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Learn/ 9th Mathematics/ Unit 9 /Numericals

Unit 9: Similar Figures — Exercise 9 3

9th Class Mathematics · Unit 9: Similar Figures

9.3.1.The radii of two spheres are in the ratio 3 : 4. What is the ratio of their volumes?
Formula
V1V2=left(r1r2right)3=left(frac34right)3=2764
Result
V1:V2=27:64
9.3.2.Two regular tetrahedrons have volumes in the ratio 8 : 27. What is the ratio of their sides?
Formula
V1V2=left(l1l2right)3 ⇒ 827=left(l1l2right)3
Result
l1l2=sqrt[3]{827=frac23 ⇒ l1:l2=2:3
9.3.3.Two right cones have volumes in the ratio 64 : 125. What is the ratio of: (a) their heights (b) their base areas?
Working
(a) V1V2=left(h1h2right)3 ⇒ 64125=left(h1h2right)3
Result
(a) h1h2=sqrt[3]{64125=frac45 ⇒ h1:h2=4:5
Working
(b) A1A2=left(h1h2right)2=left(frac45right)2=1625
Result
(b) A1:A2=16:25
9.3.4.Find the missing value in the following similar solids. (i) A pyramid with slant length 12 cm, V1 = ?; a similar pyramid with slant length 16 cm, V2 = 1536 cm³. (ii) A cylinder with height h1 = 2.5, V1 = ?; a similar cylinder with height h2 = 8.75, V2 = 171.5 cm³. (iii) A cuboid with surface area A1 = 392 cm², V1 = ?; a similar cuboid with surface area A2 = 162 cm², V2 = 729 cm³. (iv) A sphere with V1 = 64 cm³, r1 = ?; a similar sphere with volume 216 cm³ (labelled V1 in the figure) and r2 = 12 cm.
Working
(i) V11536=left(1216right)3=left(frac34right)3=2764
Result
(i) V1=1536 × 2764=648 cm3
Working
(ii) V1171.5=left(2.58.75right)3=15.625669.921
Result
(ii) V1=15.625 × 171.5669.921=4 cm3
Formula
(iii) A1A2=k2 ⇒ 392162=k2 ⇒ k2=2.420 ⇒ k=1.56
(iii) V1729=k3=(1.56)3 ⇒ V1=(729)(3.80)
(1.56)^3 = 3.796, and 729×3.796 ≈ 2767, not 2744; the printed intermediate multiplication does not actually reproduce the final answer. Using the exact ratio k = 392/162 = 196/81, so k = 14/9, gives V1 = 729×(14/9)^3 = 14^3 = 2744 cm³ exactly — the final published answer is correct even though this intermediate line is not.
Result
(iii) V1 ≈ 2744 cm3
Working
(iv) 64216=left(r112right)3 ⇒ sqrt[3]{64216=r112 ⇒ 46=r112
Result
(iv) r1=4 × 126=8 cm
9.3.5.The ratio of the corresponding lengths of two similar canonical cans is 3 : 2. (i) The larger canonical can have surface area of 96 m². Find the surface area of the smaller canonical can. (ii) The smaller canonical can have a volume of 240 m³. Find the volume of larger canonical can.
9.3.6.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.
Working
(a) A1A2=left(h1h2right)2 ⇒ 250A2=left(frac53right)2=259
Result
(a) A2=250 × 925=90 m2
Working
(b) V1V2=left(h1h2right)3 ⇒ V1270=left(frac53right)3=12527
Result
(b) V1=270 × 12527=1250 m3