Exercise 9.3
Exercise 9.3 Solutions
Volume and capacity scaling in similar solids. Proportional relationships showing Volume Ratio = (Side Ratio)³.
Question 1
Spheres Ratio
The radii of two spheres are in the ratio 3 : 4. What is the ratio of their volumes?
Solution Step-by-Step
Let r1 and r2 be the radii, and V1 and V2 be the volumes of the two spheres.
Since all spheres are similar, the ratio of their volumes is the cube of the ratio of their radii:
Since all spheres are similar, the ratio of their volumes is the cube of the ratio of their radii:
V1V2
=
(r1r2)3
Given r1 : r2 = 3 : 4, substitute these values:
V1V2
=
(34)3 = 2764
∴ The ratio of their volumes is 27:64.
Question 2
Tetrahedrons
Two regular tetrahedrons have volumes in the ratio 8 : 27. What is the ratio of their sides?
Solution Step-by-Step
Let l1 and l2 be the side lengths, and V1 and V2 be the volumes of the tetrahedrons.
Since regular tetrahedrons are similar, their volume ratio is the cube of their side length ratio:
Since regular tetrahedrons are similar, their volume ratio is the cube of their side length ratio:
V1V2
=
(l1l2)3
Substitute V1 : V2 = 8 : 27:
(l1l2)3
=
827
Take the cube root of both sides:
l1l2
=
3√(827) = 23
∴ The ratio of their sides is 2:3.
Question 3
Cones Scaling
Two right cones have volumes in the ratio 64 : 125. What is the ratio of:
(a) Their heights?
(b) Their base radii?
(c) Their surface areas?
(a) Their heights?
(b) Their base radii?
(c) Their surface areas?
Solution Step-by-Step
Since all right cones are similar, their volume ratio corresponds to the cube of their linear dimensions:
V1V2
=
(h1h2)3 = (r1r2)3 = 64125
(a) Ratio of their heights:
Taking the cube root of the volume ratio:
Taking the cube root of the volume ratio:
h1h2
=
3√(64125) = 45
∴ Height ratio = 4:5.
(b) Ratio of their base radii:
Since radii are also linear dimensions, their ratio equals the height ratio:
Since radii are also linear dimensions, their ratio equals the height ratio:
r1r2
=
45
∴ Base radii ratio = 4:5.
(c) Ratio of their surface areas:
The surface area ratio is the square of the linear ratio:
The surface area ratio is the square of the linear ratio:
A1A2
=
(r1r2)2 = (45)2 = 1625
∴ Surface area ratio = 16:25.
Question 4
Missing Volumes
Find the missing volume in the following similar figures:
(i) Pyramids: l1 = 12 cm, l2 = 16 cm, V2 = 1536 cm³, V1 = ?
(ii) Cylinders: h1 = 2.5 m, h2 = 8.75 m, V2 = 171.5 m³, V1 = ?
(iii) Cuboids: A1 = 392 cm², A2 = 162 cm², V2 = 729 cm³, V1 = ?
(iv) Spheres: V1 = 64 cm³, V2 = 216 cm³, r1 = 8 cm, r2 = ? (Wait, if V1=64 and V2=216, radii ratio is 4/6 = 2/3. If r1=8 cm, then r2 = 12 cm).
(i) Pyramids: l1 = 12 cm, l2 = 16 cm, V2 = 1536 cm³, V1 = ?
(ii) Cylinders: h1 = 2.5 m, h2 = 8.75 m, V2 = 171.5 m³, V1 = ?
(iii) Cuboids: A1 = 392 cm², A2 = 162 cm², V2 = 729 cm³, V1 = ?
(iv) Spheres: V1 = 64 cm³, V2 = 216 cm³, r1 = 8 cm, r2 = ? (Wait, if V1=64 and V2=216, radii ratio is 4/6 = 2/3. If r1=8 cm, then r2 = 12 cm).
Solution Step-by-Step
(i) Pyramids volume V1
By volume scaling rule:
V1V2
=
(l1l2)3
V11536
=
(1216)3 = (34)3 = 2764
V1
=
1536 × 2764 = 24 × 27 = 648 cm³
(ii) Cylinders volume V1
By volume scaling rule:
V1V2
=
(h1h2)3
V1171.5
=
(2.58.75)3 = (13.5)3 = (27)3 = 8343
V1
=
171.5 × 8343 = 0.5 × 8 = 4 m³
(iii) Cuboids volume V1
Given surface areas A1 = 392 cm² and A2 = 162 cm². Find side ratio:
l1l2
=
√(A1A2) = √(392162) = √(19681) = 149
Using the side ratio to find volume V1 (given V2 = 729 cm³):
V1V2
=
(l1l2)3
V1729
=
(149)3 = 2744729
V1
=
729 × 2744729 = 2744 cm³
(iv) Spheres radius r2
Given V1 = 64 cm³, V2 = 216 cm³, and r1 = 8 cm. Using volume ratio:
(r1r2)3
=
V1V2 = 64216
Take the cube root:
r1r2
=
46 = 23
8r2
=
23 ⇒ 2r2 = 24 ⇒ r2 = 12 cm
Question 5
Conical Cans
The ratio of corresponding lengths of two similar conical cans is 3:2.
(i) If the larger conical can has a surface area of 96 m², find the surface area of the smaller can.
(ii) If the smaller conical can has a volume of 240 m³, find the volume of the larger can.
(i) If the larger conical can has a surface area of 96 m², find the surface area of the smaller can.
(ii) If the smaller conical can has a volume of 240 m³, find the volume of the larger can.
Solution Step-by-Step
Given length ratio l1 : l2 = 3 : 2 (where 1 is larger, 2 is smaller).
This means scale factor k = 3/2 = 1.5.
This means scale factor k = 3/2 = 1.5.
(i) Find smaller Surface Area A2
Using area ratio:
A1A2
=
(l1l2)2
96A2
=
(32)2 = 94
9A2
=
384 ⇒ A2 = 3849 ≈ 42.67 m²
(ii) Find larger Volume V1
Using volume ratio:
V1V2
=
(l1l2)3
V1240
=
(32)3 = 278
V1
=
240 × 278 = 30 × 27 = 810 m³
Question 6
Water Tanks
The ratio of the heights of two similar cylindrical water tanks is 5:3.
(i) If the surface area of the larger tank is 250 square metres, find the surface area of the smaller tank.
(ii) If the volume of the smaller tank is 270 cubic metres, find the volume of the larger tank.
(i) If the surface area of the larger tank is 250 square metres, find the surface area of the smaller tank.
(ii) If the volume of the smaller tank is 270 cubic metres, find the volume of the larger tank.
Solution Step-by-Step
Given height ratio h1 : h2 = 5 : 3 (where 1 is larger, 2 is smaller).
(i) Find smaller Surface Area A2
Using area ratio:
A1A2
=
(h1h2)2
250A2
=
(53)2 = 259
25A2
=
250 × 9 = 2250 ⇒ A2 = 90 m²
(ii) Find larger Volume V1
Using volume ratio:
V1V2
=
(h1h2)3
V1270
=
(53)3 = 12527
V1
=
270 × 12527 = 10 × 125 = 1250 m³
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