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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.

Finding Rational Numbers Between 1 and 2

Rational numbers are those that can be expressed as a fraction of two integers. Here's how we can find five rational numbers between 1 and 2.

Method 1: Using Averaging

1. Step 1: Average 1 and 2 to find the first rational number:

   • (1 + 2) / 2 = 3 / 2 = 1.5

2. Step 2: Repeat the process to find more rational numbers:

   • (1 + 1.5) / 2 = 2.5 / 2 = 1.25
   • (1.25 + 1.5) / 2 = 2.75 / 2 = 1.375
   • (1.25 + 1.375) / 2 = 2.625 / 2 = 1.3125
   • (1.3125 + 1.375) / 2 = 2.6875 / 2 = 1.34375

Method 2: Using Reciprocals

1. Step 1: Take the reciprocal of 2:

   • 1 / 2 = 0.5

2. Step 2: Repeat the process to find more rational numbers:

   • 1 / (2 + 1) = 1 / 3 ≈ 0.333
   • 1 / (3 + 1) = 1 / 4 = 0.25
   • 1 / (4 + 1) = 1 / 5 = 0.2
   • 1 / (5 + 1) = 1 / 6 ≈ 0.167

Summary:

• Rational numbers between 1 and 2: 1.5, 1.25, 1.375, 1.3125, 1.34375 (using averaging method)
• Rational numbers between 1 and 2: 0.5, 0.333, 0.25, 0.2, 0.167 (using reciprocal method)

These methods provide us with a variety of rational numbers between 1 and 2, demonstrating the flexibility and diversity of such numbers.

Locating √3 on the Number Line

Introduction Locating √3 on the number line is an essential concept in mathematics, particularly in understanding irrational numbers and their placement in relation to rational numbers.

Understanding √3 √3 represents the square root of 3, which is an irrational number. An irrational number cannot be expressed as a fraction of two integers and has an infinite non-repeating decimal expansion.

Steps to Locate √3 on the Number Line

1. Identify Nearby Perfect Squares:

   • √3 lies between the perfect squares of 1 and 4.
   • √1 = 1 and √4 = 2.

2. Estimation:

   • Since 3 is between 1 and 4, the square root of 3 will be between 1 and 2.
   • By estimation, √3 is approximately 1.732.

3. Plotting √3 on the Number Line:

   • Start at 0 on the number line.
   • Move to the right until you reach approximately 1.732 units.

4. Final Position:

   • Mark the point on the number line corresponding to √3.

Conclusion Locating √3 on the number line involves understanding its position between perfect squares and accurately plotting its approximate value. This skill is fundamental for comprehending the continuum of real numbers and their relationships.

Solutions for 2x + 3y = 8

Introduction: In this problem, we're tasked with finding solutions to the equation 2x + 3y = 8. There are multiple solutions that satisfy this equation. Let's explore four of them:

Solution 1: Using Integer Values

• Choose a set of integer values for x and solve for y.
• Let's say x = 2.
• Substitute x = 2 into the equation: 2(2) + 3y = 8.
• Solve for y: 4 + 3y = 8.
• 3y = 8 - 4.
• 3y = 4.
• y = 4/3.
• So, one solution is (2, 4/3).

Solution 2: Using Fractional Values

• Choose fractional values for x and solve for y.
• Let's say x = 1/2.
• Substitute x = 1/2 into the equation: 2(1/2) + 3y = 8.
• Solve for y: 1 + 3y = 8.
• 3y = 8 - 1.
• 3y = 7.
• y = 7/3.
• Another solution is (1/2, 7/3).

Solution 3: Using a Variable for y

• Express y in terms of x and a constant.
• Rearrange the equation to isolate y: 3y = 8 - 2x.
• Divide both sides by 3: y = (8 - 2x)/3.
• So, a solution can be represented as (x, (8 - 2x)/3).

Solution 4: Using Graphical Method

• Graph the equation on a coordinate plane.
• Plot the points where the line intersects the x-axis and the y-axis.
• Determine the coordinates of these points as solutions.
• By plotting, we find that two points of intersection are (4, 0) and (0, 8/3).
• Thus, solutions are (4, 0) and (0, 8/3).

Conclusion: The equation 2x + 3y = 8 has multiple solutions, including both integer and fractional values of x and y. Additionally, solutions can also be represented using variables. Graphically, the solutions are the points where the line intersects the axes.

Graph of the Equation 2x – 3y = 12

To draw the graph of the equation 2x−3y=122x−3y=12, let's 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.

Rewrite Equation in Slope-Intercept Form

2x−3y=122x−3y=12
−3y=−2x+12−3y=−2x+12
y=23x−4y=32x−4

Plotting the y-intercept and Slope

1. Y-intercept: When x=0x=0,
   y=23(0)−4y=32(0)−4
   y=−4y=−4
   So, the y-intercept is (0, -4).

2. Slope: The coefficient of xx is 2332, which represents the slope.
   For every increase of 1 in xx, yy increases by 2332.
   For every decrease of 1 in xx, yy decreases by 2332.

Plotting Points and Drawing the Graph

Now, let's plot some points to draw the graph:

• x = 3: y=23(3)−4=2−4=−2y=32(3)−4=2−4=−2
  Point: (3, -2)

• x = 6: y=23(6)−4=4−4=0y=32(6)−4=4−4=0
  Point: (6, 0)

• x = -3: y=23(−3)−4=−2−4=−6y=32(−3)−4=−2−4=−6
  Point: (-3, -6)

Plotting the Graph

With these points, we can draw a straight line passing through them.

Points where the Graph Intersects the Axes

X-axis

To find where the graph intersects the x-axis, we set y=0y=0 and solve for xx:

0=23x−40=32x−4
23x=432x=4
x=4×32x=24×3
x=6x=6

So, the graph intersects the x-axis at x=6x=6, which corresponds to the point (6, 0).

Y-axis

To find where the graph intersects the y-axis, we set x=0x=0 and solve for yy:

y=23(0)−4y=32(0)−4
y=−4y=−4

So, the graph intersects the y-axis at y=−4y=−4, which corresponds to the point (0, -4).

Summary

• X-axis intersection: (6, 0)
• Y-axis intersection: (0, -4)

This information helps us visualize and understand the behavior of the equation 2x−3y=122x−3y=12 on the coordinate plane.

Graph of 9x – 5y + 160 = 0

To graph the equation 9x – 5y + 160 = 0, we'll first rewrite it in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

Step 1: Rewrite the equation in slope-intercept form

9x – 5y + 160 = 0

Subtract 9x from both sides:

-5y = -9x - 160

Divide both sides by -5 to isolate y:

y = (9/5)x + 32

Now we have the equation in slope-intercept form.

Step 2: Identify the slope and y-intercept

The slope (m) is 9/5 and the y-intercept (b) is 32.

Step 3: Plot the y-intercept and use the slope to find additional points

Now, let's plot the y-intercept at (0, 32). From there, we'll use the slope to find another point. The slope of 9/5 means that for every 5 units we move to the right along the x-axis, we move 9 units upwards along the y-axis.

So, starting from (0, 32), if we move 5 units to the right, we move 9 units up to get the next point.

Step 4: Plot the points and draw the line

Plot the y-intercept at (0, 32) and the next point at (5, 41). Then, draw a line through these points to represent the graph of the equation.

Finding the value of y when x = 5

To find the value of y when x = 5, we'll substitute x = 5 into the equation and solve for y.

9x – 5y + 160 = 0

9(5) – 5y + 160 = 0

45 – 5y + 160 = 0

Combine like terms:

-5y + 205 = 0

Subtract 205 from both sides:

-5y = -205

Divide both sides by -5 to solve for y:

y = 41

So, when x = 5, y = 41.

Problem Analysis: Given equations:

1. 3x+2y=123x+2y=12
2. xy=6xy=6

We need to find the value of 9x2+4y29x2+4y2.

Solution:

Step 1: Find the values of xx and yy

To solve the system of equations, we can use substitution or elimination method.

From equation (2), xy=6xy=6, we can express yy in terms of xx: y=6xy=x6

Substitute this expression for yy into equation (1): 3x+2(6x)=123x+2(x6)=12

Now solve for xx:

3x+12x=123x+x12=12 3x2+12=12x3x2+12=12x 3x2−12x+12=03x2−12x+12=0

Divide the equation by 3: x2−4x+4=0x2−4x+4=0

Factorize: (x−2)2=0(x−2)2=0

So, x=2x=2.

Now, substitute x=2x=2 into equation (2) to find yy: 2y=62y=6 y=3y=3

So, x=2x=2 and y=3y=3.

Step 2: Find the value of 9x2+4y29x2+4y2

Substitute the values of xx and yy into the expression 9x2+4y29x2+4y2: 9(2)2+4(3)29(2)2+4(3)2 9(4)+4(9)9(4)+4(9) 36+3636+36 7272

Conclusion: The value of 9x2+4y29x2+4y2 is 7272.