Warm Up Section 4.10 1. Find the area of a circle with diameter 12 in.

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Warm Up Section 4.10 1. Find the area of a circle with diameter 12 in. Find the radius of a circle with circumference 46 ft. Find the surface area of a sphere with diameter 10 m. Find the volume of a sphere with radius 8 cm. Find the area of a sector of a circle whose central angle measures 80o and whose radius measures 5 m.

1. diameter 12 in. 2. C = 2r A= (6)2 = 36 46 = 2r Answers to Warm Up Section 4.10 1. diameter 12 in. 2. C = 2r A= (6)2 = 36 46 = 2r A  113.1 in2 23 ft = r diameter 10 m. 4. 5. SA = 4r2 SA = 4(5)2 SA = 100 SA  314.2 m2

Application of Circles Section 4.10 Standard: MM2G3 a-d MM2G4ab Essential Question: How do I use concepts of circles and spheres to solve problems?

the measure of one section = = 30o In the clock face shown at the right, the positions of the numbers determine congruent arcs along the circle. What is the measure of the arc between any two consecutive numbers?   Since there are 12 equal sections on the clock, the measure of one section = = 30o An arc is traced out by the end of the second hand as it moves from the 12 to the 4. Is this a minor arc or a major arc? Minor arc (less than 180o)

Reasoning: 180o = 6(30o), so count six In the clock face shown at the right, the positions of the numbers determine congruent arcs along the circle. Starting at 2, what number does the end of the second hand reach as it completes a semicircle? Answer: 8 Reasoning: 180o = 6(30o), so count six sections around the clock from 2.

Reasoning: Count from 8 to 3. There are 7 sections, so 7(30o) = 210o. In the clock face shown at the right, the positions of the numbers determine congruent arcs along the circle. d. When the second hand moves from the 8 to the 3, what is the measure of the arc? Answer: 210o Reasoning: Count from 8 to 3. There are 7 sections, so 7(30o) = 210o.

So, the clock moved from 15o to 24o which is a distance of 9o. In the clock face shown at the right, the positions of the numbers determine congruent arcs along the circle. e. When the second hand moves from halfway between the 1 and the 2 to 4/5 of the way from the 1 to the 2, what is the measure of the arc? Answer: 9o Each section is 30o. So, the clock moved from 15o to 24o which is a distance of 9o.

From 3 to 7 there are 4 sections. 4(30o) = 120o. In the clock face shown at the right, the positions of the numbers determine congruent arcs along the circle. The second hand moves from the 3 to the 7. What is the measure of the corresponding major arc? Answer: 240o From 3 to 7 there are 4 sections. 4(30o) = 120o. So, the major arc measures 360o – 120o = 240o.

2. A play is being presented on a circular stage 2. A play is being presented on a circular stage. The two main characters are positions A and B at the back of the stage. Use the diagram to answer the following questions. What angle of view between the main characters does an actor at position C at the center stage have?   b. What angle of view between the main characters does the orchestra conductor at point D have? c. What angle of view between the main characters does an audience member at point E have? A B D E 80° 38° C 80o ½ (80)o = 40o ½ (80 – 38)o = 21o

50 cm AC = AD = 80 cm 50(x) = (80)(80) 50x = 6400 EA = 128 cm x = 128 3. An arch over a doorway is an arc that is 160 centimeters wide and 50 centimeters high. You are curious about the side of the entire circle containing the arch. By drawing and labeling a circle passing though the arc as shown in the diagram, you can use the following steps to find the radius of the circle. a. Find AB.   b. Find AC and AD. c. Use AB, AC, and AD to find EA. 160 cm B 50 cm 50 80 A 80 50 cm C D O AC = AD = 80 cm x E 50(x) = (80)(80) 50x = 6400 x = 128 EA = 128 cm

3. An arch over a doorway is an arc that is 160 centimeters wide and 50 centimeters high. You are curious about the side of the entire circle containing the arch. By drawing and labeling a circle passing though the arc as shown in the diagram, you can use the following steps to find the radius of the circle. d. Find EB   e. Find the length of the radius, EO. 160 cm B 128 cm + 50cm = 178 cm 50 80 A 80 50 cm C D O 128 Radius = ½ (178 cm) = 89 cm E

4. You want to estimate the diameter of a circular water tank. You stand at a location 10.5 feet from the edge of the circular tank. From this position, your distance to a point of tangency on the tank is 23 feet. a. Draw a diagram of the situation. Label your position as C and the radius of the tank as r.   r 10.5 r C 23

4. You want to estimate the diameter of a circular water tank. You stand at a location 10.5 feet from the edge of the circular tank. From this position, your distance to a point of tangency on the tank is 23 feet.   b. Find the length of the radius to the nearest tenth of a foot.    r2 + (23)2 = (r + 10.5)2 r2 + 529 = r2 + 21r + 110.25 r 10.5 r 529 = 21r + 110.25 C 23 418.75 = 21r 19.9 = r The radius of the tank is 19.9 feet.

5. A satellite is about 100 miles above Earth’s surface 5. A satellite is about 100 miles above Earth’s surface. The satellite is taking photographs of Earth. Earth’s diameter is about 8000 miles. a. Draw a diagram of the situation, representing Earth as a circle. b. In the diagram, draw a segment to show one of the farthest possible points on Earth that can be photographed from the satellite. What type of geometric figure is the segment?       100 x 4000 8000 tangent

c. Find the length of the segment drawn in part b. x 4000 100 x2 + (4,000)2 = (4,100)2 x2 + 16,000,000 = 16,810,000 4000 x2 = 810,000 x = 900 One of the farthest points on earth that can photographed by the satellite is 900 miles from the satellite.

6. The chain of a bicycle travels along the front and rear sprockets, as shown. The circumference of each sprocket is given. a. About how long is the chain?   10 in 160° 185° rear sprocket C = 12 in front sprocket C = 20 in 35.6 inches long Rear: Front: Total: 10 5.3 + 10.3 35.6

6. The chain of a bicycle travels along the front and rear sprockets, as shown. The circumference of each sprocket is given.   On a chain, the teeth are spaced in ½ inch intervals. About how many teeth are there on this chains. 10 in 160° 185° rear sprocket C = 12 in front sprocket C = 20 in There are 2 teeth for every inch. So, There are about 71 teeth on the chain.

You need about 56.5 feet of fencing. 7. You have planted a circular garden with its center at one of the corners of your garage, as shown. You want to fence in your garden. About how much fencing do you need? 12 ft You need about 56.5 feet of fencing.

A = r2 A = (21)2 A = 441 A 1385.4 ft2 d = 2r 42 = 2r 21 = r 8. A circular water fountain has a diameter of 42 feet. Find the area of the fountain. A = r2 A = (21)2 A = 441 A 1385.4 ft2 42 ft d = 2r 42 = 2r 21 = r The area of the fountain is 1385.4 ft2.

a. What is the area of the lawn that is covered by the sprinkler? 9. The diagram below shows the area of a lawn covered by a water sprinkler.   a. What is the area of the lawn that is covered by the sprinkler? 135° 16 ft The sprinkler covers about 301.6 ft2 of lawn.

b. The water pressure is lowered so that the radius is 10 feet. 9. The diagram below shows the area of a lawn covered by a water sprinkler.   b. The water pressure is lowered so that the radius is 10 feet. What is the area of lawn that will be covered? 135° 16 ft The sprinkler now covers about 117.8 ft2 of lawn.

10. The window shown is in the shape of a semicircle. Find the area of the glass in the shaded region.   45° 3 m Area shaded = area large sector – area small sector Large: r = 6 Small: r = 3 The area of the glass in the shaded region is 10.6 m2