ASSIGNMENT #1
SOLID MENSURATION -
PLANES, LINES AND ANGLES
A. Complete each statement using "Always", "Sometimes", or "Never".
1. Two distinct points are _______ collinear.
2. A line and a point not on the given line are ______ non-coplanar.
3. Three distinct non-collinear points are ______ coplanar.
4. Three points are _____ collinear.
5. Two intersecting lines _______ lie in one plane.
6. Three distinct collinear points are _______ contained in only one plane.
7. Four non-coplanar points _______ determine a space.
8. A line and a plane _________ intersect at a point.
9. Two distinct planes ______ intersect at a line.
10. Two straight lines perpendicular to the same straight line are _______ parallel.
B. Determine the theorem that justifies the following practical applications.
1. Two nails were driven against a wall and a line was drawn to connect them. The line drawn is also on the wall.
2. Karen and Gina were trying to find the location of Manila on the world map. They used the imaginary latitude and longitude lines to do so.
3. Shaggy, a carpenter, decided he had enough of his wobbling four-legged chair and built a three-legged stool that is more suitable.
4. Stan folded his sheet of paper into two and left a crease on where he folded it.
5. Steven, a professional photographer uses tripod for his camera so that he can take picture of his subject better.
C. Solve the following problems completely.
1. How can two pencils (representing lines) be held so that a plane can passed containing both; so that no plane can be passed containing both?
2. What is meant by the statement, "Two planes determine a line" Is this universally true?
3. Can a plane intersect two planes that are not parallel so that the intersections are parallel? Illustrate your answer by passing a plane through the edges of an open book. How must this plane be passed so that the intersections are parallel lines?
4. If two lines do not intersect, can a third line be drawn perpendicular to them?
5. Why are the corner edges of a building perpendicular to the plane of a level street?
6. How many positions can a lightning rod erected at a point occupy w out being vertical? How many positions may it assume and be vertical?
7. In erecting a vertical column a carpenter may use a carpenter square to make certain that the column is perpendicular to the floor. In how in different positions must he places the square against the column to ascertain whether it is vertical? Why?
8. Can a line be perpendicular to both of two planes if they are not parallel?
9. Can one line be perpendicular to two other lines that intersect? Explain.
10. A pole 26 ft. long leans against a wall at a point 10 ft. from the ground. What is the length of the projection of the pole on the ground?
ASSIGNMENT #2
MENSURATION OF PLANE FIGURES
1. If three lines intersect, then there is exactly one plane that contains them.
2. If two lines intersect, then there is exactly one plane that contains them.
3. A line can be in two different planes
Answer the following problems completely.
1. If the perimeter of a square is 20, what is its area
2. A parallelogram has sides 9 cm and 15 cm lone. If the shorter altitude is 3 cm, what is the length of the longer altitude?
3. If the area of an equilateral triangle is√3, what is the length of each side?
4. The area of a rhombus is 126 square meters, and one of its diagonals measures 14 meters, what is the length of the other diagonal?
5. If an isosceles trapezoid has bases with lengths 12 and 28 and an area of 300, what is its perimeter?
6. Find the area of a regular hexagon with perimeter of 72.
7. If the perimeter of an octagonal stop sign is 80 inches and its height is about 24 inches, what is its area.
8. In a right triangle, the altitude to the hypotenuse separates the hypotenuse into two equal pieces. If the whole hypotenuse is 150, what is the length of the altitude?
9. A rectangle filed has a length of 1200 yards and a diagonal of 1300 yards. What is the width of the field in yards?
10. Two sides of a right triangle are 5 and 15. Which of the following could be the third side?
11. The diagonals of a rhombus measure 16 and 30. What is the perimeter of the rhombus?
12. A commuter drives 8 blocks west and 6 blocks south to get to work. If a block is 50 yards, how many yards would a crow have fly between the two locations?
13. A kite has diagonals measuring 16 and 21. The former divides the latter into pieces of 15 and 6. What is the perimeter of the kite?
14. A pair of corresponding sides of two similar polygons measures 15 and 21. What could be their perimeters?
15. The perimeters of two similar triangles are 54 and 30. The shortest side on the smaller triangle is x, and the corresponding side on the larger triangle is 3x-6. What is the length x?
16. Could 8, 9, 0.1 be the measures of the sides of a triangle?
17. Find the range of the values for the third side of a triangle with sides of length 3 cm
18. A man trapped inside a barrel with a diameter of 36 inches rolls 50 yards down a hill. How many complete revolutions does the man make?
19. If the area of a circle is increased by 21%, by what percent does its circumference increase?
20. If a certain shape is both a rectangle and a rhombus, and one side of it is 3, what is sum of the remaining side measures?
21. Two angles in a parallelogram have a sum of 154. What is the measure of the angle between them?
22. A parallelogram's diagonals have lengths of 24 and 17, and the angle between them measures 90. What is it?
23. If one base of a trapezoid is 10 units long and the midsegment is 13 units long, what is the length of the other base?
ASSIGNMENT #3
MEASUREMENT AND SCALING
1. In each of the following, give the numeral that can be written in the bank space so that the resulting statement will the true.
a) 70 km. = ___mi. b) 240 cm. = in. c) 40 m. = yd.
d) 500 ft. = ___m. e) 30 in. = m. f) 400 mi = m.
g) 5g. = oz. h) 8 kg. = lb. i) 448 g = lb.
j) 3lb. = g. k) 4.2 kg = oz. l) 500 lb = g.
m) 2da. 8hr. 15min. = min. n) 2 yr. 8 mo. = da. o) 4 wk. 3da = hr.
2. Perform the indicated operations.
a) 15 m 2 mm
+ 3 m 4 mm
b) 8 kg 2 g.
+ 13 kg 5 g.
c) 6 m
+ 1 yd
d) 12 kg
- 4 lbs
e) 66 m
- 42 cm
f) 312 ft
- 750 in
g) 3 m 4 mm
× 3.2
h) 12 kg 2 ??g
× 2
i) 2 g 1mg
× 5.2
j) 6 kg 9 g
÷ 3
k) 8 ft 20 in
÷ 12
l) 45 yd
÷ 1.88
3. Each week Charles spends 135 minutes in shopworks. How much time in hours does he spend in this work?
4. On a parking meter was printed: "12 minutes for P5. Maximum deposit P24." How many hours may a driver legally park at one of the meters for P25?
5. Which is a longer race, a mile run or a 1500-meter run, and by how much?
6. A plane is flying at the rate of 450 km per hour. How many miles per hour is the plane flying?
7. A mountain is 15,400 feet high. What is its height in meters, correct to the nearest meter?
8. A butcher bought 2400 kilograms of meat. How many pounds did he buy?
9. If 120 grams of meat are used to make a hamburger, about how many ounces is this?
10. The contents of a jar weigh 126 g. How many ounces do the contents weigh?
11. What is the cost of 3 lb. 7 oz. of peaches at $.49 per pound?
ASSIGNMENT #4
SOLIDS FOR WHICH V=BH
1. Atrenchis200ft.loegandl2ft.deep,Sft.wideatthetOPand4ft. wide at the bottom. How many cubic yards of earth have been removed?
2. When an irregular-shaped rock is placed in a cylindrical vessel of water whose radius is 4.l8~ in., the water rises 6.85 in. What is the volume of the
3. rock if it is completely submerged?
4. A dam is 40 ft. long, 12 ft. high, 7 ft. wide at the bottom, and 4 ft. wide at the top. How many cubic yards of material were used in constructing it?
5. L 4. Two rectangular water tanks with tops on the same level are connected
a. by a pipe through their bottoms. The base of one is 6 in. higher than that of
b. the other. Their dimensions are 4 by 5 by 2~ ft. and 4 by 7 by 3 ft., respec
6. r ~ tively. How deep is the water in the larger tank when the water they contain equals half their combined capacity, if the 2fft. and 3-ft. edges are vertical?
1. A cylindrical tin can holding 2 gal, has its height equal to the diameter of its base. Another cylindrical tin can with the same capacity has its height equal to twice the diameter of its base. Find the ratio of the amount of tin required for making the two cans with covers.
2. How much wood is wasted in turning out the largest possible cylindrical rod from a stick whose uniform square cross- sectional area is 10 sq. in. and whose length is 5 ft.?
3. How long a wire 0.1 in. in diameter can be drawn from a block of copper 2 by 4 by6 in.?
4. A piece of lead pipe is 50ft, long. Its outer radius is 2 in., and it is ~ in. thick. Into how many spherical bullets ~ in. in diameter can it be melted?
5. A corncrib 20 ft. long has a cross section ‘which is represented in the figure. The crib is entirely filled with corn on the ear. How many bushels bf corn does it contain, if 2 bu. of corn on the ear are equal to I. bu. of shelled corn? (Use the approximation,~1 bu. = 1~ Cu. ft.)
6. How many bricks each 8 in.~ by 2~ in. by 2 in. will be required to build a wall 22 ft. by 3 ft. by 2 ft. (not alkwing for mortar)?
7. The height of a hot-water boiler attached tG a furnace is 5~ ft. The circumference of the boiler is 48 in. If the metal is j in. thick, how many gallons will the boiler contain when full? (One gal. 231 cii. in.)
8. A cylindrical tank~ diameter 1 ft., length 6 ft., is placed so that it~áxis is horizontal. How many pounds of water will be used in filling it to ~depth of 9 in., if water weighs 62.4 lb. per cu. ft.?
9. How many washers can be made from a cube of metal 4 in. on a side, ii the washers are ~ in. in diameter and j in. thick? The hole in the center of the washers is ~ in. in diameter.
10. What are the dimensions of a gallon can of uniform square cross section whose height is twice the length of an edge of its base? (One gal. equals 231 cu. in.)
11. A wood in an makes a wedge-shaped cut in the trunk of a tree. Assume that the trunk is a right circular cylinder of radius 4 in., that the lower surface of the cut is a horizontal plane, and that the upper surface is a plane inclined at an angle of 450 to the'horizontal and intersecting the lower surface of the out in a ‘diameter. This wedge is then cut into two equal pieces by a vertical cut. What is the area of this vertical section? The woodnian now wishes to divide one of these pieces by another section 4. parallel to the first section. If the new section is to have an area equal to one-fourth that of the original section, where should the cut be made?
12. How deep a cut should be milled off one side of a 2k-in. shaft to make a flat (a rectangular flat area) i~ in. wide?
13. The base of a right cylinder is shown in the figure. it is formed by describing semicircular arcs within the square upon the four sides as diani-eters. If the altitude of the cylinder is 12 in., find the volume and total area.
ASSIGNMENT #5 SOLIDS OF WHICH V = 1/3 Bh
1. The lateral edge of a pyramidal church spire is 61 ft. Each side of its octagonal base is 22 ft. What will be the cost of painting the spire at 24 cents a square foot?
2. Each element of a circular conical pile of sand 6 ft. high is inclined 45° to the horizontal. How many cubic feet of sand does the pile contain?
3. The inside dimensions Of a trunk are 4 ft., 3 ft., 2 ft. Find the dimensions of a trunk similar in shape that will hold 4 times as much.
4. How many square feet of canvas are required for a conical tent 18 ft. high and 10 ft. in diameter if 10 per cent of the material is wasted?
5. A model steamboat is 2 ft. ‘3 in. long and it displaces 1.5 lb. of water. The ship of which the model is made 15 720 ft. long. What is its displacement in tons?
6. Among the interesting applications of similarity is the ease of a shadow, as here shown, where the, light is the center of similitude. If a man's profile is 12 in. in height and the similar shadow is 16 in. in height, find the ratio of the area of the profile to the area of the shadow.
7. How far from the top must you cut a conical tent in order to cut the cloth in half?
8. A well 40 ft. deep and 6.5 ft. in diameter is lined with stone 1.5 ft. thick so that the inner diameter of the well becomes
3.5 ft. Find the number of cubic feet of stone required.
9. A pyramidal roof 16 ft. in height, standing on a square base 24 ft. on a side, is covered with sheet lead ~ in. thick. (a) Find the weight of the lead if 1 cu~ in. of lead weighs 7 oz. (&) If the lead is stripped off and east into bullets, each of which is in the form of a cylinder ~ in. long and I~T in. in diameter, surmounted by a cone of the same diameter and ~ in. high, find how many bullets there will be.
10. The monument erected in Babylon by Queen Semiramis at her husband Ninus's tomb is said to have been one block of solid marble in the form of a square pyramid, the sides of whose base were 20 ft. and the height of the monument 150 ft. If the marble weighed 185 lb. per cu. ft., find the weight of the monument.
11. Find the volume of the largest pyramid which can be cut from a rectangular parallelepiped whose edges are 2 ~n. by 3 in. by 4 in. Discuss fully.
12. Find the least waste in cutting two conical blocks from a block of wood in the form of a right circular cylinder of radius 4 in. and altitude 7 in.
13. A mound of earth in the form of the solid shown in the figure has a rectangular base 17 yd. long and 8.62 yd. wide. Its perpendicular height is 5 yd., and the length on the top is 8.56 yd. Find the number of cubic yards of earth in the mound.
14. A vessel is in the form of an inverted regular square pyramid of altitude 9.87 in. and base edge 6.27 in. The depth of the water it contains is 6 in. (a) How much will the surface rise when I pt. of water is added? (One gal, 231 Cu. in.) (b) Find the wetted surface when the depth of the water is 9.23 in.
15. Find the volume of the largest cone having its circular base circumscribed about a face of a rectangular parallelepiped of dimensions 2 ft. by 3 ft. by 4 ft. and its vertex lying in the opposite face.
16. An ink bottle is in the form of a right circular cylinder with a large conical opening as shown in the figure. When it is filled level with the bottom ef the opening, it can just be turned upside down without any ink spilling. Prove that the depth of the cone is three-fifths the depth of the bottle.
ASSIGNMENT #6 SOLIDS OF WHICH V = (mean)Bh
A Dutch windmill in the shape of the frustum of a right circular cone is 12 meters high. The outer diameters at the bottom and the top are 16 meters and 12 meters, the inner diameters 12 meters and 10 meters. Hoe many cubic meters of stone were required to built it.
An irregular pile of coal with plane faces is 16 ft. high and covers 600 sq. ft. of level ground. Its mid section and level top are estimated to contain 400 ad 200 sq. ft., respectively. Find the cost of transporting it at P30.00 per load, if the coal truck holds 110 cu. ft. of coal.
A railway embankment across a valley has the following dimensions: width at top, 20 ft.; width at base, 45 ft.; height, 11 ft.; length at top, 1020 yd.; length at base, 960 yd. Find the volume.
A chimney in the shape of a frustum of a regular pyramid is 186.3 ft. high. Its upper bas is a square 10 ft. on a side, and its lower base is a square 16 ft. on a side. The flue is of uniform square cross section, 71/4by 71/4ft. Find the weight of the chimney if the material weighs 112.8 lb. per cu. ft.
A lighthouse consists of a tall tapering constructed of brick and has a circular cross section. The tower is surmounted by a conical top consisting of a tin roof with an overhang such that the eave is 1 ft. The over all
An irregular pile of earth is 15 ft. high and covers 500 sq. ft. Its mid-section and level top are estimated to contain 400 and 200 sq. ft., respectively. Find the cost of removing it at 60 cents per load, if the truck measures 3 by 4 by 9 ft.
Find the total area of the frustum of a regular square pyramid which is inscribed in the frustum of a cone whose upper and lower diameters are 4 ft. and 6 ft., respectively, and whose altitude is 12 ft.
The space occupied by the water in a reservoir is the frustum of a right circular cone. Each axial section of this frustum has an area of 880 sq. ft., and the diameters of the upper and the lower bases are in the ratio 6:5. If the reservoir contains 13,6000, 000 gal. , find the depth of the water.
Find the difference between the volume of the frustum of a square pyramid whose lower and upper base edges are 8 and 6 ft., respectively, and the volume of a prism of the same altitude whose base is a mid-section of the frustum.
If in the frustum of a cone the diameter of the upper base equals the slant height, find the lateral area (a) if the altitude is 4 in. and the perimeter of a vertical section through the axis is 26 in., (b) if the altitude is 7.2 in. and the perimeter is 39.2 in.
ASSIGNMENT #7 THE SPHERE
Zone
1. A wooden ball 11.15 in diameter sinks to a depth of 9.37 in. in water. Find the area of the wetted Surface
2. A candle 8 ft 6 in. from the surface of a sphere 12 ft. in diameter Find the area of the surface illuminated.
3. Find the volume of the earth's atmosphere if it extends 50 miles in height and the earth is considered a sphere of 2960- mile radius.
4. What will it cost to gild the surface of a globe whose radius is 11/2 decimeters, at an average cost of 3/5 cent per square centimeter?
5. A top consist of a spherical segment and a cone. If the altitude of the segment is 1 in., the radius of the common base 3 in., and the altitude of the cone 6 in., find: (a) the total surface of the top; (b) the volume of the top.
6. A terrestrial globe is 20 in. in diameter. A plane is passed so as to cut from the globe a circular section 16 in. in diameter. Find the area of the smaller zone determined by this section.
ASSIGNMENT #8 SOLID OF REVOLUTION
Finding Volume of Solid of Revolution
A. Find the volume of the solid formed by revolving the region bounded by the graph(s) of the equation(s) about the x-axis.
1. y = √(4 -x2) 2. y = x2
3. y = √x 4. y = √(4 - x2)
5. y = -x + 1, y = 0, x = 0 6. y = ex+1 + 1, y = x, x = 0
7. y = √x + 1, y = 0, x = 0, x = 9 8. y = ex, y = 0, x = 0, x = 1
B. Find the volume of the solid formed by revolving the region bounded by the graphs of the equation(s) about the y-axis.
1. y = x2, y = 4, 0 ≤ x ≤ 2 2. y = √(16 - x2), y = 0, 0 ≤ x ≤ 4
3. y = x2/3 4. x = -y2 + 4y
Problems
Direction: Apply Disk method or Washer Method to solve the following worded problems.
1. The line segment from (0,0) to (6,3) is revolved about the x-axis to form a cone. What is the volume of the cone?
2. Use the Disk method to verify the volume of a right circular come is 1/3Πr2h, where r is the radius of the base and h is the height.
3. Use the Disk Method to verify that the volume of a sphere of radius r is 4Πr3.
4. The upper half of the ellipse 9x2 + 16y2 = 114 is revolved about the x axis to form a prolate spheroid (shaped like a football). Find the volume of the spheroid.
5. A tank on the wings of a jet airplane is modeled by revolving the region bounded by the graph of y = 1/8x2√(2- x) and the x-axis about the x- axis, where x and y are measured in meters (see figure). Find the volume of the tank.
6. A soup bowl can be modeled as a solid of revolution formed by revolving the graph of y = √(x/2 + 1), 0 ≤ x ≤ 4 about the x-axis. Use this model, where x and y are measured in inches, to find the volume of the soap bowl.
7. A pond is to be stocked with a species of fish. The food supply in 500 cubic feet of pond water can adequately support one fish. The pond is nearly circular, is 20 feet deep at its center, and has a radius of 200 feet. The bottom of the pond can be modeled by y = 20[(0.005x)2 - 1].
a. How much water is in the pond?
b. How many fish can the pond support?
8. A regulation-size football can be modeled as a solid of revolution formed by revolving the graph of f(x) = -0.0944x2 + 3.4 and -5.5 ≤ x ≤ 5.5 about the x-axis. Use this model to find the volume of a football. (In the model, x and y are measured in inches.)