Part 3 Module 9 Real-world problems involving volume

Useful facts - volume The following facts will be provided on a formula sheet when you take quizzes or tests on this material in lab.

Exercise #1 The pedestal on which a statue is raised is a rectangular concrete solid measuring 9 feet long, 9 feet wide and 6 inches high. How much is the cost of the concrete in the pedestal, if concrete costs $70 per cubic yard? A. $34,020 B. $105 C. $315 D. $2835

Solution #1 The pedestal on which a statue is raised is a rectangular concrete solid measuring 9 feet long, 9 feet wide and 6 inches high. How much is the cost of the concrete in the pedestal, if concrete costs $70 per cubic yard? We just calculate the volume (in cubic yards) and then multiply by the cost factor of $70 per cubic yard. V = LWH L = 9 feet, W = 9 feet, H = 6 inches We must convert all three measurements to yards before we use the volume formula. L = 9 feet = 3 yards W = 9 feet = 3 yards H = 6 inches = 6/36 yard = 1/6 yard V = 3 yd x 3 yd x 1/6 yd = 9/6 cubic yards = 1.5 cubic yards Cost = 1.5 cubic yards x $70 per cubic yard = $105

Exercise #2 Gomer stores his iguana food in a can that is 8 inches tall and has a diameter of 6 inches. He stores his hamster food in a can that is 10 inches tall and has a diameter of 5 inches. Which can is larger? A. The iguana food can. B. The hamster food can. C. They are the same size. D. There is insufficient information to answer this question.

Solution #2 Gomer stores his iguana food in a can that is 8 inches tall and has a diameter of 6 inches. He stores his hamster food in a can that is 10 inches tall and has a diameter of 5 inches. The volume of the iguana food can is π(32)(8) = 72π cubic inches. The volume of the hamster food can is π(2.52)(10) = 62.5π cubic inches. The iguana food can is bigger. Some of you would get this problem wrong because you calculated the “size” of the cans by just using Area. Some of you would get this problem wrong because you entered 6 inches and 5 inches respectively into the Volume formulas for radius, but those numbers were diameters, not radii.

Exercise #3 www.math.fsu.edu/~wooland/GeoNew/Geo26.html Plato the candle-maker has a bunch of round (spherical) scented candles which are 14 centimeters in diameter. Plato wants to melt some of them down to make cylindrical candles with a diameter of 30 centimeters and a height of 26 centimeters. How many of the spherical candles are needed to make one of the cylindrical candles? A. 12.79 B. 27.41 C. 7.93 D. 17.06

Solution #3 Round (spherical) candles: 14 cm in diameter, so radius is 7 cm. cylindrical candles: diameter of 30 cm (so radius is 15 cm) and a height of 26 cm. How many of the spherical candles are needed to make one of the cylindrical candles? We divide the volume of the large (cylindrical) candle by the volume of the small (spherical) candle. Cylindrical candle: V = π(152)(26) = 18,378.32 cubic cm Spherical candle: V = (4/3)π (73) = 1436.76 cubic cm (18,378.32 cubic cm)/(1436.76 cubic cm) = 12.79 The cylindrical candle is 12.79 times as big as the spherical candle, so it takes 12.79 spherical candles to make one cylindrical candle.

Exercise #4 (Based on a true story) Gomer has been working out by lifting weights. He has been using spherical lead-alloy weights with a radius of 3 inches. Each sphere weighs 20 pounds. Now he wishes to lift 100 pound weights, so he special-orders spherical weights, made of the same material, but with a radius of 15 inches. Why is Gomer in intensive care? Why is Gomer bankrupt?

Solution #4 We will divide the volume of the larger sphere by the volume of the smaller sphere. Larger sphere: V = (4/3)π (153) = 14,137.166941 cubic inches Smaller sphere: V = (4/3)π (33) = 113.09733552923 cubic inches Divide: 14,137.166941/ 113.09733552923 = 125 This tells us that the large spherical weight is 125 times as big as the small spherical weight, so it weighs 125 times as much. 125 x 20 lb. = 2500 lb. Gomer is intensive care because he thought was lifting a 100 pound weight, when in fact he was trying to lift a 2500 weight. He hurt himself. Gomer is bankrupt because he ordered several of these 2500 pound weights from a company on the west coast, and he went broke paying the shipping charges. He should have clicked the button that says “Estimate shipping charges.

Exercise #5 www.math.fsu.edu/~wooland/GeoNew/Geo12.html The formula N = 0.06125DV can be used to determine the number (N) of five-pound bags of ice needed to reduce the temperature of a small body of water by D degrees Fahrenheit where V is the volume of water (in cubic feet). Archimedes has a cone-shaped water garden with a radius of 9.5 feet filled to a depth of 17 inches. How many bags of ice are needed to reduce the temperature of the water from 91 degrees to 82 degrees? A. 74 B. 11 C. 886 D. 1205