1 yd 2 1 yd 2 3 ft 3 ft = ))))) x ))))) x ))))) = 9 sq. ft yd yd. 60 yd 2 60 yd 2 3 ft 3 ft = ))))) x ))))) x ))))) = 540 ft 2 yd yd.

Transcription

1 MATH 310 TEST Measure April ANSWERS (6)1a. 1 yard (or 3 feet) Draw a sket ch w hich illust rat es the relat ionship betw een square yards and square feet. 1b. Show the dimensional analysis f or t he conversion of 60 square yards t o square feet. Complete the conversion. In t he sket ch w e can see t hat 1 square y ard is 9 square f eet. 1 y ard (or 3 feet) Dimensional analysis says t his: 1 yd 2 1 yd 2 3 ft 3 ft = ))))) x ))))) x ))))) = 9 sq. ft yd yd 60 yd 2 60 yd 2 3 ft 3 ft = ))))) x ))))) x ))))) = 540 ft 2 yd yd (12) 2. Convert each of the following, showing your work. a. 5.2 km = cm cm 5.2 km 1000m 100cm 5.2 km = ))))) x )))))) x ))))) = cm km m b. 5.3 m 3 = cm 3 cm m 3 100cm 100cm 100cm 5.3 m 3 = ))))) x )))))) x ))))) x ))))) = cm 3 m m m c ml water (at 4 C) = kg. 1 kg 2550 ml = 2550 cc = 2550 g = 2550 g x ))))) = kg 1000 g (5) 3. How many liters of w at er are needed to f ill a t ank 1 met er w ide, 1 met er high and 1 met er deep? Show the dimensional analysis that leads to your answer. 1 m 3 10dm 10dm 10dm 1 m 3 = ))))) x )))))) x ))))) x ))))) = 1000 dm 3 = 1000 L m m m OR 1000 dm 3 = 1000 dm cm x )))))) = cm 1 dm 3 = g If w e know 1 dm 3 = 1 L = ml If w e know: = 1000 L 1 cc = 1 g = 1 ml (of WATER at 4 C)

2 (10) 4. Find t he volume of the pyramid show n at right, given that the base of the pyramid has area 84 0 m 2 and the height (h ) is 60m. Show your work. Volume of Cone or Pyramid is 1/3 t he Volume of t he corresponding cylinder. Thus Volume = (1/3) (Area ofbase) (height) = (1/3) ( 840 m 2 ) ( 60 m ) = 16,800 m 3 (10) 5.The Flatiron Building (oldest remaining skyscraper in NYC) is shaped like a triangle, as seen from above. (Well, OK, a slightly rounded triangle.) The dimensions are as shown. The height of the fairly flat-topped building is 285 ft (from ground level). Using t his inf ormation, f ind the volume (as near as possible) of t he Flatiron Building (that is abov e ground level). 173ft 190 ft The Flatiron Building, then, is a triangular-based prism. Thus its volume is: Volume = (Area of base) (height) = (87 ft x 173 ft /2) (285 ft) = (½) (87)(173) (285) ft 3 87 ft = ft picture of Flatiron Building from... (See a good modern pict ure of the Flatiron Building at: htt p:// le.pl/budow le/grafika/ameryka_polnocna/flatiron_building.jpg ) (5) 6. Find the ent ire surf ace area of the solid object illustrated below. Assume all angles betw een connected segments are right angles. 20m 20m 22m The roof, as seen from straight above, is a 50 m x 50m square Surface Area = SA(roof + base) + SA(Big Laterals) + SA (small laterals) 50 m = 2 (50m) m 60m m 22m 60 m = m m m 2 50 m = m 2 The roof as seen f rom above:

3 (4,3) 6 (4, 3) (-4, -3) 8 In all the above, the area shown here.. is one square unit... (5) 7. Estimate the area of t he figure in #7. # of whole square units w ithin region + (½) #of partial units w ithin region 14 unit 2 + (½) (18 unit 2 ) 23 unit 2 (5) 8. Find the area enclosed by the figure in #8. (Curve turns at point s: (1,1) & (6,7) & (1,7) & (2,5) ) 5 6 u 2 (½) 6 1 u 2 (½) 5 6 u 2 30 u 2 3 u 2 15 u 2 12 u 2 (8) 9. Find the PERIMETER of t he triangle in figure #9 above Perimeter = Distance from ( 4, 3) t o (4,-3) + Distance to (4,3) + Length of Hypot enuse = 4 ( 4) + 3 ( 3) + square root ( ) = = 24 (units) = c = c = c 2 (8) 10. Find the AREA of the figure at right, given all arcs are semicircular. First w e look at the region, and det ermine it is just a semic ircular region w it h a semicircular bite taken out! Thus t he area cont ained is: 7mm 10mm 7mm Area of t he larger semicircular region area of t he smaller semicircular region...w here R is t he radius of the = ½ of R 2 ½ o f r 2 outer semicircle, and r is the radius of the inner semicircle. = (½ ) (12mm) 2 (½ ) (5mm) 2 = (½) 144 mm 2 (½ ) 25 mm 2 = (½) 119 mm 2

4 (5) 11. Find the area of a 60 sector of a circle with radius 3 m. The area of a 60 sect or of a circle is just 60 /3 60, or 1/ 6 of the area of the ent ire circle. (1/6) of R 2 = (1/6) of R 2 = (1/6) of (3m) 2 = (3m) 2 /6 = (3/2) m 2 = 3 /2 m 2 OR 1.5 m 2 (4) 12. Write an expression for the Volume of a sphere of radius r: (4/3) r 3 Write an expression for the Surface Area of a sphere of radius r: 4 r 2 Volume MUST be 3-dimensional. Area MUST be 2-dimensional. (6) 13. Find the area of the shaded region (w ithin t he square, out side the circle). 2m AREA of SQUARE AREA of CIRCLE 2m Side 2 R 2 (2m) 2 (1m) 2 4 m 2 m 2 ( 4 ) m m 2

5 Addit ional M easure Questions (1 6 pts) Name: 1. A circular cylinder was designed to hold 500mL. If the diameter of a cylinder is doubled, and the height is tripled, what is the new capacity? V = 500 ml V = r 2 h If t he diameter is doubled, then t he radius is doubled; so t he new radius = 2r If t he height is t ripled, then the new height is 3h. NV = (2r) 2 (3h) NV = 4r 2 3h = 12 r 2 h = 12 (original volume) So the volume is multiplied by 12. (Increased twelve-fold) 2. A very tall (Oops, make that very SHORT) right circular cone has dimensions measured as shown: Using this information, determine the volume of the cone, showing any intermediate steps and computations needed. 3. Find t he complete surf ace area of the cone. 2. Circumf erence of base = 49 cm tells us the radius of the base is 24.5 (ugh!) (Since 2 r = 49, r = 49 /(2 ) ) 25cm Then the height of the cone is the vertical arm of the right triangle (...t he SHORT right triangle, contrary t o the illust ration!): 25 cm H = 25 2 H H = 625 C = 49 cm H 2 = cm H = square root of So now w e know the height of the cone is (square root of 24.75) cm cm and the radius of the cone is 24.5 cm and the slant height of the cone is 25cm. Volume = (1/ 3) of Area of Base Height of cone = (1/3) r 2 H (1/3) (24.5cm) 2 ( cm) (1/3) (24.5cm) 2 ( cm) cm 3 3. Surf ace Area of cone = Area of Base + area of the lateral surface = r 2 + (½) Perimeter of base Slant height = (24.5 cm) 2 + (½) 49 cm 25 cm = cm cm 2 = cm cm 2 (I think this was a short cone that had a tall self-image!)