Solids: Surface Area and Volume. 1. Use four cubes of the same size to build shapes such as the ones shown

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1 Solids: Surface Area and Volume 1. Use four cubes of the same size to build shapes such as the ones shown below (others are possible sketch them after you have built them). In each case, nd the surface area using the square face of a cube as a unit of area. Note: the surface includes the bottom of the solid, but does not include those squares that are joined together to form the sold. (Note: This is a van Hiele Level 1 activity.) 2. Suppose that you have 10 separate unit cubes. The total surface area id 60 square units and the total volume is 10 cubic units. If you arrange the cubes as shown, the volume is till 10 cubic units, but the surface area of the solid is only 36 square units (check it). For each of the following problems, assume that all 10 cubes are stacked to form a single solid sharing complete faces (no loose cubes allowed). How can you arrange 10 cubes to get a surface are of 34 square units? Sketch your answer. What is the greatest possible surface area you can obtain with 10 cubes? Sketch your answer. What is the smallest possible surface area you can obtain with 10 cubes? Sketch your answer. 1

2 What is the greatest possible surface area you can obtain with 27 cubes? The smallest? Sketch your answers. What is the greatest possible surface area you can obtain with 36 cubes? The smallest? Sketch your answers. What arrangement seems to give the greatest surface area for a given number of cubes? What arrangement seems to give the least surface area for a given number of cubes? Biologists have found that an animal's surface area is an important factor in its regulation of body temperature. Why do you think desert animals such as snakes are long and thin? Why do furry animals curl up in a ball to hibernate? 3. Take 30 congruent cubes. By stacking the small cubes, build a cube with the greatest volume. How many of the given cubes did you need to build your cube? (This is a van Hiele Level 1 activity) 2

3 Solids: Surface Area and Volume. Part II 1. Take one of the small unit cubes. What is the total length of its edges (L)? Its total surface area (A)? Its volume (V)? Form a cube that is 2 units on an edge. What is the total length of the edges of the large cube (L)? Its total surface area (A)? Its volume (V)? Make the next larger cube and record L, A and V in the table below L A V square square square square Note any patterns. Predict L, A and V for the 4 4 square. Check your prediction. 2. In the following gure, the drawing on the left shows the solid. The drawing on the right tells you howmany cubes are in each stack forming the solid. It is called a base design. What is the correct base design for the following solid? 3

4 Write the base design for the following solid. Make and then draw the solid for this base design. 3. Given are the following three base designs for stacks of unit cubes. For each one, determine the volume and total surface area (including the bottom). 4

5 Moldy Cheese A grocer had a large cube of cheese, which he forgot on the shelf. When he found it again it had grown a nice layer of blue mold on all its outside faces (including the one on the bottom). Since it was a "friendly" type of mold, he decided to go ahead and cut the cube of cheese into smaller cubes. 1. Suppose the cube was sliced by cutting each edge into two equal lengths. Use the set of cubes to build a model of this situation. After slicing, how many cubes did he have? How many slices or cuts have tobemadetocutthelargecube into smaller cubes? How many of the smaller cubes have faces covered with mold? For each smaller cube determine how many faces have mold on them. 2. Next, suppose the cube was sliced by cutting each edge into three equal lengths. Use the set of cubes to build a model of this situation. After slicing, how many cubes did he have? How many slices or cuts have tobemadetocutthelargecube into smaller cubes? How many of the smaller cubes have faces covered with mold? How many of the smaller cubes have mold on no face? One face? Two faces? Three faces? Four faces? 3. Complete the following table: n size of cube number number of Small cubes with moldy faces. of slices small cubes n n n n 5

6 The volume of a prism is given by Finding Volumes V = Bh where B is the area of the base and h is the height. Example 1. Find the volume and the surface area of a prism whose base is a right triangle with legs 3 ft. and 4 ft. and whose height is 8 ft. The volume of a right circular cylinder is given by V = Bh What is B here? How canyou nd it? Example 2. Find the volume and the surface area of a right circular cylinder of radius 3 cm. and height 8 cm. 1. Consider three boxes (right rectangular prisms) as indicated: Box 1 is Box 2 is Box 3 is Find the volume of each box. Find the surface area of each box. Do boxes with the same volume always have the same surface area? Explain. Which box used the least amount of cardboard to make? 2. Find two boxes with the same surface arae but dierent volumes. 3. The volume of an object with an irregular shape can be determined by measuring the volume of water it displaces. Arock placed in an aquarium measuring 2.5 ft. long by 1 ft. wide causes the water level to rise 1 in. What is the volume of the rock? WARNING: Be careful 4 with the units! 6

7 4. A swimming pool has shape and dimensions as shown (the units are given in meters). How many square meters of tile are needed to tile the sides and the bottom? How much water does it hold if lled to the top? The volume of a pyramid is given by V = 1 3 Bh where B is the area of the base and h is the height. Example 3. Find the volume and the surface area of the square pyramid shown. The side faces are isosceles triangles with base 24 in. and height 13 in. The volume of a right circular cone is given by V = 1 3 Bh What is B here? What is the formula for B? 7

8 Example 4. Find the volume of a right circular cone where the base has radius 6 ft. and the height measures 10 ft. Give a sketch. The volume of a sphere is given by where r is the radius of the sphere. 4 3 r2 Example 5. Find the volume of a sphere whose diameter is 12 ft. 1. A box ispacked with six soda pop cans as shown. The box measures 5 in. by 5in.by 7.5 in. What percentage of the volume of the interior of the box isnotoccupied by the cans? 2. A scoop of icecream is a perfect square with radius 5 cm. (a big scoop!) It ts exactly along the "equator" in a sugar cone as shown. Suppose the icecream melts and the cone does not absorb any of it. If the cone is 10 cm. tall, will it hold all of the melted icecream? If not, how tall would the cone have to be to hold the melted ice cream? 8

9 Volume and Surface Area Problems Note: Be sure you indicate the units in each problem and show howyou got your answers. 1. The top of a rectangular box has area 10 sq.ft., its right side has area 12 sq.ft. and the front has area 30 sq.ft. Find the volume of the box. 2. A right circular cylinder has a surface area of 66 sq. in. and a height of 8 in. Find the diameter of the base of the cylinder. 3. The Pyramid of Cheops (in Egypt) has a square base 240 yd. on a side and a height of 160 yd. Find the volume of the pyramid. Find the surface area of the four exterior sides of the pyramid. 4. A sphere of radius 2 m., a right circular cylinder of radius 2 m. and a right circular cone of radius 2 m. all have the same volume. Find the height of the cylinder and the height of the cone. 5. A concrete slab 4 in. thick is to be poured for a circular patio 12 ft in diameter. If the concrete costs $ 50 for each cubic yard, nd the cost of the concrete (to the nearest cent) for the patio. 6. Suppose we have 45 cubes, each ofwhich measures 4 in. on a side. We have to build a box in the shape of a large cube, so that we can store the small cubes in the box. How large does the box have tobeto guarantee that we can store all the cubes while keeping the size of the box as small as possible? Explain the method you used to nd the size of the box. Would there be any space left in the box to store additional 4-in. blocks later? If the answer is yes, how many more? 7. An aquarium measures 26 in. long by 15 in. wide by 12 in. high. If the aquarium is lled to the top with water, how much doesthewater weigh? (One cubic foot of water weighs 62.4 pounds.) 9