Volumes Of Prisms And Cylinders

ders.27. SOIL A soil scientist wants to determine the bulkdensity of a potting soil to assess how well a specificplant will grow in it. The densityof the soil sample is the ratio of its weight to itsvolume.a. If the weight of the container with the soil is 20pounds and the weight of the container alone is 5pounds, what is the soil’s bulk density? b. Assumingthat all other factors are favorable, how well should aplant grow in this soil if a bulk density of 0.018 poundper square inch is desirable for root growth? Explain.c. If a bag of this soil holds 2.5 cubic feet, what is itsweight in pounds?Find the volume of each composite solid. Roundto the nearest tenth if necessary.28.SOLUTION:The solid is a combination of two rectangular prisms.The base of one rectangular prism is 5 cm by 3 cmand the height is 11 cm. The base of the other prismis 4 cm by 3 cm and the height is 5 cm.SOLUTION:a. First calculate the volume of soil in the pot. Thendivide the weight of the soil by the volume.29.SOLUTION:The solid is a combination of a rectangular prism anda right triangular prism. The total volume of the solidis the sum of the volumes of the two rectangularprisms.The weight of the soil is the weight of the pot withsoil minus the weight of the pot.W 20 – 5 15 lbs.The soil density is thus:b. 0.0018 lb/in3 is close to 0.0019 lb/in3 so the plantshould grow fairly well.c.30.eSolutions- Poweredofby eachCogneroFindManualthe volumecompositeto the nearest tenth if necessary.solid. RoundPage 8SOLUTION:The solid is a combination of a rectangular prism andtwo half cylinders.

12-4 Volumes of Prisms and Cylinders30.SOLUTION:The solid is a combination of a rectangular prism andtwo half cylinders.31. MANUFACTURING A can 12 centimeters tall fitsinto a rubberized cylindrical holder that is 11.5centimeters tall, including 1 centimeter for thethickness of the base of the holder. The thickness ofthe rim of the holder is 1 centimeter. What is thevolume of the rubberized material that makes up theholder?SOLUTION:The volume of the rubberized material is thedifference between the volumes of the container andthe space used for the can. The container has a31. MANUFACTURING A can 12 centimeters tall fitsinto a rubberized cylindrical holder that is 11.5centimeters tall, including 1 centimeter for thethickness of the base of the holder. The thickness ofthe rim of the holder is 1 centimeter. What is thevolume of the rubberized material that makes up theholder?radius ofand a height of 11.5 cm.The empty space used to keep the can has a radiusof 3.25 cm and a height of 11.5 – 1 10.5 cm. The2volume V of a cylinder is V πr h, where r is theradius of the base and h is the height of the cylinder.Therefore, the volume of the rubberized material is3about 304.1 cm .SOLUTION:The volume of the rubberized material is thedifference between the volumes of the container andthe space used for the can. The container has aradius ofFind each measure to the nearest tenth.32. A cylindrical can has a volume of 363 cubiccentimeters. The diameter of the can is 9centimeters. What is the height?SOLUTION:and a height of 11.5 cm.The empty space used to keep the can has a radiusof 3.25 cm and a height of 11.5 – 1 10.5 cm. The2volume V of a cylinder is V πr h, where r is theradius of the base and h is the height of the cylinder.Therefore, the volume of the rubberized material is3about 304.1 cm .eSolutions Manual - Powered by CogneroFind each measure to the nearest tenth.32. A cylindrical can has a volume of 363 cubic33. A cylinder has a surface area of 144π square inchesand a height of 6 inches. What is the volume? Page 9SOLUTION:

12-4 Volumes of Prisms and Cylinders33. A cylinder has a surface area of 144π square inchesand a height of 6 inches. What is the volume?SOLUTION:Use the surface area formula to solve for r.34. A rectangular prism has a surface area of 432square inches, a height of 6 inches, and a width of 12inches. What is the volume?SOLUTION:Use the surface area formula to find the length of thebase of the prism.Find the volume.The radius is 6. Find the volume.34. A rectangular prism has a surface area of 432square inches, a height of 6 inches, and a width of 12inches. What is the volume?SOLUTION:Use the surface area formula to find the length of thebase of the prism.35. ARCHITECTURE A cylindrical stainless steelcolumn is used to hide a ventilation system in a newbuilding. According to the specifications, the diameterof the column can be between 30 centimeters and 95centimeters. The height is to be 500 centimeters.What is the difference in volume between the largestand smallest possible column? Round to the nearesttenth cubic centimeter.SOLUTION:The volume will be the highest when the diameter is95 cm and will be the lowest when it is 30 cm.That iswhen the radii are 47.5 cm and 15 cm respectively.Find the difference between the volumes.Find the volume.eSolutions Manual - Powered by Cognero36. CCSS MODELING The base of a rectangularswimming pool is sloped so one end of the pool is 6feet deep and the other end is 3 feet deep, as shownin the figure. If the width is 15 feet, find the volumePage 10of water it takes to fill the pool.

12-4 Volumes of Prisms and Cylinders36. CCSS MODELING The base of a rectangularswimming pool is sloped so one end of the pool is 6feet deep and the other end is 3 feet deep, as shownin the figure. If the width is 15 feet, find the volumeof water it takes to fill the pool.SOLUTION:The swimming pool is a combination of a rectangularprism and a trapezoidal prism. The base of therectangular prism is 6 ft by 10 ft and the height is 15ft. The bases of the trapezoidal prism are 6 ft and 3ft long and the height of the base is 10 ft. The heightof the trapezoidal prism is 15 ft. The total volume ofthe solid is the sum of the volumes of the twoprisms.37. CHANGING DIMENSIONS A soy milk companyis planning a promotion in which the volume of soymilk in each container will be increased by 25%. Thecompany wants the base of the container to stay thesame. What will be the height of the new containers?SOLUTION:Find the volume of the original container.The volume of the new container is 125% of theoriginal container, with the same base dimensions.Use 1.25V and B to find h.37. CHANGING DIMENSIONS A soy milk companyis planning a promotion in which the volume of soymilk in each container will be increased by 25%. Thecompany wants the base of the container to stay thesame. What will be the height of the new containers?38. DESIGN Sketch and label (in inches) three differentdesigns for a dry ingredient measuring cup that holds1 cup. Be sure to include the dimensions in each3drawing. (1 cup 14.4375 in )SOLUTION:Sample answers:For any cylindrical container, we have the followingequation for volume:SOLUTION:Find the volume of the original container.The volume of the new container is 125% of theoriginal container, with the same base dimensions.Use 1.25V and B to find h.eSolutions Manual - Powered by CogneroThe last equation gives us a relation between theradius and height of the cylinder that must be fulfilledto get the desired volume. First, choose a suitablePage 11radius, say 1.85 in, and solve for the height.

last equationgivesandus aCylindersrelation between the12-4TheVolumesof Prismsradius and height of the cylinder that must be fulfilledto get the desired volume. First, choose a suitableradius, say 1.85 in, and solve for the height.If we choose a height of say 4 in., then we can solvefor the radius.For any rectangular container, the volume equation is:39. Find the volume of the regular pentagonal prism bydividing it into five equal triangular prisms. Describethe base area and height of each triangular prism.Choose numbers for any two of the dimensions andwe can solve for the third. Let l 2.25 in. and w 2.5 in.SOLUTION:The base of the prism can be divided into 5congruent triangles of a base 8 cm and thecorresponding height 5.5 cm. So, the pentagonalprism is a combination of 5 triangular prisms of height10 cm. Find the base area of each triangular prism.Therefore, the volume of the pentagonal prism is40. PATIOS Mr. Thomas is planning to remove an oldpatio and install a new rectangular concrete patio 20feet long, 12 feet wide, and 4 inches thick. Onecontractor bid 2225 for the project. A secondcontractor bid 500 per cubic yard for the new patioand 700 for removal of the old patio. Which is theless expensive option? Explain.eSolutions Manual - Powered by CogneroSOLUTION:Convert all of the dimensions to yards.Page 12

volumeandof thepentagonal prism is12-4Therefore,Volumes theof PrismsCylinders40. PATIOS Mr. Thomas is planning to remove an oldpatio and install a new rectangular concrete patio 20feet long, 12 feet wide, and 4 inches thick. Onecontractor bid 2225 for the project. A secondcontractor bid 500 per cubic yard for the new patioand 700 for removal of the old patio. Which is theless expensive option? Explain.c. ANALYTICAL Describe which change affectsthe volume of the cylinder more: multiplying theheight by x or multiplying the radius by x. Explain.SOLUTION:a. The oblique cylinder should look like the rightcylinder (same height and size), except that it ispushed a little to the side, like a slinky.SOLUTION:Convert all of the dimensions to yards.b. Find the volume of each.20 feet yd12 feet 4 yd4 in. ydFind the volume.The volume of the square prism is greater.c. Do each scenario.The total cost for the second contractor is about.Therefore, the second contractor is a less expensiveoption.41. MULTIPLE REPRESENTATIONS In thisproblem, you will investigate right and obliquecylinders.a. GEOMETRIC Draw a right cylinder and anoblique cylinder with a height of 10 meters and adiameter of 6 meters.b. VERBAL A square prism has a height of 10meters and a base edge of 6 meters. Is its volumegreater than, less than, or equal to the volume ofthe cylinder? Explain.c. ANALYTICAL Describe which change affectsthe volume of the cylinder more: multiplying theheight by x or multiplying the radius by x. Explain.Assuming x 1, multiplying the radius by x makes2the volume x times greater.2For example, if x 0.5, then x 0.25, which is lessthan x.42. CCSS CRITIQUE Franciso and Valerie eachcalculated the volume of an equilateral triangularprism with an apothem of 4 units and height of 5units. Is either of them correct? Explain yourreasoning.SOLUTION:a. The oblique cylinder should look like the righteSolutions Manual - Powered by Cognerocylinder (same height and size), except that it ispushed a little to the side, like a slinky.Page 13SOLUTION:

32container is 60π in .the volume x times greater.2example,x 0.5,andthenx 0.25, which is less12-4ForVolumesof ifPrismsCylindersthan x.a. Choose some basic values for 2 of the sides, andthen determine the third side. Base: 3 by 5.42. CCSS CRITIQUE Franciso and Valerie eachcalculated the volume of an equilateral triangularprism with an apothem of 4 units and height of 5units. Is either of them correct? Explain yourreasoning.3 by 5 by 4πb. Choose some basic values for 2 of the sides, andthen determine the third side. Base: 5 by 5.SOLUTION:Francisco; Valerie incorrectly usedas thelength of one side of the triangular base. Franciscoused a different approach, but his solution is correct.Francisco used the standard formula for the volumeof a solid, V Bh. The area of the base, B, is onehalf the apothem multiplied by the perimeter of thebase.5 by 5 byc. Choose some basic values for 2 of the sides, andthen determine the third side. Base: Legs: 3 by 4.43. CHALLENGE A cylindrical can is used to fill acontainer with liquid. It takes three full cans to fill thecontainer. Describe possible dimensions of thecontainer if it is each of the following shapes.a. rectangular prismb. square prism3 by 4 by 10πc. triangular prism with a right triangle as the base44. WRITING IN MATH Write a helpful response tothe following question posted on an Internetgardening forum.I am new to gardening. The nursery will deliver atruckload of soil, which they say is 4 yards. Iknow that a yard is 3 feet, but what is a yard ofsoil? How do I know what to order?SOLUTION:3The volume of the can is 20π in . It takes three fullcans to fill the container, so the volume of the3container is 60π in .a. Choose some basic values for 2 of the sides, andthen determine the third side. Base: 3 by 5.eSolutions Manual - Powered by CogneroSOLUTION:Sample answer: The nursery means a cubic yard,3which is 3 or 27 cubic feet. Find the volume of yourgarden in cubic feet and divide by 27 to determinethe number of cubic yards of soil needed.45. OPEN ENDED Draw and label a prism that has avolume of 50 cubic centimeters.SOLUTION:Page 14Choose 3 values that multiply to make 50. Thefactors of 50 are 2, 5, 5, so these are the simplest

SOLUTION:Sample answer: The nursery means a cubic yard,3which is 3 or 27 cubic feet. Find the volume of yourin cubicfeet anddivideby 27 to determine12-4gardenVolumesof PrismsandCylindersthe number of cubic yards of soil needed.45. OPEN ENDED Draw and label a prism that has avolume of 50 cubic centimeters.SOLUTION:Choose 3 values that multiply to make 50. Thefactors of 50 are 2, 5, 5, so these are the simplestvalues to choose.Sample answer:Both formulas involve multiplying the area of thebase by the height. The base of a prism is a polygon,so the expression representing the area varies,depending on the type of polygon it is. The base of a2cylinder is a circle, so its area is πr .48. The volume of a triangular prism is 1380 cubiccentimeters. Its base is a right triangle with legsmeasuring 8 centimeters and 15 centimeters. What isthe height of the prism?A 34.5 cmB 23 cmC 17 cmD 11.5 cmSOLUTION:46. REASONING Determine whether the followingstatement is true or false . Explain.Two cylinders with the same height and the samelateral area must have the same volume.SOLUTION:True; if two cylinders have the same height (h 1 h 2)and the same lateral area (L1 L2), the circularbases must have the same area.49. A cylindrical tank used for oil storage has a heightthat is half the length of its radius. If the volume of3the tank is 1,122,360 ft , what is the tank’s radius?F 89.4 ftG 178.8 ftH 280.9 ftJ 561.8 ftSOLUTION:The radii must also be equal.47. WRITING IN MATH How are the formulas forthe volume of a prism and the volume of a cylindersimilar? How are they different?SOLUTION:Both formulas involve multiplying the area of thebase by the height. The base of a prism is a polygon,so the expression representing the area varies,depending on the type of polygon it is. The base of a2cylinder is a circle, so its area is πr .48. The volume of a triangular prism is 1380 cubiccentimeters. Its base is a right triangle with legsmeasuring 8 centimeters and 15 centimeters. What isthe height of the prism?A 34.5 cmB 23 cmeSolutions Manual - Powered by CogneroC 17 cmD 11.5 cm50. SHORT RESPONSE What is the ratio of the areaof the circle to the area of the square?SOLUTION:Page 15The radius of the circle is 2x and the length of eachside of the square is 4x. So, the ratio of the areas can

where is the slant height and P is the perimeter ofthe base.The slant height is the height of each of thecongruent lateral triangular faces. Use thePythagorean Theorem to find the slant height.12-4 Volumes of Prisms and Cylinders50. SHORT RESPONSE What is the ratio of the areaof the circle to the area of the square?SOLUTION:The radius of the circle is 2x and the length of eachside of the square is 4x. So, the ratio of the areas canbe written as shown.51. SAT/ACT A county proposes to enact a new 0.5%property tax. What would be the additional taxamount for a landowner whose property has ataxable value of 85,000?A 4.25B 170C 425D 4250E 42,500SOLUTION:Find the 0.5% of 85,000.Find the perimeter and area of the equilateraltriangle for the base. Use the Pythagorean Theoremto find the height h of the triangle.Therefore, the correct choice is C.Find the lateral area and surface area of eachregular pyramid. Round to the nearest tenth ifnecessary.52.SOLUTION:The lateral area L of a regular pyramid is,where is the slant height and P is the perimeter ofthe base.The perimeter is P 3 10 or 30 feet.The slant height is the height of each of thecongruent lateral triangular faces. Use thePythagorean Theorem to find the slant height.eSolutions Manual - Powered by CogneroSo, the area of the base B isft2.Page 16Find the lateral area L and surface area S of theregular pyramid.

Therefore, the surface area of the pyramid is about12-4 Volumes of Prisms and CylindersSo, the area of the base B is2255.4 ft .ft2.Find the lateral area L and surface area S of theregular pyramid.53.SOLUTION:The lateral area L of a regular pyramid is,where is the slant height and P is the perimeter ofthe base.Here, the base is a square of side 7 cm and the slantheight is 9 cm.2So, the lateral area of the pyramid is about 212.1 ft .So, the lateral area of the pyramid is 126 cm2.Therefore, the surface area of the pyramid is about2255.4 ft .The surface area S of a regularpyramid is, whereL is the lateral area and B is the area of the base.53.SOLUTION:The lateral area L of a regular pyramid is,Therefore, the surface area of the pyramid is 1752cm .where is the slant height and P is the perimeter ofthe base.Here, the base is a square of side 7 cm and the slantheight is 9 cm.54.So, the lateral area of the pyramid is 126 cm2.SOLUTION:The pyramid has a slant height of 15 inches and thebase is a hexagon with sides of 10.5 inches.A central angle of the hexagon isor 60 , so theangle formed in the triangle below is 30 .The surface area S of a regularpyramid is, whereL is the lateral area and B is the area of the base.eSolutions Manual - Powered by CogneroPage 17

SOLUTION:The pyramid has a slant height of 15 inches and thebase is a hexagon with sides of 10.5 inches.12-4 Volumes of Prisms and CylindersA central angle of the hexagon isor 60 , so theangle formed in the triangle below is 30 .Therefore, the surface area of the pyramid is about758.9 in2.55. BAKING Many baking pans are given a specialnonstick coating. A rectangular cake pan is 9 inchesby 13 inches by 2 inches deep. What is the area ofthe inside of the pan that needs to be coated?Use a trigonometric ratio to find the measure of theapothem a.SOLUTION:The area that needs to be coated is the sum of thelateral area and one base area.Therefore, the area that needs to be coated is 2(13 29)(2) 13(9) 205 in .Find the indicated measure. Round to thenearest tenth.56. The area of a circle is 54 square meters. Find thediameter.Find the lateral area and surface area of thepyramid.SOLUTION:So, the lateral area of the pyramid is 472.5 in2.The diameter of the circle is about 8.3 m.57. Find the diameter of a circle with an area of 102square centimeters.SOLUTION:Therefore, the surface area of the pyramid is about758.9 in2.55. BAKING Many baking pans are given a specialnonstick coating. A rectangular cake pan is 9 inchesby 13 inches by 2 inches deep. What is the area ofthe inside of the pan that needs to be coated?The diameter of the circle is about 11.4 m.58. The area of a circle is 191 square feet. Find theradius.SOLUTION:The area that needs to be coated is the sum of thelateral area and one base area.Therefore, the area that needs to be coated is 2(13 eSolutions Manual

volumes of the rectangular prism and the cylinders. SANDCASTLES In a sandcastle competition, . the diameter of the cylinders should be about 2.52 ft. Find the volume of the solid formed by each net. . is the sum of the volumes of the two rectangular prisms. 62/87,21 The solid is a combination of a rectangular prism and