f.3 math 好急!!!!!! part 6 20點

1. Two tanks, similar in shape, are respectively 20cm and 30cm high. If the smaller tank holds 24m3 of water, find the capacity of the large one.
The linear ratio of the tanks = 30/20 = 3/2
The volume ratio of the tanks = (3/2)3 = 27/8
The capacity of the large tank = (27/8)(24m3) = 81m3
2. A metal sphere of radius 3.5 cm is melted and recast to form two identical smaller spheres. Find, correct to 3 sig. fig.
a)the radius of the smaller sphere,
The volume ratio = 1/2
The linear ratio = 3√(1/2) = 0.7937
Radius of smaller sphere = (0.7939)(3.5cm) = 2.78cm
b)the surface area of the smaller sphere = 4πr2
= 4π(2.78)2
= 97.1cm2
c) the percentage change in the total surface area.
The area ratio = 0.79372 = 0.630
Since there are 2 smaller spheres, the total area ratio is 0.630 x 2 = 1.260
There is an increase of surface area by (1.26 - 1)/(1) * 100% = 26%
3. The heights of two similar cylindrical containers are 100cm and 40cm.
a) Find the ratio of the base radius of the smaller container to that of the large container.
The linear ratio is 40/100 = 2/5
This is the same for the ratio of the radius
The ratio of the base radius of the smaller container to that of the large container is 2:5
b) If the surface area of the larger container is 12.5 m2 , what is the surface area of the smaller one?
The area ratio = (2/5)2 = 4/25
Surface area for the smaller one = (4/25)(12.5m2) = 2m2

(1) let height of smaller tank = h1 and height of large tank = h2
volume ratio smaller to larger tank = (h1)^3 / (h2)^3
= 20^3 / 30^3 = 8/27
volume of smaller tank / volume of large tank = volume ratio = 8 /27
24 / volume of large tank = 8/27
volume of large tank = 81 cu m.
(2) volume of metal sphere = (4/3)πr^3 = (4/3)π(3.5^3) = 179.59 cu. cm
volume of smaller sphere = volume of large sphere / 2 = 179.59 / 2
= 89.797 cu. cm.
(a) radius of smaller sphere = [3(89.797) / 4π]^(1/3) = 2.78 cm
(b) surface area of smaller sphere = 4πr^2 = 4π(2.78)^2 = 96.975sq. cm
(c) surface area of original sphere = 4πr^2 = 4π(3.5)^2 = 153.938 sq. cm
percentage change = (original surface area - new orginal surface areas) / original surface area x 100% = [153.938 - 2(96.975)]/153.938 x 100% = 26.0%
(3) height ratio of smaller to larger cylinder = base radius ratio of smaller to larger cylinder = 40/100 = 2 / 5
(b) surface area ratio = square of height ratio = (2/5)^2 = 4/25
surface area of tge smaller cylinder = surface area of the larger cylinder x surface area ratio
= 12.5 x 4/ 25 = 2 sq. m