480,184 = 38 feet.
Other investigations:
1.
When
Thomas enters a pizza store, he notices that a 14 inch circular pizza has a
price of $7 and a 16 inch circular pizza has a price of $12. He complains the
prices are unfair? Why? What should be the "fair" price of the 16
inch pizza?
2.
The
trails in a park resemble the following diagram. Find the total length of the
trails, given that all the circles have the same area and circle 2 is tangent
to the midpoint of L.
3.
A
bicycle wheel has a radius of 13 inches. How many revolutions must the wheel
make to go 1 mile?
4.
Suppose
a string is wrapped around the Earth at the equator. (Assume the Earth is a
sphere with a smooth surface.) Now pretend that we pull the string outward from
the earth's surface so that the string is always one-inch away from the earth's
surface all the way around the equator. How much more string will be needed to
complete this new circle?
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Four (Ellipse Introduction)