Pretest Please complete the pretest for this standard on your own. Try to remember all you can from our first discussion of this topic. Explain and use formulas for lateral area, surface area, and volume of solids. MA.912.G.7.5 1
Surface Area and Volume Surface Area: What it takes to COVER something Volume: What it takes to FILL something Lateral Area & Surface Area Surface Area covers the ENTIRE object. Lateral Area does not include any of the BASES. (tops and boloms) Surface area will ALWAYS be GREATER than lateral area. 2
Use the reference sheet as a guide when solving area and volume problems!!! Use the reference sheet as a guide when solving area and volume problems!!! 3
Find the volume of the cone shown as a decimal rounded to the nearest tenth. A. 124.6 yd 3 B. 1308.5 yd 3 C. 2118.5 yd 3 D. 872.3 yd 3 V = 1 3 πr2 h V = 1 3 π ( 7 ) 2 ( 17) V = 1 3 π ( 49 )( 17) Since the radius is squared and the height is not, you need to be careful where you plug in your values in this formula. V = 1 ( 3 833 )π Your answer may differ slightly if you use the approximaaon 3.14 rather than the pi key on your calculator. V 872.3 yd 3 Volume 1. Find the surface area of the cylinder in terms of π. Since this problem asks for the surface area and not the lateral surface area, we will include the two bases. S.A. = 2πrh + 2πr 2 πr 2 is the area of a circle. Since there are two circular bases, we see this reflected in the S.A. formula S.A. = 2π ( 6) ( 19) + 2π ( 6) 2 S.A. = 228π + 72π S.A. = 300π in 2 4
2. Find the volume of the sphere shown. Give your answer rounded to the nearest cubic cenameter. V = 4 3 πr3 V = 4 3 π ( 7 ) 3 V = 4 ( 3 343 )π V =1436 cm 3 Find both the lateral area and the surface area of the square pyramid at the right. S.A. = 1 2 P + B We are given the alatude and not the slant height so we need to apply the Pythagorean Theorem. Remember that the first leg is the alatude and the second leg is half of the base s edge. = ( 19) 2 + ( 20) 2 = 761 P = 38+ 38+ 38+ 38 =152 B = 38 38 =1444 Remember, that lateral area does not include any bases, so you can leave B out of the formula. Surface Area S.A. = 1 2 P + B S.A. = 1 ( 2 152 ) 761 +1444 3540.6 yd 2 S.A. = 1 ( 2 152 ) 761 2096.6 yd 2 5
Determine how changes in dimensions affect the surface area and volume of common geometric solids MA.912.G.7.7 Kendra has a compost box that has the shape of a cube. She wants to increase the size of the box by extending every edge of the box by half its original length. Afer the box is increased in size, which of the following statements is true? A. The volume of the new compost box is exactly 112.5% of the volume of the original box. B. The volume of the new compost box is exactly 150% of the volume of the original box. C. The volume of the new compost box is exactly 337.5% of the volume of the original box. D. The volume of the new compost box is exactly 450% of the volume of the original box. Since we do not know the length of the original edge, we can make our own. Let s say that the edge was 4 units. Then the new edge would be increased by half of that which is 2. The new edge would be 6 units. Old volume is 64 units 3. New volume is 216 units 3 216 100 = 337.5% 64 Using a Concrete Example to Solve 6
A city is planning to replace one of its water storage tanks with a larger one. The city s old tank is a right circular cylinder with a radius of 12 feet and a volume of 10,000 cubic feet. The new tank is a right circular cylinder with a radius of 15 feet and the same height as the old tank. What is the maximum number of cubic feet of water the new storage tank will hold? Strategy: Make a table Comparing Volume Old Tank New Tank Formula Missing Dimensions r=12; V=10,000 r=15 h=? Vnew=? V = πr 2 h Strategy: Make a table Old Tank New Tank Formula Missing Dimensions r=12; V=10,000 ( ) 2 h 10000 = π 12 10000 144π = h r=15 h=? Vnew=? V = πr 2 h Begin by using what you know about the old tank to find the height of both tanks. Now that we know the height and the radius of the new tank, we can calculate the volume. V = πr 2 h V = π 15! $ & " 144π % ( ) 2 # 10000 V =15, 625 ft 3 Comparing Volumes 7
Rodney s dad wants him to paint a box that he was making. Rodney calculated that he needed 128 cubic inches of paint to cover the box. Later, Rodney s dad changed the size of the box by mulaplying each of its edges by ¾. How much paint does Rodney need to buy now? S.A. = 2bh + 2bw + 2hw Surface area is the sum of the areas of the sides. Since the area is squared, we need to square ¾. 2! 3$ # & " 4 % = 9 16 This means that the surface area of the box will be reduced by a factor of 9/16. 128 9 = 72 in3 16 Comparing Surface Area 1. A rectangular fish tank is 7 feet wide by 3 feet long and is filled to a depth of 3 feet. All of the water is poured into a second tank that is 6 feet wide by 2 feet long. What is the depth of the water in the new tank? Find the volume of the old tank. V = lwh V = 3 ( )( 7) ( 3) = 63 ft 3 The water in the new tank will have the same volume. Use it to determine the new depth (height). 63 = ( 2) ( 6)h 63 =12h h = 63 12 = 21 4 ft 8
2. A rectangular prism with dimensions 2 in by 3 in by 6 in has the same volume as a right square pyramid that has a 6 in by 6 in base. What is the height of the pyramid? Volume of the rectangular prism: V = lwh V = 2 ( )( 3) ( 6) = 36 in 3 Volume of a right square pyramid: V = 1 3 Bh 36 = 1 ( 3 6 6 )h 36 = 1 ( 3 36 )h 36 =12h h = 3 in 3. A firework explodes and is spreading from its center in a spherical palern. The sphere currently covers a surface area of 240 sq. meters. The diameter of the firework is expected to double in the next 30 seconds. What will be the new surface area of the sphere? S.A. = 4πr 2 S.A. = 4π ( 2r) 2 If the diameter doubles, so does the radius. S.A. = 4π 4r 2 So the surface area increases by a factor of 4. 240 4 = 960 square meters 9