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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

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≈15412.56r2≈12.56154
     • r2≈12.27r2≈12.27
     • r≈12.27r≈12.27

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.

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

2. • )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

1. • .
   • Thus, s=6 cms=6cm.

2. 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.