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Surface Area and Volumes

Surface Area and Volumes relates to CBSE/Class 9/Mathematics

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Surface Area and Volumes Questions

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Answered on 18/04/2024 Learn CBSE/Class 9/Mathematics/Surface Area and Volumes/Sphere

Nazia Khanum

Calculating the Longest Pole Length for a Room Introduction: In this scenario, we aim to determine the longest possible length of a pole that can fit inside a room with given dimensions. Given Dimensions: Length (l) = 10 cm Breadth (b) = 10 cm Height (h) = 5 cm Approach: To find the longest pole that... read more

Calculating the Longest Pole Length for a Room

Introduction: In this scenario, we aim to determine the longest possible length of a pole that can fit inside a room with given dimensions.

Given Dimensions:

  • Length (l) = 10 cm
  • Breadth (b) = 10 cm
  • Height (h) = 5 cm

Approach: To find the longest pole that can fit inside the room without sticking out, we need to consider the diagonal length of the room.

Calculations:

  1. Diagonal Length of the Room (d):

    • We'll use the Pythagorean theorem to calculate the diagonal length (d) of the room.
    • Formula: d=l2+b2+h2d=l2+b2+h2

 

  • Substituting the given values: d=102+102+52d=102+102+52
  • d=100+100+25=225=15d=100+100+25

=225

 

    • =15 cm
  1. Longest Pole Length:

    • The longest pole that can fit inside the room without protruding is equal to the diagonal length of the room.
    • Therefore, the longest pole length = 15 cm.

Conclusion: Hence, the longest pole that can be put in a room with dimensions l=10l=10 cm, b=10b=10 cm, and h=5h=5 cm is 15 cm.

 
 
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Answered on 18/04/2024 Learn CBSE/Class 9/Mathematics/Surface Area and Volumes/Sphere

Nazia Khanum

Problem: Finding the Volume of a Sphere Given Data: Total surface area of the sphere: 154 cm² Solution Steps: Find the Radius of the Sphere: The formula for the surface area of a sphere is 4πr24πr2, where rr is the radius. Given 4πr2=1544πr2=154 cm², we solve for rr. Calculate... read more

Problem: Finding the Volume of a Sphere

Given Data:

  • Total surface area of the sphere: 154 cm²

Solution Steps:

  1. Find the Radius of the Sphere:

    • The formula for the surface area of a sphere is 4πr24πr2, where rr is the radius.
    • Given 4πr2=1544πr2=154 cm², we solve for rr.
  2. Calculate the Volume of the Sphere:

    • Once we find the radius, we can use the formula for the volume of a sphere, 43πr334πr3, to find the volume.

Detailed Solution:

  1. Finding the Radius of the Sphere:

    • Surface area formula: 4πr2=1544πr2=154 cm²
    • Solving for rr:
      • r2=1544πr2=4π154
      • r2=1544×3.14r2=4×3.14154 (Using ππ approximately as 3.14)
      • r2≈15412.56r212.56154
      • r2≈12.27r2≈12.27
      • r≈12.27r12.27
      • r≈3.5r≈3.5 cm (approximated to one decimal place)
  1. Calculate the Volume of the Sphere:

    • Using the radius r=3.5r=3.5 cm in the volume formula:
      • Volume V=43πr3V=34πr3
      • V=43×3.14×(3.5)3V=34×3.14×(3.5)3
      • V≈43×3.14×42.875V34×3.14×42.875
      • V≈43×3.14×42.875V34×3.14×42.875
      • V≈43×3.14×42.875V34×3.14×42.875
      • V≈179.59V≈179.59 cm³

Conclusion:

  • The total volume of the sphere is approximately 179.59179.59 cubic centimeters.
 
 
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Answered on 18/04/2024 Learn CBSE/Class 9/Mathematics/Surface Area and Volumes/Sphere

Nazia Khanum

Introduction In this explanation, I'll guide you through the process of finding the volume of a sphere when its radius is given as 3r. Formula for the Volume of a Sphere The formula for calculating the volume of a sphere is: V=43πr3V=34πr3 Where: VV = Volume of the sphere ππ = Pi (approximately... read more

Introduction

In this explanation, I'll guide you through the process of finding the volume of a sphere when its radius is given as 3r.

Formula for the Volume of a Sphere

The formula for calculating the volume of a sphere is:

V=43πr3V=34πr3

Where:

  • VV = Volume of the sphere
  • ππ = Pi (approximately 3.14159)
  • rr = Radius of the sphere

Given Information

Given that the radius of the sphere is 3r, we'll substitute r=3rr=3r into the formula.

Calculation

Substituting r=3rr=3r into the formula, we get:

V=43π(3r)3V=34π(3r)3

V=43π27r3V=34π27r3

V=36πr3V=36πr3

Conclusion

The volume of the sphere when the radius is 3r is 36πr336πr3.


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Answered on 18/04/2024 Learn CBSE/Class 9/Mathematics/Surface Area and Volumes/Sphere

Nazia Khanum

Finding the Volume of a Cube Understanding the Problem To find the volume of a cube, we first need to understand the given information: Total surface area of the cube is 216 cm². Solution Steps Determine the Side Length of the Cube Since a cube has six equal square faces, the total surface... read more

Finding the Volume of a Cube

Understanding the Problem

To find the volume of a cube, we first need to understand the given information:

  • Total surface area of the cube is 216 cm².

Solution Steps

  1. Determine the Side Length of the Cube

    • Since a cube has six equal square faces, the total surface area can be expressed as 6s26s2, where ss is the side length of the cube.
    • So, 6s2=216 cm26s2=216cm2.
    • Solving for ss, we get s=2166s=6216

 

    • .
  • Calculate the Volume of the Cube

    • Once we have the side length, we can calculate the volume of the cube using the formula V=s3V=s3.
    • Substituting the value of ss, we get V=(2166)3V=(6216

 

    • )3.
    • Simplify to find the volume.

Detailed Calculation

  1. Determine the Side Length of the Cube

    • Given: Total surface area (AA) = 216 cm².
    • Formula for total surface area: A=6s2A=6s2.
    • Substitute the given value: 216=6s2216=6s2.
    • Solve for ss: s=2166s=6216

 

  • .
  • Calculate: s=36s=36

 

    • .
    • Thus, s=6 cms=6cm.
  1. Calculate the Volume of the Cube

    • Formula for volume: V=s3V=s3.
    • Substitute the value of ss: V=63V=63.
    • Calculate: V=216 cm3V=216cm3.

Final Answer

  • The volume of the cube is 216 cm3216cm3.
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Answered on 18/04/2024 Learn CBSE/Class 9/Mathematics/Surface Area and Volumes/Sphere

Nazia Khanum

Finding the Base Area of a Right Circular Cylinder Understanding the Problem To find the base area of a right circular cylinder, we need to utilize the given information about its circumference. Given Information: Circumference of the base: 110 cm Solution Steps: Determine the Radius: The circumference... read more

Finding the Base Area of a Right Circular Cylinder

Understanding the Problem To find the base area of a right circular cylinder, we need to utilize the given information about its circumference.

Given Information:

  • Circumference of the base: 110 cm

Solution Steps:

  1. Determine the Radius:
    • The circumference of a circle CC is given by the formula: C=2πrC=2πr.
    • Given C=110C=110 cm, we can rearrange the formula to solve for the radius rr: 110=2πr110=2πr Solving for rr: r=1102πr=2π110
  2. Calculate the Base Area:
    • The formula for the area AA of a circle is: A=πr2A=πr2.
    • Plug in the value of rr obtained from step 1 into the formula: A=π(1102π)2A=π(2π110)2 Simplify: A=π(11024π2)A=π(4π21102) A=11024πA=4π1102
  3. Final Calculation:
    • Calculate the value of AA: A=121004πA=4π12100 A≈3035.5πA≈π3035.5 A≈964.88A≈964.88 sq. cm (rounded to two decimal places)

Conclusion:

  • The base area of the right circular cylinder is approximately 964.88964.88 square centimeters.
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