2. In the diagram shown, how much more water can be poured into the container before it overflows?

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1 Blue Surface Area and Volume (Note: 1 cubic centimeter = 1 milliliter) 1. The container shown is filled with water to a depth of 9 cm. How much water is it holding? 17 cm 12 cm 10 cm 12 cm 18 cm 10 cm 2. In the diagram shown, how much more water can be poured into the container before it overflows? 9 cm 14 cm 15 cm 27 cm. The diagram shows the steel base for a vice. The measurements are in cm. Calculate the volume of steel used to make this The electricity to Jason's electric water heater is shut off because he needed to drain the tank to clean out mineral deposits. Jason now needs to refill his water heater. He has to be very sure that it is completely full of water before the electricity is turned back on because there are two heating elements in the heater that must be completely covered with water, or they will burn out and cause an expensive repair. The inside of the tank, which is a cylinder, measures 5 decimeters in diameter and is 1 1/2 meters high. When Jason runs water, it fills a liter container in 6 seconds. It is now 6:5 P.M. and Jason has just turned on the water to fill the tank. At what time will the tank be full so Jason can turn the electricity to the water heater back on. (Round to the nearest minute.) 1

2 5. What is the volume of the water in a 50-foot long garden hose that has a 2-inch inside diameter, assuming the hose is entirely filled with water?? Unit 6 Area, and Volume 6. If a cylinder of height 8' and a diameter of 4' is dropped slowly into a full tank of water measuring 10 x 10 x 10, what is the volume of the water that overflows? 7. If a cubic meter is dropped into a full tank of water, what is the mass (in kilograms) of the water that overflows? 1 cm equals 1 ml, and 1 ml equals 1 gram. 8. Concrete is ordered by the cubic yard. If a contractor says that he needs 10 yards, that means he wants 10 cubic yards of concrete. Kathy is going to make a cement driveway 12' wide, 81' long, and 6 inches thick. How many yards of concrete will she need? 9. A contractor has been asked to pave 15 miles of new road. Before he can set his price, he needs to know how much the cement will cost. If the road is 45 feet wide and 9 inches thick and the cost of cement is $5 per cubic yard, what will be the cost of the cement needed for the road? 10. The moon and the earth are almost perfect spheres. The moon is 2000 miles in diameter compared to the earth's 8000 mile diameter. The diameter of the earth is 4 times that of the moon. The volume of the earth is how many times larger than the moon's volume? 11. Three cubes are dropped into a container of water that is filled to the top. They are a 1 inch cube, a 1 foot cube, and a yard cube. What is the volume of water that spills out? 12. A waterfall has a flow rate of 2700 cubic feet of water per second. How many cubic yards of water go over the waterfall in a year? 1. Heather is building a parking area that will be 6 feet long and 21 feet wide. She is trying to decide whether to make the concrete 5 inches or 6 inches thick. If concrete costs $5 per cubic yard, how much extra money would it cost to build the parking area 6 inches deep? 2

3 14. Something Fishy The students at Highland Middle School are designing a schoolyard garden, to include a small pond. They have decided that they will build their pond to hold 1000 gallons of water. They have also selected two types of fish to occupy their pond: goldfish and koi. The students want to have an equal number of goldfish and koi. Goldfish can grow up to 12 inches in length, while a koi can reach a length of 24 inches. Both types of fish require two gallons of water per inch of fish length. What is the maximum number of each breed they can purchase, making sure they have enough space in the water for each fish? Extra: They decide to make the pond 4 feet deep to discourage raccoons from eating the fish. Find the approximate area of the pond's surface. [There are about 7.5 US gallons of water in a cubic foot.] 15. Let's do a little more work with measurements this week. We'll begin with a cube that measures two centimeters on each edge. This cube will have a surface area of 24 square centimeters, and it will have a volume of 8 cubic centimeters. What will happen to the area and volume if the edge of the cube is doubled? Bonus: Let's see if we can generalize. What will happen to the area and volume of any cube if the length of its edges is doubled? 2 cm Solutions ml depth of 9 cm à 8 cm à 9 1 x 10 x 17) + (1 x 10 x 12) + ( x 8) à à 570 cm³ à 570 ml

4 ml cm³ entire thing is steel à (16 x 14 x 10) 2240 cm³ à (14 x x 6) 252 cm³ = 1988 cm³ 4. 7:04 P.M. Change measurements to decimeters: Radius 2.5 decimeters Height: 15 decimeters Area of top circular part:.14 x 2.5 x 2.5 = square decimeters Volume: square decimeters x 15 (height) = cubic decimeters If it takes 6 seconds to fill one liter, it takes 6 x = seconds to fill the tank seconds 60 seconds in a minute = minutes 6: =7: cubic inches Change measurements to inches: Radius is one inch Area of circular top:.14 x 1 x 1 =.14 square inches Volume:.14 x 600 = 1884 cubic inches. Height is 600 inches cubic feet Area of circular top:.14 x 2 x 2 = square feet Volume: x 8 feet = cubic feet kg. A cubic meter is 100cm x 100cm x 100cm = 1,000,000 cm, which equals 1,000,000ml or 1,000,000g, which is equivalent to 1000 kg yards Change all measurements to yards 12 feet = 4 yards 81feet = 27 yards 6 inches are equal to 6/6 =.1666 yard Volume: 4 x 27 x.1666 = 18 cubic yards 4

5 9. $,465,000 Change all measurements to yards. 45 feet = 15 yards 9 inches = 1/4 yard 15 miles: 15 x 1760 yards per mile = 26,400 yards Volume of cement needed for road: 15 x.25 x 26,400 = cubic yards 99,000 x $5 = $2,465, times Volume of Earth: 4 x x 4000 x 4000 x 4000 = Volume of the moon: 4 x x 1000 x 1000 x 1000 = When you reduce, the 4 s cancel. The s cancel. The rest reduces to ,85 cubic inches Change all to inches. Volume of 1 inch cube: 1 x 1 x 1 = 1 cubic inch Volume of 1 foot cube: 12 x 12 x 12 = 1728 cubic inches Volume of 1 yard cube: 6 x 6 x 6 = cubic inches 46, = 48,85 cubic inches 12.,15,600,000 cubic yards There are feet x feet x feet = 27 cubic feet in a cubic yard = 100 cubic yards per second. Seconds in one year: 6o seconds x 60 minutes/hour x 24 hours/day x 65 days/year = 1,56,000 seconds/year 1,56,000 seconds x too cubic yards per second = , $81.67 Change measurements to yards: 6 feet: 12 yards 21 feet: 7 yards One inch thickness: 1/6 or Volume of one inch of thickness: 12 x 7 x = Cost: x $5 = $ The maximum number of each breed they can purchase for the pond is 1, that is 1 goldfish and 1 koi. 1. Each goldfish grows to 12 inches in length. It requires two gallons per inch, so at the full 12 inches, it will require 2 x 12 gallons, that is 24 gallons. 2. Each koi can reach 24 inches. At that length it will require 24 x 2 gallons, that is 48 gallons. 5

6 . A single pair of koi and gold fish will need gallons=72 gallons. 2 pairs = 2 X72 = 144 gallons pairs = X72 = 216 gallons 4 pairs = 4X72 = 288 gallons 5 pairs = 5 X72= 60 gallons 6 pairs = 6X72 = 42 gallons 7 pairs = 7 X72 = 504 gallons 8 pairs = 8 X 72 = 576 gallons 9 pairs = 9 X72 = 648 gallons 10 pairs = 10 x72 = 720 gallons 11 pairs = 11 x72 = 792 gallons 12 pairs = 12 x72 = 864 gallons 1 pairs = 1 x 72 = 96 gallons 14 pairs = 14 x72 = 1008 gallons Unit 6 Area, and Volume Since the pond can hold only 1000 gallons, the maximum number of pairs of gold fish and koi is 1. I cross checked my answer using a second method. Each pair requires 72 gallons. We have 1000 gallons maximum. Divide that by 72 and you get 1 remainder 64. So they have enough space for 1 koi and 1 goldfish. EXTRA 1. I am assuming the pond is in the form of a rectangle so the volume is equal to the length X width X depth. But length X width is equal to the area so the volume = area x depth. The volume is 1000 gallons. Since each cubic foot is 7.5 gallons, the volume in cubic feet is 1000 divided by 7.5 = 1. cubic feet. Thus 1.= area X depth. Now the depth is 4 feet so 1.=Area x 4. Therefore the area= 1./4 =. square feet. Let us check the answer. Volume = area x depth =. x 4 cubic feet =1.2 cubic feet Each cubic foot is 7.5 gallons, so the volume in gallons is 1.2 x 7.5 =1000 gallons. 15. If the edge length of a cube is doubled from 2 cm to 4 cm, then the surface area changes from 24 to 96cm², and the volume of the cube goes from 8 to 64cm³. Bonus: For all edge lengths n, if it is doubled then the surface area of the cube increases by a factor of 4, and the volume is increased by a factor of 8. To find the surface area of a cube I multiplied length times the width of one side of the cube, and since there are 6 sides I then multiplied the product by 6. 2 x 2 x 6 = 24cm² To find the volume of a cube I multiplied length times width times height. 2 x 2 x 2 = 8 cm³ Now if I doubled the length of all of the edges of the cube the surface area will be: 4 x 4 x 6 = 96cm² and the volume will be 4 x 4 x 4 = 64cm³ Bonus: If the edge length is the general term n then the surface of the cube is n x n x 6 = 6n² The volume is n x n x n = n³ 6

7 Now if I double the length of the sides of our cube the new side length is 2n The surface area of the cube becomes 2n x 2n x 6 =4 x (6n²) or it increases by a factor of 4 The volume becomes 2n x 2n x 2n = 8n³ or the volume increases by a factor of 8. 7

8 Bibliography Information Unit 6 Area, and Volume Teachers attempted to cite the sources for the problems included in this problem set. In some cases, sources may not have been known. Problems Bibliography Information The Math Drexel ( 4-1 Zaccaro, Edward. Challenge Math (Second Edition): Hickory Grove Press,