Chapter 10B

Chapter 10B

Problem Solving with Geometry

Fill in the Blank

Convert the given degree measure into degrees, minutes, and seconds of arc.

15.  $32.5\degree$   

15.

•  $32\degree$   Pull out the whole
  

•  $0.5\degree\times\frac{60'}{1\degree}$  = 30' convert the fractional degrees

•  $32\degree30'$  

Fill in the Blank

Convert the given degree measure into degrees, minutes, and seconds of arc.

19.  $149.83\degree$   

19.

 $149\degree$  Pull out the whole deg
 $0.83\degree\times\frac{60'}{1\degree}$  =49.8' Pull out whole min
 $0.8'\times\frac{60''}{1'}$  = 48''

Fill in the Blank

Convert the given angle measure into degrees and decimal fractions of a degree. For example, 30° 30′ = 30.5°. (Round to two decimal places.)

25. 8° 59′ 10′′

25.

 $8+\frac{59}{60}+\frac{10}{60\cdot60}=8+\frac{59}{60}+\frac{10}{3600}$  

Fill in the Blank

31. Find the latitude and longitude of the location on Earth precisely opposite Toronto, Canada (latitude 44°N, longitude 79°W).

Latitude and Longitude

• Buenos Aires, Argentina 34.6°S, 58.4°W

• Cape Town, South Africa 33.9°S, 18.4°E

• Which is farther from the North Pole? Explain.

Open Ended

33. Which is farther from the North Pole: Buenos Aires, Argentina, or Cape Town, South Africa? Explain.

Fill in the Blank

35. Buffalo, New York, is at nearly the same longitude as Miami, Florida, but Buffalo’s latitude is 43°N while Miami’s latitude is 26°N. About how far away is Buffalo from Miami? (Use the fact that 1° of latitude north or south is about 69.4 miles.)

Buffalo to Miami

• 43°N to 26°N

• 43 - 26 = 17° difference

•  $17\degree\times\frac{69.4\ miles}{1\degree}$  

• 17 x 69.4 = 1179.8 miles

Walking Around the Sun

• 39. The Sun has an angular diameter of about 0.5° viewed from Earth and a distance from Earth of about 150 million kilometers. What is its true diameter?

•  $angular\ size=\ physical\ size\times\frac{360\degree}{2\pi\times dist\alpha nce}$  

•  $angular\ size\ \div\frac{360\degree}{2\pi\times dist\alpha nce}=physical\ size$  

•  $physical\ size\ =\ angular\ size\times\frac{2\pi\times dist\alpha nce}{360\degree}$  

Fill in the Blank

39. The Sun has an angular diameter of about 0.5° viewed from Earth and a distance from Earth of about 150 million kilometers. What is its true diameter?

 $physical\ size\ =\ angular\ size\times\frac{2\pi\times dist\alpha nce}{360\degree}$  

Multiple Choice

Determine which of the following pairs of surfaces is steeper.

43. A railroad track with a 3% grade or a railroad track with a slope of 1/25.

• 3% grade

• 1/25

Fill in the Blank

45. What is the slope of a roof with an 8 in 12 pitch?

Fill in the Blank

45. (cont.) How much does the roof rise in 15 horizontal feet?

45

•  $\frac{2\ rise}{3\ run}=\frac{?}{15\ run}$  

• 3x = 30 or multiply by  $\frac{5}{5}$  

Fill in the Blank

Refer to the map in Figure 10.39. Assume that the length of each east-west block is 1/8 mile and the length of each north-south block is 1/5 mile.

a. Find the shortest possible walking distance (following the streets) between the two locations.

51. The bus stop and the library.

51.

 $\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{5}$  
 $6\left(\frac{1}{8}\right)+\frac{1}{5}=\frac{6}{8}+\frac{1}{5}$  = 0.95

Fill in the Blank

Refer to the map in Figure 10.39. Assume that the length of each east-west block is 1/8 mile and the length of each north-south block is 1/5 mile.

b. Find the straight-line distance (“as the crow flies”) between the two locations.

51. The bus stop and the library.

Multiple Choice

56. How many different paths between the bus stop and the library in Figure 10.39 have the shortest possible walking distance?

• 5

• 6

• 7

• 8

• 9

Find the area in acres

• 57. The stream frontage is 200 feet in length and the property line is 800 feet in length.

•  $a^2+b^2=c^2$  

•  $200^2+b^2=800^2$  

•  $A=\frac{1}{2}bh$  

•  $1\ acre\ =43560\ ft^2$  

Fill in the Blank

Acreage Problems. Refer to Figure 10.33, but use the lengths given in each exercise. Find the area in acres of the property for each set of lengths.

57. The stream frontage is 200 feet in length and the property line is 800 feet in length.

Fill in the Blank

65. Determine the lengths of the unknown sides in the following pairs of similar triangles.

Determine x.

Fill in the Blank

65. (cont.) Determine y.

69.

Assume that the solar access policy given in Example 8 is in force, and find the maximum allowed height for each house.

A 12-foot fence on the property line casts a 25-foot shadow on the shortest day of the year, and the north side of the house is set back 60 feet from the property line.

Fill in the Blank

69. Assume that the solar access policy given in Example 8 is in force, and find the maximum allowed height for each house.

A 12-foot fence on the property line casts a 25-foot shadow on the shortest day of the year, and the north side of the house is set back 60 feet from the property line.

79. Maximizing area -> max volume

• P = 2w + 2l and A = lw

• Give a set perimeter and graph on Desmos

• Squares maximizes the area of a rect. prism. (Equilateral triangles for triangles. Circles for all 2D figures.)

• Add a dimension. What sort of rectangular 3D figure do you think would maximize volume?

Fill in the Blank

79. Suppose you are designing a cardboard box that must have a volume of 8 cubic feet. The cost of the cardboard is $0.15 per square foot. What is the most economical design for the box (the one that minimizes the cost), and how much will the material for each box cost?

86.

An old principle of public landscaping says “Build sidewalks last.” In other words, let the people find their chosen paths and then build the sidewalks on the paths. Figure 10.40 shows a campus quadrangle that measures 40 meters by 30 meters. The locations of the doors of the library, chemistry building, and humanities building are shown. What is the combined length of the new sidewalks (gold lines) that connect the three buildings?

Fill in the Blank

86. Find the length of the lower right sidewalk.

86.

• Left side?

• 40 - 15 = 25

• Bottom?

• 30-15 = 15

• hypotenuse?

Fill in the Blank

86. Find the length of the lower left portion of the sidewalk.

86.

• Top?

• 30 m

• Right?

• 40 - 15 -15 = 10

Fill in the Blank

86. Find the length of the upper portion of the sidewalk.

Open Ended

89. A rancher must design a rectangular corral with an area of 400 square meters. She decides to make a corral that measures 10 meters by 40 meters (which has the correct total area). How much fencing is needed for this corral? Has the rancher found the most economical solution? Can you find another design for the corral that requires less fencing but still provides 400 square meters of corral?

89.

• A square maximizes the area. Therefore

   $A=s^2$  

• For a set area of 400,  $400=s^2$  

•  $s=\sqrt{400}=20$  

• If each side is 20 meters and there are 4 sides, then 4 x 20 = 80 ft total fencing.

96.

• It is a dark and stormy night. Through the beat of the wipers on your truck, you see a rickety country bridge marked “Load Limit 40 Tons.” You know your truck, White Lightning, like the back of your hand; it weighs 16.3 tons with you and your gear. Trouble is, you are carrying a cylindrical steel water tank that is full of water. The empty weight of the tank is printed on its side: 1750 pounds. But what about the water? Fortunately, the Massey-Fergusson trucker’s almanac in your glove box tells you that every cubic inch of water weighs 0.03613 pound. So you dash out into the rain with a tape measure to find the dimensions of the tank: length = 22 ft, diameter = 6 ft 6 in. Back in the truck, dripping and calculating, do you risk crossing? Explain.

• 22 ft = 264 in and 6ft 6in = 78 in

•  $V=\pi r^2h$  

•  $V=\left(3.14\right)\left(78^2\right)\left(264\right)=5043392.64\ in^3$  

96

•  $50433.92.64\ in^3\times\frac{0.03613\ lb}{in^3}\approx182217.78\ lb$  

•  $182217.78\ lb+1750\ lb\left(empty\ t\alpha nk\right)\ \approx183,957.78\ lb$  

• Convert to tons ( 1 ton = 2000 lbs)

•  $183,957.78\div2000\approx91.97$ tons with a 40 ton weight limit.

• NO.

• Still no.