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1. • =15 cm

2. 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.

1. • • r≈3.5r≈3.5 cm (approximated to one decimal place)

2. 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≈43×3.14×42.875V≈34×3.14×42.875
     • V≈43×3.14×42.875V≈34×3.14×42.875
     • V≈43×3.14×42.875V≈34×3.14×42.875
     • V≈179.59V≈179.59 cm³

Conclusion:

• The total volume of the sphere is approximately 179.59179.59 cubic centimeters.

Are the square roots of all positive integers irrational?

Introduction: The question probes into the nature of square roots of positive integers, whether they are exclusively irrational or if there are exceptions.

Explanation: The statement that the square roots of all positive integers are irrational is false. While there are indeed many examples of square roots that are irrational, there are also instances where the square root of a positive integer results in a rational number.

Example:

• Square root of 4:
  • Integer: 4
  • Square root: √4 = 2
  • Nature: Rational

Explanation of the Example:

• The square root of 4 is 2, which is a rational number.
• This contradicts the notion that all square roots of positive integers are irrational.

Conclusion: In conclusion, not all square roots of positive integers are irrational. The square root of 4, for instance, is a rational number, demonstrating that exceptions exist to the notion that all such roots are irrational.

Decimal Expansions of Fractions

1. Decimal Expansion of 10/3:

• Calculation:

  • Divide 10 by 3.
  • The result will be 3.3333...

• Decimal Expansion:

  • 103=3.3‾310=3.3

2. Decimal Expansion of 7/8:

• Calculation:

  • Divide 7 by 8.
  • The result will be 0.875.

• Decimal Expansion:

  • 78=0.87587=0.875

3. Decimal Expansion of 1/7:

• Calculation:

  • Divide 1 by 7.
  • The result will be 0.142857142857...

• Decimal Expansion:

  • 17=0.142857‾71=0.142857

Conclusion:

• The decimal expansions for the given fractions are:
  • 103=3.3‾310=3.3
  • 78=0.87587=0.875
  • 17=0.142857‾71=0.142857

Expressing 0.3333… as a Fraction:

Understanding the Repeating Decimal:

• When we write 0.3333…, the 3's continue indefinitely, indicating a repeating decimal.

Notation:

• Let x = 0.3333…

Multiplying by 10:

• If we multiply both sides of x by 10, we get 10x = 3.3333…

Subtracting Original Equation:

• Now, let's subtract the original equation (x) from the new equation (10x):
  • 10x - x = 3.3333... - 0.3333...
  • 9x = 3

Solving for x:

• Dividing both sides by 9, we find:
  • x = 3/9

Simplifying the Fraction:

• Both 3 and 9 can be divided by 3:
  • x = 1/3

Conclusion:

• Therefore, 0.3333… can be expressed as 1/3.

Graph of the Equation x - y = 4

Graphing the Equation:

To draw the graph of the equation x−y=4x−y=4, we'll first rewrite it in slope-intercept form, which is y=mx+by=mx+b, where mm is the slope and bb is the y-intercept.

Given equation: x−y=4x−y=4

Rewriting in slope-intercept form:

y=x−4y=x−4

Now, let's plot the graph using this equation.

Plotting the Graph:

1. Find y-intercept:
   Set x=0x=0 in the equation y=x−4y=x−4
   y=0−4y=0−4
   y=−4y=−4
   So, the y-intercept is at the point (0,−4)(0,−4).

2. Find x-intercept:
   To find the x-intercept, set y=0y=0 in the equation y=x−4y=x−4.
   0=x−40=x−4
   x=4x=4
   So, the x-intercept is at the point (4,0)(4,0).

Drawing the Graph:

Now, plot the points (0,−4)(0,−4) and (4,0)(4,0) on the Cartesian plane and draw a straight line passing through these points. This line represents the graph of the equation x−y=4x−y=4.

Intersecting with the x-axis:

To find where the graph line meets the x-axis, we need to find the point where y=0y=0.

Substitute y=0y=0 into the equation x−y=4x−y=4:

x−0=4x−0=4

x=4x=4

So, when the graph line meets the x-axis, the coordinates of the point are (4,0)(4,0).

Graphing the Equation x + 2y = 6

To graph the equation x+2y=6x+2y=6, we'll first rewrite it in slope-intercept form (y=mx+by=mx+b):

x+2y=6x+2y=6 2y=−x+62y=−x+6 y=−12x+3y=−21x+3

Plotting the Graph

To plot the graph, we'll identify two points and draw a line through them:

1. Intercept Method:

   • y-intercept (when x = 0): y=−12(0)+3=3y=−21(0)+3=3 Therefore, the y-intercept is (0, 3).
   • x-intercept (when y = 0): 0=−12x+30=−21x+3 −12x=3−21x=3 x=−6x=−6 Therefore, the x-intercept is (-6, 0).

2. Slope Method: From the slope-intercept form y=−12x+3y=−21x+3, the slope is -1/2, meaning the line decreases by 1 unit in the y-direction for every 2 units in the x-direction.

Plotting the Points and Drawing the Line

Using the intercepts and the slope, we plot the points (0, 3) and (-6, 0), then draw a line through them.

Finding the Value of x when y = -3

Given y=−3y=−3, we substitute this value into the equation y=−12x+3y=−21x+3 and solve for x:

−3=−12x+3−3=−21x+3 −12x=−3−3−21x=−3−3 −12x=−6−21x=−6 x=−6×(−2)x=−6×(−2) x=12x=12

Conclusion

• The graph of the equation x+2y=6x+2y=6 is a straight line passing through points (0, 3) and (-6, 0).
• The value of xx when y=−3y=−3 is x=12x=12.