Introduction. Rectangular Volume. Module #5: Geometry-Volume Bus 130 1:15

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1 Module #5: Geometry-olume Bus 0 :5 Introduction In this module, we are going to review the formulas for calculating volumes, rectangular, cylindrical, and irregular shaped. To set the stage, let s suppose we re the project manager in charge of ordering sand for a job that requires 5 cubic yards of sand. Our company has an 8-ft open storage corral that has sand leftover from a previous project. A quick estimate indicates that the sand has a slope of 0 to 5 feet (run of 0 ft and rise of 5 ft). First, we need to estimate how much sand is in storage so that we can order the appropriate amount. Then, when the delivery has been made at the site, we will need to determine whether the two circular piles of sand provide the correct amount. A quick measurement indicates that each of the two circular cone shaped piles of sand that have a diameter of 4 ft and a height of 4.5 ft. In this module, we will go over the formulas that are needed to calculate volumes. Rectangular olume From prior experience, we may already know that to calculate the volume of a rectangular box, we just multiply the lengths of the three edges: length, width and depth. The formula for rectangular volume of boxes, cubes, and bricks is box = length width depth = l w d Note, sometimes the situation uses depth and sometimes it uses height. So, if we want to use the letter h instead of the letter d, we can do that. Example #: A sidewalk is ft wide and 5 ft long. It s in deep. Determine the volume in terms of cubic yards. We have already done this problem in previous modules when the focus was on converting the units. The first step is to convert the inches into feet. The shortcut to go from inches to feet is to divide by. in = ft = 0.5 ft

2 Module #5: Geometry-olume Bus 0 :5 So, we ve got that the width of the sidewalk is ft, the length is 5 ft, and the depth is 0.5 ft. Now, we can substitute these numbers into the formula for the appropriate letters. Where we see an l in the formula, we ll replace it with 5 ft. Where we see a w, we replace it with ft. And, we replace the d with 0.5 ft. Substituting the numbers into the formula gives = l w d ( ) ( ) ( ( )( = 5 ft ft 0.5 ft = 5.05 ft ft ft = 6.5 ft ) ) After we substituted the numbers into the formula, we were able to separate the numbers and units. This separation can be done when we are multiplying everything together. Three units of feet multiplying together will give us cubic feet. Next, we need to convert the cubic feet to cubic yards. The shortcut is to divide the number of cubic feet by 7: ft = yd 0.98 yd The volume of concrete needed for this section of sidewalk is almost cubic yard. Example #: A brick used for a patio is 4 inches wide, 8 inches long, and inches thick. Will 800 bricks fit into the bed of a truck that has only.5 cubic yards of carrying capacity? First, we need to find the volume of a brick. Then we need to multiply by 800 to get the total volume. Our last step is to convert the units from cubic inches to cubic yards so that we can compare the volume of bricks to the carrying capacity of the truck. Since a brick has a rectangular shape, we can use the formula = l w d. The volume of a single brick is = 4in 8in in = 64in

3 Module #5: Geometry-olume Bus 0 :5 Each brick is 64 cubic inches. So, 800 bricks has a total volume of 5,00 cubic inches ( in = 5, 00 in ). To convert cubic inches into cubic yards, we ll use two relationships: cubic foot = 78 cubic inches and cubic yard = 7 cubic feet. Using the technique covered in Module #, let s calculate 5, 00 in ft yd 78 in 7 ft 5, 00 in ft yd 78 in 7 ft 5, yd.5 yd The volume of the 600 bricks is approximately.5 cubic yards. So, there is room to spare in the truck for the bricks. (Let s hope that the truck can handle the weight.) Check Your Understanding (a) If a conference room is 0 ft by 5 ft and has 5 ft walls, how many cubic feet is the volume of the room? (Information needed for heating and ventilation.) (b) For construction of a pool, a hole with dimensions 0 ft by 40 ft and 5 ft deep needs to be dug. How much soil (cubic yards) needs to be excavated? (c) For construction of a pool, a hole with dimensions 5 ft by 5 ft and 5 ft deep needs to be dug. How much loose soil (cubic yards) needs to be hauled? (Use the factor of.5 for loose soil. Hint: recall Module #) Answers: (a) 0,500 cu ft, (b) 48. cu yds, (c) cu yds

4 Module #5: Geometry-olume Bus 0 4:5 Basic Concept of olumes Before we cover the volume of a cylinder, let s take a moment to look at the basic concept of volumes. The basic strategy is as easy as sliced bread. Actually, it can be described using sliced bread. Let s take a loaf of bread and cut slices that are inch thick. Then, let s say that the cross-sectional area of one slice (that s the area on which we spread the butter) is approximately 0 square inches. With the thickness being inch thick, the volume of each slice is 0 cubic inches. Notice that the volume each slice is really about the area rather than the width. That s because we made the thickness to be a nice inch. Next, let s start stacking the slices of bread. For each slice that s added to the stack, the volume gets larger by 0 cubic inches. To calculate the volume, we are in essence adding areas. When we stack the slices the areas begin to create the volume of the loaf of bread. The key to calculating volumes is determining how to slice it up or how it s layered. Next, we figure out the cross-sectional area. Last, we multiply by the depth (or height). We can see these steps in the formula for a box, = l w d. When we multiply the numbers for length and width, we are actually finding the cross-sectional area. Then, we multiply by the depth to get volume. So, the formula for volume is actually = area depth or = area height Cylindrical olume Instead of rectangular bricks, suppose we were dealing with circular stepping stones. For cylinders, instead of thinking of a loaf of bread, imagine cutting up a large salami or pepperoni. If we stack the pepperoni slices, the cross-sectional area is the area of a circle. We are adding up the areas a bunch of little circles to make the volume of a cylinder. So, the volume of a cylinder has the formula for the area of a circle in it:

5 Module #5: Geometry-olume Bus 0 5:5 = area of circle height = π rh.4 r r h Example #: A silo has diameter of 0 ft and a height of 0 ft. What is the volume of the silo (cubic yards)? Since the formula for cylindrical volume uses radius, we need use 0 ft instead of 0 ft. Recall, the radius is half the diameter. So, substitute these values into the formula and calculate: = π r h 0 0 = π.4 0 ft 0 ft 0 ft = 680 ft Next, we need to convert from cubic feet to cubic yards. The shortcut for this conversion is to divide by 7. (This shortcut puts together the steps of multiplying by the fraction fraction.) yd, canceling the units, multiplying straight across, and calculating the 7 ft ft = yd.6 yd The silo has a volume that is approximately.6 cubic yards. Example #4: Will 600 circular stepping stones with diameter of 8 in and thickness of in fit into the bed of a truck that has only.5 cubic yards of carrying capacity? First, we ll find the volume of a single stepping stone brick. Then, we ll multiply by 800 for the total volume. And, our last step is to convert the units from cubic inches into to cubic yards so that we can compare to the carrying capacity of the truck. With a diameter of 8 in, the stepping stones have a radius of 4 in. The volume of a single stepping stone is = π r h ( ) = π 4 in in.4 4 in 4 in in = in

6 Module #5: Geometry-olume Bus 0 6:5 Each stepping stone is cubic inches. So, 800 bricks have a total volume of 80,864 cubic inches ( in = 80,864in ). Again, to convert cubic inches into cubic yards, let s use the two relationships: cubic foot = 78 cubic inches and cubic yard = 7 cubic feet. To make the conversion, let s calculate 80,864 in ft yd 78 in 7 ft 80,864 in ft yd 78 in 7 ft 80, yd.9 yd For 800 stepping stones with an 8 in diameter and in thickness, the truck is not big enough to make the delivery. Check Your Understanding (a) A silo has a diameter of 5 ft and a height of 4 ft. What is the volume of the silo in terms of cubic yards? (b) A milk delivery truck has a trailer that is shaped as a cylinder on it s side. The cylinder has a length of 0 ft and an inside diameter of 5 ft. How many gallons does the trailer hold when full? (Hint: cu ft = 7.48 gal) (c) A storage drum has a diameter of 8 in and a height of ft. Determine the number of gallons that the drum holds. (Hint: cu ft = 7.48 gal) Answers: (a) 57 cu yds, (b) 96 gal, (c) 9.6 gal

7 Module #5: Geometry-olume Bus 0 7:5 Irregular olumes For volumes that are not composed of a rectangular or circular base, the strategy of determining the cross sectional area is important. To find the volume, all we need to do is multiply the cross sectional area by the height or depth. That s as long as the sides are perpendicular to the base. If the sides angle in or out, then we can t use the formula = area of base height. Example #5: Figure 5. shows the plan of a room. The room has 8 ft walls. To determine the type of ventilation system for the room, we need to determine the volume of the room (cu ft). Figure 5. 9 ft 0 ft 6 ft 8 ft For an irregular shaped room whose walls are perpendicular to the floor, the volume of the room is room = area of floor wall height Using the method in Module #4, let s break the shape into two rectangles. We have a rectangle that is 9 ft by 8 ft and another one that is ft by ft. The large rectangle has an area of 9 ft 8 ft = 7 ft. The small rectangle has an area of ft ft = 6 ft. The room has a total area of 7 ft + 6 ft = 78 ft. To find the volume, we need to multiply the cross-sectional area by the height: = 78 ft 8 ft = 64 ft The room has a volume of 64 cubic feet.

8 Module #5: Geometry-olume Bus 0 8:5 Example #6: For the following path, pine mulch will be spread inches thick. How much pine mulch is required? Figure 5. Figure 5. ft 5 ft ft 5 ft 5 ft ft 5 ft ft 4 ft 4 ft To find the volume, we ll find the area of the path first. Then, we ll multiply by the thickness. The key to finding the area of the path is to break the shape up into basic shapes. One method is to break the shape up into two rectangles and a triangle (as seen in Figure 5.). The triangle at the bend will have a base of 4 ft and a height of ft. So, the area of the triangular part is 4 ft ft 6 ft A = =. One of the rectangles has a length of ft and a width of 4 ft. So, the area of this rectangular part is A = ft 4 ft = 48 ft. The second rectangle has a length of ft and a width of 5 ft. So, the area of the second rectangular part is A = ft 5 ft = 65 ft sum of the three parts: A = 6 ft + 48 ft + 65 ft = 9 ft.. Thus, the total area of the path is the To calculate the volume, we ll convert the inches in terms of feet by dividing by. The shortcut for going from inches to feet is to divide by. The depth of the mulch is in = ft 0.7 ft. Now that we ve got the area of the path and we ve got the units of depth in terms of feet, we can multiply to get the volume. = 9 ft 0.7 ft 0. ft

9 Module #5: Geometry-olume Bus 0 9:5 Lastly, we ll convert form cubic feet to cubic yards by dividing by ft = yd 0.75 yd To cover the path with inches of pine mulch, we ll need approximately three-quarters of a cubic yard of mulch. Cones and Spheres The last shapes that we will review are cones and spheres (balls). Both of the formulas for the volume of these shapes can be developed using calculus. Don t worry, we won t go into their development. However, do notice that both of the formulas involve the radius. That s because they have a circular component. The following are the two formulas for the volumes: cone sphere = π r h 4 = π r Example #6: A circular cone shape pile of gravel has a diameter of 0 ft and a height of ft. How many cubic yards of gravel do we have? Since the pile is shaped like a circular cone, we can use the formula for the volume of a cone. Because the radius is half of the diameter, we have a radius that is 5 ft. Substituting the numbers into the formula gives us the volume in terms of cubic feet: cone = π ( 5 ft) ( ft) ft ft ft ( ) 78.5 ft

10 Module #5: Geometry-olume Bus 0 0:5 Next, we ll convert 78.5 cubic feet into cubic yards by dividing 7: ft = yd.9 yd. Thus, a circular pile of gravel that has a diameter of 0 ft and height of 5 ft will contain approximately.9 cubic yards of gravel. Example #7: When a tree is transplanted, the tree will be dug up and the root system (roots and soil) will be wrapped in burlap (called a root ball). Typically, a cubic foot of soil weighs about 0 pounds. If the root ball has a diameter of feet, how much does the root ball weigh? To determine the weight of the root ball, we will need to use two steps. First, we need to find the volume of the root ball. Then, we will need to convert the volume into weight using the relationship 0 lbs = cu ft. Since a root ball is spherical, we can use the formula for the volume of a sphere. Because the diameter is ft, the radius is ft. Substituting this number into the formula we get sphere = 4 π r = 4 π ( ft ) 4.4 ft ft ft ( ) 4. ft The volume of root ball is 4. cubic feet. Notice, the power indicates that we multiply the radius three separate times. Nest, we ll convert the volume into weight by multiplying with the fraction, 0 lbs. This fraction will cancel the cubic feet and put in pounds. ft

11 Module #5: Geometry-olume Bus 0 :5 4. ft 0 lbs ft 4. ft 0 lbs ft 4. 0 lbs 46 lbs A root ball that has a diameter of ft will weigh approximately 46 pounds. Check Your Understanding (a) If a circular cone shaped pile of mulch has a diameter of 8 ft and a height of.5 ft, how many cubic yards of mulch is in the pile? (b) At the end of a project, a pile of trash 6 ft high with a radius of.5 ft needs to be hauled away. If it costs $50 per cubic yard of trash, how much would it cost for the pile of trash to be hauled away? (c) What is the weight of a root ball with a.5 ft diameter? (Use cu ft = 0 pounds) Answers: (a).6 cu yds, (b) $7.69, (c) 900 pounds Now, that we ve looked at the formulas for volumes and the strategy for irregular shapes, let s take a look at the problem given in the introduction. Recall, that we have some sand in storage and need to order enough sand so that we have a total of 5 cubic yards of sand for the project. Also, when the sand is delivered, we want to confirm that the amount delivered is the amount ordered. Let s start with how much sand we need to order. The sand that was leftover from a previous job is stored in open corral bin that has a width of 8 ft. The sand is piled up at the back of the bin and slopes down to the front of the bin (see Figure 5.4). For estimating purposes, let s assume that the sand slopes consistently across the bin.

12 Module #5: Geometry-olume Bus 0 :5 Figure 5.4 Top iew 8 ft Side iew 5 ft 0 ft To find the volume of sand, let s imagine that we can slice the pile of sand into slices of triangles. So, the steps to determine the volume are find the area of the triangle, then multiply the area by the width of the corral. The area of the triangle is 0 ft 5 ft 5 ft A = =. So, the volume of the sand is = area width = 5 ft 8 ft = 00 ft Next, let s convert the units from cubic feet to cubic yards. We ll use the shortcut of dividing by 7 to get the number of cubic yards: ft = yd 7.4 yd Thus, we have approximately 7.4 cubic yards of sand in the storage corral. Since we need a total of 5 cubic yards of sand, we need to order an additional 7.6 cubic yd 7.4 yd = 7.6 yd yards ( 5 ). After the delivery has been made, we see two piles of sand that have a diameter of 4 ft and a height of about 4.5 ft. We want to know if the pile of sand is at least 7.6 cubic yards. First, we ll calculate the volume in terms of cubic feet. Then, we ll convert the units to cubic yards. Lastly, we ll double the amount for the two piles.

13 Module #5: Geometry-olume Bus 0 :5 Because the diameter of each pile is 4 ft, the radius is 7 ft. We can substitute the radius and height into the formula for the volume of a cone: cone = π ( 7 ft) ( 4.5 ft) ft ft ft ( ) 0.79 ft Next, we ll convert the units from cubic feet to cubic yards using the shortcut of dividing by 7: ft = yd 8.5 yd Since each pile is approximately 8.5 cubic yards, together the two piles contain approximately 7 cubic yards. Thus, for the project, we are short about 0.6 cubic yards of sand. Looks like it s a time to get on the cell phone and make some calls.

14 Module #5: Geometry-olume Bus 0 4:5 Questions. A landscaper wants 4 inches of mulch applied to a shrub bed that has dimensions of 6ft by 8 ft. How many cubic yards of mulch needs to be ordered?. How many cubic yards of gravel are in a circular cone shaped pile that has a diameter of 0 ft and a height of 4 ft?. How many cubic yards of soil need to be excavated to make a hole with dimensions 0ft by 5 ft by 6ft? 4. The homeowner removed a dead tree along with the root system from a local park. This removal left a 4 ft square hole that was.5 ft deep. (a) To fill this hole, how many cubic yards of soil must the homeowner have hauled in? (Don t account for the shrinkage factor.) (b) Suppose the shrinkage factor for the soil is.0. How many cubic yards of soil must the homeowner have hauled in? 5. When planting a tree, it is recommended that we should dig a pit that is.5 times the diameter of the root ball. How many cubic yards of fill soil will be needed for a tree that has a root ball with a diameter of inches? 6. A crushed stone border.5 feet wide is to be installed around a reflecting pool. The border of stones is to be inches deep. (a) How many cubic yards of stone should you order delivered to the site? (b) How many gallons of water are needed to make the pool have a depth of 8 inches? (Assume a flat bottom to the pool and cubic foot of water = 7.48 gal.) 7. In order to determine the correct exhaust fan to purchase, the salesperson needs to determine the volume of the room. For the given floor design, find the volume (cubic feet) of the room if the walls are 5 ft. 0 ft pool 0 ft 8 ft 44 ft crushed stone 6 ft

15 Module #5: Geometry-olume Bus 0 5:5 8. For the given floor design, determine how many cubic yards of concrete need to be 44 ft ordered in order to have the floor be 4 inches thick. 6 ft 4 ft 0 ft 0 ft 4 ft 6 ft 4 ft 9. Jack has a pick up truck that has a bed size of 4 x 6 x -6. He needs to deliver a truck load of mulch to a job site. If the bob cat loader has dimensions of x 5 x 8, how many scoops from the bob cat will it take to fill the truck once? 0. For the following garden, how many cubic yards of mulch is needed to cover the ground, not including the reflection pool, with 4 inches of pine mulch? The reflection pool has a diameter of 5 ft. 6 ft ft 5 ft. To build a home on a hillside, the contractor needs to excavate soil for the foundation that will be 0 ft into the hill and 0 ft along the hill. The hill has a slope of to (a horizontal change of with a vertical change of. Assuming the cross-sectional area can be estimated with a triangle, determine the number of cubic yards of soil that needs to be excavated.. At the job site, there is a pile of sand that has a diameter of 8 ft and a height of.5 ft. At the yard, there is sand stored in a 6 ft open storage corral and the sand has a slope with horizontal of 6 ft and vertical of 4 ft. The project just got modified and now requires 7.5 cubic yards of sand. How many cubic yards of sand do you need to order to meet the demands of the change in plans? 6 ft 7 ft ft ft