10. Modeling With Circles
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Chapter 5
10.
Modeling With Circles
This lesson explores the application of mathematical concepts like area calculations and trigonometric ratios to circular objects in real-life situations. For example, it helps you decide which pizza size offers more food by calculating the area. It also delves into how the Knights of the Round Table could determine the radius of their chairs using trigonometric ratios. These mathematical tools are not just theoretical but have practical applications in various aspects of life, such as food, furniture, and even timekeeping.
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| | 7 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Modeling With Circles
Slide of 7
Based on the features and theorems of the circles, this lesson will relate the circles with the real-life examples.
Catch-Up and Review
Here are some recommended readings before getting started with this lesson.
Explore
Investigating Circular Shaped Objects
Tennis balls are sold in cylindrical containers that contain three balls each. The tennis balls and container can be modeled as follows.
The container, lying on its side, passes through an X-ray scanner in Tallahassee International Airport, where Jordan works as a security guard. Depending on the opacity of the container and the balls, what should Jordan expect to see when she reads the screen of the X-ray scanner? Explore the applet by adjusting the opacity of the container and the tennis balls.
Example
Modeling the Area of Circular Objects
Izabella and her friends are ordering pizza from their favorite spot. They can afford to buy either two 9-inch pizzas or one 13-inch pizza. The thickness of the pizza is the same for each.
They cannot quite decide which option means more food. By modeling each pizza as a circle, help Izabella and her friends determine the best option.
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Hint
Since both pizzas have the same thickness, the area of the pizzas will help to make a decision.
Solution
It is given that the pizzas have the same thickness. Therefore, areas of the pizzas will determine the best option. Begin by modeling the pizzas as circles with diameters of 9 and 13 inches.
9-inch Pizza29=4.5 in13-inch Pizza213=6.5 in
Now that the radii of the pizzas were found, the formula for the area of a circle can be used to find their areas. Begin by finding the area of 9-inch pizza.
The area of the pizza is about 63.62 square inches. Following the same procedure, the area of 13-inch pizza can also be found. | Radius | Substitution | Area | |
|---|---|---|---|
| 9-inch pizza | 4.5 | A=π(4.5)2 | ≈63.62 |
| 13-inch pizza | 6.5 | A=π(6.5)2 | ≈132.73 |
Area of two 9-inch Pizzas: ≈127.24Area of one 13-inch Pizza: ≈132.73
As a result, the best option is to buy a 13-inch pizza. One of the friends was so close to ordering the two 9-inch pizzas. Thanks for helping them make the right choice. Example
Modeling the Radius of Circular Objects
The Knights of the Round Table are the highly-esteemed knights of the mythical fellowship of King Arthur. In the beginning, there were only twelve knights. They met at the Round Table that is a symbol of equality among all of its legendary members.
Assume that each knight has the same circular chair around the Round Table whose radius is 10 feet. Given that all chairs and the Round Table are externally tangent to each other, what is the radius of each chair? Round the answer to the nearest tenth.
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Hint
Begin by modeling the Round Table and the chairs using circles. Draw a triangle whose vertices are the center of the inner circle and two adjacent outer circles. Then use the trigonometric ratios.
Solution
The Round Table and the chairs can be modeled by using circles.
By labeling the centers of the inner circle and two adjacent outer circles, a triangle passing through these centers can be drawn.
Segments that connect the centers of the tangent circles pass through the points of tangency. Assume that the radius of each chair is r feet.
12360∘=30∘
Now that the measure of the vertex angle was found, the altitude of the base can be drawn to use the trigonometric ratios. Remember that the altitude of the base of an isosceles triangle bisects the vertex angle and the base. 
sinθ=ABBD
SubstituteValues
Substitute values
sin15∘=10+rr
▼
Solve for r
MultEqn
LHS⋅(10+r)=RHS⋅(10+r)
10sin15∘+rsin15∘=r
SubEqn
LHS−rsin15∘=RHS−rsin15∘
10sin15∘=r−rsin15∘
FactorOut
Factor out r
10sin15∘=r(1−sin15∘)
DivEqn
LHS/(1−sin15∘)=RHS/(1−sin15∘)
1−sin15∘10sin15∘=r
RearrangeEqn
Rearrange equation
r=1−sin15∘10sin15∘
UseCalc
Use a calculator
r=3.491981…
RoundDec
Round to 1 decimal place(s)
r≈3.5
Example
Modeling Bodies Using Cylinders
Jordan and Ali are comparing their weights. Jordan is 5 feet tall, and Ali is 6 feet tall. Their shoulder widths are 1 foot and 1.4 feet, respectively.
Consider their bodies to be roughly cylindrical, and their shoulder widths could represent the diameter. What is the ratio of Jordan's weight to Ali's weight?
Answer
4:9 or 94
Hint
Volume of a cylinder is the product of the base area and height of the cylinder.
Solution
When a body is modeled using a cylinder, one's height and shoulder width will represent the cylinder's height and diameter, respectively.
By comparing the volumes of these cylinders, a relation between the weights of Jordan and Ali can be set up. To do this, the radii of the cylinders should be found. Recall that the radius of a cylinder is half of its diameter.
Shorter Cylinder21=0.5 ftTaller Cylinder21.4=0.7 ft
Now that the radii have been found, the formula for the volume of a cylinder can be used to find their volumes.
| h | r | πr2h | V | |
|---|---|---|---|---|
| Shorter Cylinder | 5 | 0.5 | π(0.5)2(5) | ≈4 |
| Taller Cylinder | 6 | 0.7 | π(0.7)2(6) | ≈9 |
The volumes of the cylinders imply that if Jordan weighs 80 pounds, Ali weighs 180 pounds. Therefore, the ratio of Jordan's weight to Ali's weight is 4:9 or 94.
Example
Modeling Arc Measures
Arc measure is also an important concept to interpret the situations modeled by circles. For example, consider a time zone wheel.
A time zone wheel is a tool used to find the time in different locations across the world. For example, to find the time in Tokyo when it is 1 P.M. in Amsterdam, rotate the small wheel until the labels of 1 P.M. and Amsterdam align.
As can be seen, it is 9 P.M. in Tokyo.
a What is the arc measure between each time zone on the wheel?
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b What is the measure of the minor arc from the Istanbul zone to the Los Angeles zone?
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Hint
a How many time zones are there on the wheel?
b How many time zones are there between the Istanbul zone and Los Angeles zone?
Solution
a It can be seen that the wheel is divided into 24 time zones. Since the measure of the whole wheel is 360∘, the arc measure between each time zone on the wheel can be found by dividing 360∘ by 24.
24360∘=15∘
b Notice that the minor arc between the Istanbul zone and Los Angeles zone contains 11 time zones. Since the arc measure between each time zone is 15∘, the measure of the minor arc can be found by multiplying 15∘ by 11.
15∘×11=165∘
Closure
Modeling Real Life Situations Using Circles
In real life, there are plenty of situations where circles can be appreciated. In this lesson, a few common situations have been covered. How about some other fun ones? Take, for example, one of the most complex applications of circles — the global positioning system.
Modeling With Circles
Exercises
Consider the roll of toilet paper in the diagram. What is the volume of the toilet paper? Round to the nearest cubic centimeter.
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The toilet roll can be viewed as a large cylinder with a smaller cylindrical hole in the middle. By calculating the volume of the cylinder, including the hole, and then subtracting the volume of the hole, we get the volume of the toilet paper.
The diameter of the toilet paper roll, including the hole, is 8 centimeters. Therefore, the radius is 4 centimeters. With this information, we can find its volume.
The cylindrical hole has a diameter of 2 centimeters, which means its radius is 1 centimeters. With this information, we can calculate the volume of the hole.
Finally, we subtract the volume of the hole from the volume of the large cylinder including the hole. 160π- 10π ≈ 471cm^3
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Kevin is building a 10 meter tall cylindrical silo. What radius will the silo have for its volume to be 785 cubic meters? Round the answer to the nearest meter.
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Let's recall the formula for the volume of a cylinder. V=π r^2 h Let's substitute the known values for volume and height into the formula and then solve for r.
The silo will have a radius of about 5 meters.
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Kevin has bought an ice cream with the following dimensions.
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The ice cream contained within the waffle cone can roughly be modeled by a cone and a hemisphere. Since the hemisphere sits on the base of the cone, they both have a diameter of 6 centimeters and therefore a radius of 3 centimeters.
To determine the volume of a hemisphere and the volume of a cone, we use the following formulas. Hemisphere:& V=2/3π r^3 [0.7em] Cone:& V=1/3π r^2 h Let's calculate the volume of the hemisphere.
Let's also calculate the volume of the cone.
Finally, we will add the volumes of the hemisphere and cone to get the volume of the ice cream. 18π+ 12π≈ 94 cm^3
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A rectangular piece of paper can be rolled to form a cylinder, as shown in the illustration.
A cylinder is made from a square piece of paper with a side length of 10 centimeters.
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To calculate the volume of a cylinder, we can use the following formula. V = π r^2 h We know the height is 10 centimeters since it matches the length of the paper's side. We also know that the diameter is 3.2 centimeters, which means the radius is 1.6 centimeters. Let's substitute these values into the formula and simplify.
The volume is about 80.4 cubic centimeters.
When the paper is rolled, one of the square's sides becomes the circumference of the circle. The length of the side is 20 centimeters which means the circumference will have this length as well. We can calculate the circumference of a circle using the following formula. C = π d In this formula, d is the circle's diameter. Let's substitute C = 20 and solve for d.
The cylinder has a diameter of about 6.4 centimeters.
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Modeling With Circles
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