Mathematics--mensuration－thhzpvx的部落格

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Mathematics--mensuration

發問:

1. 圖片參考：http://imgcld.yimg.com/8/n/HA05298198/o/701102030025613873376680.jpg Find the curved surface area of the right circular cone as shown in the figure 2. A metallic sphere with a radius of R is melted. How many identical right circular cones with radii and height R/8 can be recast?3.... 顯示更多 1. 圖片參考：http://imgcld.yimg.com/8/n/HA05298198/o/701102030025613873376680.jpg Find the curved surface area of the right circular cone as shown in the figure 2. A metallic sphere with a radius of R is melted. How many identical right circular cones with radii and height R/8 can be recast? 3. 圖片參考：http://imgcld.yimg.com/8/n/HA05298198/o/701102030025613873376691.jpg The figure shows a solid which is composed of a right circular cone and a hemisphere with the same base. If the ratio of the volume of the cone to that of the hemisphere is 5:3, find the ratio of r to h. 4. 圖片參考：http://imgcld.yimg.com/8/n/HA05298198/o/701102030025613873376692.jpg In the figure, the height of the pyramid VPQRS is 8h. ABCD adn PQRS are squares with sides a and 8a respectively. Find the ratio of the volume of the frustum ABCDSPQR to that of the pyramid VPQRS. 5. If the surface area of a sphere decreases by 51%, find the percentage change in its volume. 6. The base radius of a right circular cone is 18cm. If the total surface area of the cone is 675pi sq.cm, find the slant height of the cone. 7. If a solid sphere is fitted exactly into a cubical container with sides 7 cm, find the volume of the empty space in the container after the sphere is put inside. (Correct your answer to 2 decimal places.) 8. 圖片參考：http://imgcld.yimg.com/8/n/HA05298198/o/701102030025613873376703.jpg Find the volume of the right rectangular pyramid as shown in the figure. (Correct your answer to 3 sig.fig.) Just try your best to answer! You don't have to answer all the questions if you don't know some of them! I need steps for all of them. Try Q1-4 first as I don't know how to do them. I just want to check my answer for Q5-8. THX!!!

最佳解答:

Question 1: Cos 60o = r/d where r = radius and s = slant height s = r/cos Cos 60o = 22 cm/0.5 = 22 cm curved surface area = (pi) x (r) x (s) curved surface area = (pi) x (11 cm) x (22 cm) = 243 pi sq. cm = 760.3 cm^2 Question 2: Volume of sphere = 4/3 (pi) (R^3) Volume of cone = 1/3 (pi) (r^2)(h) = 1/3 (pi) ((R/8)^2)(R/8) = 1/3 (pi) ((R/8)^3) Number of cones = Volume of sphere/ Volume of cone Number of cones = 4/3 (pi) (R^3) / 1/3 (pi) ((R/8)^3) Number of cones = 4/3 (pi) (R^3) / 1/3 (pi) (0.125^3)(R^3) = 4/(0.125^3) = 2048 Question 3: Volume of cone = 1/3 (pi)(r^2)(h) Volume of hemisphere = ? volume of sphere = ? (4/3) (pi)(r^3) Volume of cone / volume of hemisphere = 5/3 1/3 (pi)(r^2)(h)/ ? (4/3) (pi)(r^3)= 5/3 1/3 (pi)(r^2)(h)/ 2/3 (pi)(r^3)= 5/3 h/ 2r = 5/3 2r/h = 3/5 r /h = 3/10 The ratio of r to h 3:10 Question 4: Volume of frustum = Volume of pyramid VPQRS – Volume of pyramid VABCD Volume of a square base pyramid = 1/3(s^2)(h) Volume of pyramid VPQRS = 1/3[(8a)^2](8h) Ratio of height to half side of square base is a constant Height of pyramid VABCD/ a = 8h/8a Height of pyramid VABCD = h Volume of pyramid VABCD = 1/3[(a)^2](h) Volume of frustum = 1/3[(8a)^2](8h) – 1/3[(a)^2](h) Volume of frustum = 1/3[512(a)^2](h) – 1/3[(a)^2](h) Volume of frustum = 1/3[511(a)^2](h) The ratio of Volume of frustum to Volume of pyramid VPQRS = 1/3[511(a)^2](h)/ 1/3[(8a)^2](8h) = 1/3[511(a)^2](h)/ 1/3[512(a)^2](h) = 511/512 The ratio of Volume of frustum to Volume of pyramid VPQRS 511 : 512 Question 5: Surface of a sphere = 4 (pi) (r^2) 0.49 = 4 (pi) ((kr)^2)/ 4 (pi) (r^2) 0.49 = k^2 k = 0.7 radius decreases by 30% Volume of a sphere = 4/3 (pi) (r^3) Change in volume of a sphere = 4/3 (pi) [(0.7r)^3]/ 4/3 (pi) (r^3) = 0.34.3 Volume decreases by (1 – 0.343)100 = 65.7% Volume decreases by 65.7% Question 6: A = pir^2 + pi r s 675 pi cm^2 = pi(18^2) +pi 18s 675 = 324 + 18s slant height, s = (675 – 324)/18 = 19.5 cm Run out of space... Let other reader do the rest. 2011-02-04 11:26:51 補充： Ques 7: V = 7^3 - 4/3 (pi)(3.5)^3 = 163.41 cm^3 Ques 8: V = 1/3(10)(6)(15.198) = 304 cm^3

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