Solving Geometric Applications


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1 1.8 Solving Geometric Applications 1.8 OBJECTIVES 1. Find a perimeter 2. Solve applications that involve perimeter 3. Find the area of a rectangular figure 4. Apply area formulas 5. Apply volume formulas One application of addition is in finding the perimeter of a figure. Definitions: Perimeter The perimeter is the distance around a closed figure. If the figure has straight sides, the perimeter is the sum of the lengths of its sides. Example 1 Finding the Perimeter We wish to fence in the field shown in Figure 1. How much fencing, in feet (ft), will be needed? 30 ft 20 ft 45 ft 18 ft 25 ft Figure 1 NOTE Make sure to include the unit with each number. The fencing needed is the perimeter of (or the distance around) the field. We must add the lengths of the five sides. 20 ft 30 ft 45 ft 25 ft 18 ft 138 ft So the perimeter is 138 ft. CHECK YOURSELF 1 What is the perimeter of the region shown? 2 50 in. 28 in. 15 in. A rectangle is a figure, like a sheet of paper, with four equal corners. The perimeter of a rectangle is found by adding the lengths of the four sides. 103
2 104 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS Example 2 Finding the Perimeter of a Rectangle Find the perimeter in inches (in.) of the rectangle pictured below. 8 in. 5 in. 5 in. 8 in. The perimeter is the sum of the lengths 8 in., 5 in., 8 in., and 5 in. 8 in. 5 in. 8 in. 5 in. 26 in. The perimeter of the rectangle is 26 in. CHECK YOURSELF 2 Find the perimeter of the rectangle pictured below. 1 7 in. 7 in. 1 In general, we can find the perimeter of a rectangle by using a formula. A formula is a set of symbols that describe a general solution to a problem. Let s look at a picture of a rectangle. Length Width Width Length The perimeter can be found by adding the distances, so Perimeter length width length width To make this formula a little more readable, we abbreviate each of the words, using just the first letter.
3 SOLVING GEOMETRIC APPLICATIONS SECTION Rules and Properties: Formula for the Perimeter of a Rectangle P L W L W (1) There is one other version of this formula that we can use. Because we re adding the length (L) twice, we could write that as 2 L. Because we re adding the width (W) twice, we could write that as 2 W. This gives us another version of the formula. Rules and Properties: Formula for the Perimeter of a Rectangle P 2 L 2 W (2) In words, we say that the perimeter of a rectangle is twice its length plus twice its width. Example 3 uses formula (1). Example 3 Finding the Perimeter of a Rectangle A rectangle has length 1 and width 8 in. What is its perimeter? Start by drawing a picture of the problem. 1 NOTE We say the rectangle is 8 in. by 1 8 in. 8 in. 1 Now use formula (1) P 1 8 in. 1 8 in. 38 in. The perimeter is 38 in. CHECK YOURSELF 3 A bedroom is 9 ft by 12 ft. What is its perimeter?
4 106 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS Units Analysis What happens when we multiply two denominate numbers? The units of the result turn out to be the product of the units. This makes sense when we look at an example from geometry. The area of a square is the square of one side. As a formula, we write that as A s 2 1 ft 1 ft 1 ft 1 ft This tile is 1 foot by 1 foot. A s 2 (1 ft) 2 1 ft 1 ft 1 (ft) (ft) 1 ft 2 In other words, its area is one square foot. If we want to find the area of a room we are actually finding how many of these square feet can be placed in the room. Let s look now at the idea of area. Area is a measure that we give to a surface. It is measured in terms of square units. The area is the number of square units that are needed to cover the surface. One standard unit of area measure is the square inch, written in. 2. This is the measure of the surface contained in a square with sides of See Figure 2. NOTE The unit inch (in.) can be treated as though it were a number. So in. in. can be written in. 2. It is read square inches. NOTE The length and width must be in terms of the same unit. Other units of area measure are the square foot (ft 2 ), the square yard (yd 2 ), the square centimeter (cm 2 ), and the square meter (m 2 ). Finding the area of a figure means finding the number of square units it contains. One simple case is a rectangle. Figure 3 shows a rectangle. The length of the rectangle is 4 inches (in.), and the width is The area of the rectangle is measured in terms of square inches. We can simply count to find the area, 12 square inches (in. 2 ). However, because each of the four vertical strips contains 2, we can multiply: Area 1 2 Width One square inch Figure 2 1in. 2 Length Figure 3
5 SOLVING GEOMETRIC APPLICATIONS SECTION Rules and Properties: Formula for the Area of a Rectangle In general, we can write the formula for the area of a rectangle: If the length of a rectangle is L units and the width is W units, then the formula for the area, A, of the rectangle can be written as A L W (square units) (3) Example 4 Find the Area of a Rectangle A room has dimensions 12 feet (ft) by 15 feet (ft). Find its area. 12 ft 15 ft Use formula (3), with L 15 ft and W 12 ft. A L W 15 ft 12 ft 180 ft 2 The area of the room is 180 ft 2. CHECK YOURSELF 4 A desktop has dimensions 50 in. by 25 in. What is the area of its surface? We can also write a convenient formula for the area of a square. If the sides of the square have length S, we can write Rules and Properties: Formula for the Area of a Square NOTE S 2 is read S squared. A S S S 2 (4) Example 5 Finding Area 3' 3' You wish to cover a square table with a plastic laminate that costs 60 a square foot. If each side of the table measures 3 ft, what will it cost to cover the table?
6 108 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS We first must find the area of the table. Use formula (4), with S 3 ft. A S 2 (3 ft) 2 3 ft 3 ft 9 ft 2 Now, multiply by the cost per square foot. Cost 9 60 $5.40 CHECK YOURSELF 5 You wish to carpet a room that is a square, 4 yd by 4 yd, with carpet that costs $12 per square yard. What will be the total cost of the carpeting? Sometimes the total area of an oddly shaped figure is found by adding the smaller areas. The next example shows how this is done. Example 6 Finding the Area of an Oddly Shaped Figure Find the area of Figure in. Region 1 Region 2 Figure 4 The area of the figure is found by adding the areas of regions 1 and 2. Region 1 is a by rectangle; the area of region Region 2 is a by rectangle; the area of region 2 2 The total area is the sum of the two areas: Total area CHECK YOURSELF 6 Find the area of Figure 5. Figure 5 Hint: You can find the area by adding the areas of three rectangles, or by subtracting the area of the missing rectangle from the area of the completed larger rectangle.
7 SOLVING GEOMETRIC APPLICATIONS SECTION Our next measurement deals with finding volumes. The volume of a solid is the measure of the space contained in the solid. Definitions: Definition of a Solid A solid is a threedimensional figure. It has length, width, and height. 1 cubic inch Figure 6 Volume is measured in cubic units. Examples include cubic inches (in. 3 ), cubic feet (ft 3 ), and cubic centimeters (cm 3 ). A cubic inch, for instance, is the measure of the space contained in a cube that is on each edge. See Figure 6. In finding the volume of a figure, we want to know how many cubic units are contained in that figure. Let s start with a simple example, a rectangular solid. A rectangular solid is a very familiar figure. A box, a crate, and most rooms are rectangular solids. Say that the dimensions of the solid are 5 in. by by as pictured in Figure 7. If we divide the solid into units of, we have two layers, each containing 3 units by 5 units, or 15 in. 3 Because there are two layers, the volume is 30 in. 3 5 in. Figure 7 In general, we can see that the volume of a rectangular solid is the product of its length, width, and height. Rules and Properties: Formula for the Volume of a Rectangular Solid V L W H (5) Example 7 Finding Volume A crate has dimensions 4 ft by 2 ft by 3 ft. Find its volume. 3' 4' 2'
8 110 CHAPTER 1 OPERATIONS ON WHOLE NUMBERS NOTE We are not particularly worried about which is the length, which is the width, and which is the height, because the order in which we multiply won t change the result. Use formula (5), with L 4 ft, W 2 ft, and H 3 ft. V L W H 4 ft 2 ft 3 ft 24 ft 3 CHECK YOURSELF 7 A room is 15 ft long, 10 ft wide, and 8 ft high. What is its volume? Overcoming Math Anxiety Taking a Test Earlier in this chapter, we discussed test preparation. Now that you are thoroughly prepared for the test, you must learn how to take it. There is much to the psychology of anxiety that we can t readily address. There is, however, a physical aspect to anxiety that can be addressed rather easily. When people are in a stressful situation, they frequently start to panic. One symptom of the panic is shallow breathing. In a test situation, this starts a vicious cycle. If you breathe too shallowly, then not enough oxygen reaches the brain. When that happens, you are unable to think clearly. In a test situation, being unable to think clearly can cause you to panic. Hence we have a vicious cycle. How do you break that cycle? It s pretty simple. Take a few deep breaths. We have seen students whose performance on math tests improved markedly after they got in the habit of writing remember to breathe! at the bottom of every test page. Try breathing, it will almost certainly improve your math test scores! CHECK YOURSELF ANSWERS in in ft in $ ft 3
9 Name 1.8 Exercises Section Date Find the perimeter of each figure ft 2. ANSWERS 4 ft 7 ft yd 8 yd 5 ft 5 ft yd 10 ft in in. 8 yd 5 yd 10 yd yd 7 yd yd Multiply the following. Be sure to use the proper units in your answer ft 2 ft mi 13 mi in yd 26 yd Label the following statements true or false. 13. (10 ft) ft 14. (5 mi) 2 25 mi (8 yd) yd (9 in.) 2 9 in
10 ANSWERS Find the area of each figure yd yd 9 in ft in. 4 ft ft ft 10 ft 8 in. 25 ft 40 ft 10 in ft 5 in. 3 ft 2 ft 5 ft 7 ft in in. 18 ft 15 ft 3 ft 112
11 ANSWERS Find the volume of each solid shown yd yd yd in. 8 in. 8 in yd 3 yd 3 yd Solve the following applications. 35. Window size. A rectangular picture window is 4 feet (ft) by 5 ft. Meg wants to put a trim molding around the window. How many feet of molding should she buy? 36. Fencing material. You are fencing in a backyard that measures 30 ft by 20 ft. How much fencing should you buy? 113
12 ANSWERS Tile costs. You wish to cover a bathroom floor with 1squarefoot (1 ft 2 ) tiles that cost $2 each. If the bathroom is rectangular, 5 ft by 8 ft, how much will the tile cost? Roofing. A rectangular shed roof is 30 ft long and 20 ft wide. Roofing is sold in squares of 100 ft 2. How many squares will be needed to roof the shed? 39. House repairs. A plate glass window measures 5 ft by 7 ft. If glass costs $8 per square foot, how much will it cost to replace the window? 40. Paint costs. In a hallway, Bill is painting two walls that are 10 ft high by 22 ft long. The instructions on the paint can say that it will cover 400 ft 2 per gallon (gal). Will one gal be enough for the job? 41. Tile costs. Tile for a kitchen counter will cost $7 per square foot to install. If the counter measures 12 ft by 3 ft, what will the tile cost? 42. Carpet costs. You wish to cover a floor 4 yards (yd) by 5 yd with a carpet costing $13 per square yard (yd 2 ). What will the carpeting cost? 43. Frame costs. A mountain cabin has a rectangular front that measures 30 ft long and 20 ft high. If the front is to be glass that costs $12 per square foot, what will the glass cost? 44. Posters. You are making posters 3 ft by 4 ft. How many square feet of material will you need for six posters? 114
13 ANSWERS 45. Shipping. A shipping container is 5 ft by 3 ft by 2 ft. What is its volume? 46. Size of a cord. A cord of wood is 4 ft by 4 ft by 8 ft. What is its volume? Storage. The inside dimensions of a meat market s cooler are 9 ft by 9 ft by. What is the capacity of the cooler in cubic feet? 48. Storage. A storage bin is 18 ft long, wide, and 3 ft high. What is its volume in cubic feet? A rancher wants to build cattle pens as pictured below. Each pen will have a gate 8 ft wide on one end. What is the total cost of the pens if the fencing is $6 per linear foot and each gate is $25? 10 ft. 8 ft. 50. Approximate the total area of the sides and ends of the building shown. 30 ft 83 ft 60 ft 51. Suppose you wish to build a small, rectangular pen, and you have enough fencing for the pen s perimeter to be 3. Assuming that the length and width are to be whole numbers, answer the following. (a) List the possible dimensions that the pen could have. (Note: a square is a type of rectangle.) (b) For each set of dimensions (length and width), find the area that the pen would enclose. (c) Which dimensions give the greatest area? (d) What is the greatest area? 115
14 ANSWERS Suppose you wish to build a rectangular kennel that encloses 100 square feet. Assuming that the length and width are to be whole numbers, answer the following. (a) List the possible dimensions that the kennel could have. (Note: a square is a type of rectangle.) (b) For each set of dimensions (length and width), find the perimeter that would surround the kennel. (c) Which dimensions give the least perimeter? (d) What is the least perimeter? Answers ft yd in yd in False 15. True yd in ft in in ft 37. $ $ $ $ ft $
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