NCERT Solutions for Class 9 Maths Chapter 11 Surface Areas and Volumes पृष्ठीय क्षेत्रफल एवं आयतन
Important Questions for Class 9 Maths Chapter 11
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Exercise 11.1
Question 1:
Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curved surface area.
एक शंकु के आधार का व्यास 10.5 सेमी है और इसकी तिर्यक ऊँचाई 10 सेमी है। इसका वक्रपृष्ठीय क्षेत्रफल ज्ञात कीजिए।
Answer
Diameter, d = 10.5 cm
radius, r = \( \frac{10.5}{2} \) cm
and slant height, ℓ = 10 cm
Curved Surface Area of a cone = \( \pi r l \) = \(\frac{22}{7} \times \frac{10.5}{2} \times 10 \) = 165 cm2.
Question 2:
Find the total surface area of a cone, if its slant height is 21 m and diameter of its base is 24 m.
प्रश्न 2. एक शंकु का कुल पृष्ठीय क्षेत्रफल ज्ञात कीजिए, जिसकी तिर्यक ऊँचाई 21 मी है और आधार का व्यास 24 मी है।
Answer
Diameter, d = 24 m
radius, r =\(\frac{24}{2} \) = 12m
slant height, ℓ = 21 m
Total surface area of a cone = \( \pi r (l+r) \) = \(\frac{22}{7} \times 12 \times (21+12) \) = \(\frac{22}{7} \times 12 \times 33 \) = 1244.57 m2.
Question 3:
Curved surface area of a cone is 308 cm2 and its slant height is 14 cm.
Find (i) radius of the base and (ii) total surface area of the cone.
एक शंकु का वक्रपृष्ठीय क्षेत्रफल 308 सेमी2 है और इसकी तिर्यक ऊँचाई 14 सेमी है। ज्ञात कीजिए
(i) आधार की त्रिज्या ।
(ii) शंकु का कुल पृष्ठीय क्षेत्रफल |
Answer
Slant height, ℓ = 14 cm
(i) Curved Surface area of a cone = 308 cm2
\(\Rightarrow \) \( \pi r l \) = 308
\(\Rightarrow \) \(\frac{22}{7} \times r \times 14 \) = 308
\(\Rightarrow \) r = \(\ 308 \times \frac{7}{22} \times \frac{1}{14} \)
\(\Rightarrow \) r = 7 cm
(ii) Total surface area of the cone = \( \pi r (l+r) \) = \(\frac{22}{7} \times 7 \times (14+7) \) = \(\frac{22}{7} \times 7 \times 21 \) = 462 m2.
Question 4:
A conical tent is 10 m high and the radius of its base is 24 m. Find
(i) slant height of the tent.
(ii) cost of the canvas required to make the tent, if the cost of 1 m2 canvas is ₹ 70.
प्रश्न 4. शंकु के आकार का एक तम्बू 10 मी ऊँचा है और उसके आधार की त्रिज्या 24 मी है। ज्ञात कीजिए:
(i) तम्बू की तिर्यक ऊँचाई ।
(ii) तम्बू में लगे कैनवास की लागत, यदि 1 मी’ कैनवास की लागत ₹ 70 है।
Answer
height, h = 10 m, and radius, r = 24 m
(i) \(l^2 = r^2 + h^2 \)
\(\Rightarrow \) \(l^2 = 24^2 + 10^2 = 576 + 100 = 676 \)
\(\Rightarrow l = \sqrt{676} = 26\)
(ii) Canvas required to make the tent = Curved surface area of the cone
= \( \pi r l \) = \(\frac{22}{7} \times 24 \times 26 \) = \(\frac{13728}{7} \)m2.
cost of 1 m2 canvas = ₹ 70
Cost of the canvas = \(\frac{13728}{7} \times 70 = ₹ 13728\)
Question 5:
What length of tarpaulin 3 m wide will be required to make conical tent of height 8 m and base radius 6 m? Assume that the extra length of material that will be required for stitching margins and wastage in cutting is approximately 20 cm (Use π = 3.14).
प्रश्न 5. 8 मी ऊँचाई और आधार की त्रिज्या 6 मी वाले एक शंकु के आकार का तम्बू बनाने में 3 मी चौड़े तिरपाल की कितनी लम्बाई लगेगी? यह मानकर चलिए कि इसकी सिलाई और कटाई में 20 सेमी तिरपाल अतिरिक्त लगेगा । (𝜋 = 3.14 का प्रयोग कीजिए)
Answer
height, h = 8m; radius, r = 6 m
\(l^2 = r^2 + h^2 \)
\(\Rightarrow \) \(l^2 = 6^2 + 8^2 = 36 + 64 = 100 \)
\(\Rightarrow l = \sqrt{100} = 10\)
Curved Surface Area of conical tent = \( \pi r l \) = \(\ 3.14 \times 6 \times 10 \) = 188.4 m2.
Question 6:
The slant height and base diameter of a conical tomb are 25 m and 14 m respectively. Find the cost of white-washing its curved surface at the rate of ₹ 210 per 100 m2.
प्रश्न 6. शंकु के आधार की एक गुम्बज की तिर्यक ऊँचाई और आधार का व्यास क्रमशः 25 मी और 14 मी है। इसके वक्रपृष्ठ पर ₹210 प्रति 100 मी2 की दर से सफेदी कराने का व्यय ज्ञात कीजिए।
Answer
slant height, ℓ = 25 m; diameter,d = 14 m, radius, r = \( \frac {14}{2} = 7 m\)
Curved surface area = \( \pi r l\) = \( \frac{22}{7} \times 7 \times 25 = 550 m^2 \)
Rate of White Washing = ₹ 210 per 100 m2
Total cost of white-washing of tomb = \( \ 550 \times \frac{210}{100} \) = ₹ 1155
Question 7:
A joker’s cap is in the form of a right circular cone of base radius 7 cm and height 24 cm. Find the area of the sheet required to make 10 such caps.
प्रश्न 7. एक जोकर की टोपी एक शंकु के आकार की है, जिसके आधार की त्रिज्या 7 सेमी और ऊँचाई 24 सेमी है। इसी प्रकार की 10 टोपियाँ बनाने के लिए आवश्यक गत्ते का क्षेत्रफल ज्ञात कीजिए ।
Answer
Question 8:
A bus stop is barricaded from the remaining part of the road, by using 50 hollow cones made of recycled cardboard. Each cone has a base diameter of 40 cm and height 1 m. If the outer side of each of the cones is to be painted and the cost of painting is ₹ 12 per m2, what will be the cost of painting all these cones? (Use π = 3.14 and take √1.04 = 1.02)
प्रश्न 8. किसी बस स्टॉप को पुराने गत्ते से बने 50 खोखले शंकुओं द्वारा सड़क से अलग किया हुआ है। प्रत्येक शंकु के आधार का व्यास 40 सेमी है और ऊँचाई 1 मी है। यदि इन शंकुओं के बाहरी पृष्ठों को पेन्ट करवाना है और पेन्ट की दर ₹12 प्रति मी2 है, तो इनके पेन्ट कराने में कितनी लागत आएगी? (𝜋 = 3.14 और √1.04 = 1.02 का प्रयोग कीजिए।)
Answer
NCERT Solutions for Class 9 Maths Chapter 11 Surface Areas and Volumes पृष्ठीय क्षेत्रफल एवं आयतन
Exercise 11.2
Question 1:
Find the surface area of a sphere of radius:
(i) 10.5 cm (ii) 5.6 cm (iii) 14 cm
प्रश्न 1. निम्न त्रिज्या वाले गोले का पृष्ठीय क्षेत्रफल ज्ञात कीजिए ।
(i) 10.5 सेमी (ii) 5.6 सेमी (iii) 14 सेमी
Question 2:
Find the surface area of a sphere of diameter:
(i) 14 cm (ii) 21 cm (iii) 3.5 m
प्रश्न 2. निम्न व्यास वाले गोले का पृष्ठीय क्षेत्रफल ज्ञात कीजिए ।
(i) 14 सेमी (ii) 21 सेमी (iii) 3.5 सेमी
Question 3:
Find the total surface area of a hemisphere of radius 10 cm. (Use π = 3.14)
प्रश्न 3. 10 सेमी त्रिज्या वाले एक अर्द्धगोले का कुल पृष्ठीय क्षेत्रफल ज्ञात कीजिए । (𝜋 = 3.14 लीजिए।)
Question 4:
The radius of a spherical balloon increases from 7 cm to 14 cm as air is being pumped into it. Find the ratio of surface areas of the balloon in the two cases.
प्रश्न 4. एक गोलाकार गुब्बारे में हवा भरने पर, उसकी त्रिज्या 7 सेमी से 14 सेमी हो जाती है। इन दोनों स्थितियों में, गुब्बारे के पृष्ठीय क्षेत्रफलों का अनुपात ज्ञात कीजिए ।
Question 5:
A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of ₹ 16 per 100 cm2.
प्रश्न 5. पीतल से बने एक अर्द्धगोलाकार कटोरे का आन्तरिक व्यास 10.5 सेमी है। ₹16 प्रति 100 सेमी2 की दर से इसके आन्तरिक पृष्ठ पर कलई कराने का व्यय ज्ञात कीजिए ।
Question 6:
Find the radius of a sphere whose surface area is 154 cm2.
प्रश्न 6. उस गोले की त्रिज्या ज्ञात कीजिए जिसका पृष्ठीय क्षेत्रफल 154 सेमी2 है।
Question 7:
The diameter of the moon is approximately one fourth of the diameter of the earth. Find the ratio of their surface areas.
प्रश्न 7. चन्द्रमा का व्यास पृथ्वी के व्यास का लगभग एक-चौथाई है। इन दोनों के पृष्ठीय क्षेत्रफलों का अनुपात ज्ञात कीजिए ।
Question 8:
A hemispherical bowl is made of steel, 0.25 cm thick. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl.
प्रश्न 8. एक अर्द्धगोलाकार कटोरा 0.25 सेमी मोटी स्टील से बना है। इस कटोरे की आन्तरिक त्रिज्या 5 सेमी है ।कटोरे का बाहरी वक्रपृष्ठीय क्षेत्रफल ज्ञात कीजिए ।
Question 9:
A right circular cylinder just encloses a sphere of radius r (see Fig. 11.10). Find
(i) surface area of the sphere,
(ii) curved surface area of the cylinder,
(iii) ratio of the areas obtained in (i) and (ii).
प्रश्न 9. एक लम्बवृत्तीय बेलन की त्रिज्या वाले एक गोले को पूर्णतया घेरे हुए है। (देखिए आकृति 11.10) ज्ञात कीजिए
(i) गोले का पृष्ठीय क्षेत्रफल।
(ii) बेलन का वक्रपृष्ठीय क्षेत्रफल।
(iii) ऊपर (i) और (ii) में प्राप्त क्षेत्रफलों का अनुपात।
NCERT Solutions for Class 9 Maths Chapter 11 Surface Areas and Volumes पृष्ठीय क्षेत्रफल एवं आयतन
Exercise 11.3
Question 1:
Find the volume of the right circular cone with
(i) radius 6 cm, height 7 cm (ii) radius 3.5 cm, height 12 cm
Question 2:
Find the capacity in litres of a conical vessel with
(i) radius 7 cm, slant height 25 cm (ii) height 12 cm, slant height 13 cm
Question 3:
The height of a cone is 15 cm. If its volume is 1570 cm3, find the radius of the base. (Use π = 3.14)
Question 4:
If the volume of a right circular cone of height 9 cm is 48 π cm3, find the diameter of its base.
Question 5:
A conical pit of top diameter 3.5 m is 12 m deep. What is its capacity in kilolitres?
Question 6:
The volume of a right circular cone is 9856 cm3. If the diameter of the base is 28 cm, find
(i) height of the cone (ii) slant height of the cone (iii) curved surface area of the cone
Question 7:
A right triangle ABC with sides 5 cm, 12 cm and 13 cm is revolved about the side 12 cm. Find the volume of the solid so obtained.
Question 8:
If the triangle ABC in the Question 7 above is revolved about the side 5 cm, then find the volume of the solid so obtained. Find also the ratio of the volumes of the two solids obtained in Questions 7 and 8.
Question 9:
A heap of wheat is in the form of a cone whose diameter is 10.5 m and height is 3 m. Find its volume. The heap is to be covered by canvas to protect it from rain. Find the area of the canvas required.
NCERT Solutions for Class 9 Maths Chapter 11 Surface Areas and Volumes पृष्ठीय क्षेत्रफल एवं आयतन
Exercise 11.4
Question 1:
Find the volume of a sphere whose radius is
(i) 7 cm (ii) 0.63 m
Question 2:
Find the amount of water displaced by a solid spherical ball of diameter
(i) 28 cm (ii) 0.21 m
Question 3:
The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g per cm3?
Question 4:
The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
Question 5:
How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold?
Question 6:
A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.
Question 7:
Find the volume of a sphere whose surface area is 154 cm2.
Question 8:
A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of ₹ 4989.60. If the cost of white-washing is 20 per square metre, find the (i) inside surface area of the dome, (ii) volume of the air inside the dome.
Question 9:
Twenty seven solid iron spheres, each of radius r and surface area S are melted to form a sphere with surface area S′. Find the
(i) radius r′ of the new sphere, (ii) ratio of S and S′.
Question 10:
A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm3) is needed to fill this capsule?
NCERT Solutions for Class 9 Maths Chapter 11 Surface Areas and Volumes पृष्ठीय क्षेत्रफल एवं आयतन
