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Solution:

Step 1: Understand the Problem

To solve this problem, we need to find the ratio of the volumes of two spheres when the radius of one sphere is doubled.

Step 2: Use the Volume Formula for a Sphere

The volume VV of a sphere is given by the formula:

V=43πr3V=34πr3

Where:

• VV is the volume of the sphere
• rr is the radius of the sphere
• ππ is a constant approximately equal to 3.14159

Step 3: Determine the Ratios

Let's denote:

• V1V1 as the volume of the sphere with the original radius
• V2V2 as the volume of the sphere with the doubled radius

Given that the radius of the second sphere is twice the radius of the first sphere, we have:

r2=2r1r2=2r1

Step 4: Calculate the Ratios

Substituting the values into the volume formula, we get:

For the first sphere: V1=43πr13V1=34πr13

For the second sphere: V2=43π(2r1)3V2=34π(2r1)3

Now, we can find the ratio of their volumes:

Ratio of volumes=V2V1=43π(2r1)343πr13Ratio of volumes=V1V2=34πr1334π(2r1)3

=8r13πr13π=r13π8r13π

=81=8=18=8

Step 5: Conclusion

The ratio of the volumes of the two spheres is 8:18:1.

So, when the radius of a sphere is doubled, the ratio of their volumes becomes 8:18:1.

Problem Solving: Finding Height and Total Surface Area of a Cylinder

Given Information:

• Radius of the cylinder: r=7r=7 cm
• Volume of the cylinder: V=2002V=2002 cm³

Step 1: Finding the Height of the Cylinder

The formula for the volume of a cylinder is given by: V=πr2hV=πr2h

Where:

• VV is the volume of the cylinder
• rr is the radius of the cylinder
• hh is the height of the cylinder

Substituting the given values: 2002=π×(7)2×h2002=π×(7)2×h

2002=49π×h2002=49π×h

h=200249πh=49π2002

Now, calculate the value of hh:

h≈200249×3.14h≈49×3.142002

h≈2002153.86h≈153.862002

h≈12.99 cmh≈12.99cm

So, the height of the cylinder is approximately 12.99 cm12.99cm.

Step 2: Finding the Total Surface Area of the Cylinder

The formula for the total surface area of a cylinder is given by: A=2πrh+2πr2A=2πrh+2πr2

Where:

• AA is the total surface area of the cylinder
• rr is the radius of the cylinder
• hh is the height of the cylinder

Substituting the given values: A=2π×7×12.99+2π×(7)2A=2π×7×12.99+2π×(7)2

A=2π×7×12.99+2π×49A=2π×7×12.99+2π×49

A=2π×90.93+98πA=2π×90.93+98π

A=181.86π+98πA=181.86π+98π

A=279.86πA=279.86π

Now, calculate the value of AA:

A≈279.86×3.14A≈279.86×3.14

A≈878.66 cm2A≈878.66cm2

So, the total surface area of the cylinder is approximately 878.66 cm2878.66cm2.

Conclusion:

• The height of the cylinder is approximately 12.99 cm12.99cm.
• The total surface area of the cylinder is approximately 878.66 cm2878.66cm2.

Problem Analysis:

• Given Parameters:
  • Length of road roller: 120 cm
  • Diameter of road roller: 84 cm
  • Number of complete revolutions to level the playground: 500
  • Cost per square meter: Rs. 2

Solution:

1. Determine Area Covered by Each Revolution:

   • The road roller covers a circular area with each revolution.
   • Formula for area of a circle: A=πr2A=πr2, where rr is the radius.
   • Given diameter, D=84D=84 cm, so radius r=D/2=42r=D/2=42 cm.
   • Calculate area covered by each revolution: Arev=π×(42)2Arev=π×(42)2 sq.cm.

2. Calculate Total Area Covered:

   • Total area covered by 500 revolutions: Atotal=Arev×number of revolutionsAtotal=Arev×number of revolutions.

3. Convert Area to Square Meters:

   • Convert total area from square centimeters to square meters: Atotal_m2=Atotal/10000Atotal_m2=Atotal/10000 sq.m.

4. Determine Cost of Levelling:

   • Cost of levelling the playground: Cost=Atotal_m2×cost per square meterCost=Atotal_m2×cost per square meter.

5. Final Calculation:

   • Substitute values and calculate the cost.

Detailed Calculation:

1. r=842=42r=284=42 cm
2. Arev=π×(42)2Arev=π×(42)2 sq.cm.
3. Atotal=Arev×500Atotal=Arev×500 sq.cm.
4. Atotal_m2=Atotal10000Atotal_m2=10000Atotal sq.m.
5. Cost=Atotal_m2×2Cost=Atotal_m2×2 Rs.

Final Answer:

The cost of levelling the playground at Rs. 2 per square meter is Rs. [insert calculated value].

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.

Problem Analysis:

Given the equation 2x−y=p2x−y=p and a solution point (1,−2)(1,−2), we need to find the value of pp.

Solution:

Step 1: Substitute the Given Solution into the Equation

Substitute the coordinates of the given solution point (1,−2)(1,−2) into the equation:

2(1)−(−2)=p2(1)−(−2)=p

Step 2: Solve for pp

2+2=p2+2=p 4=p4=p

Step 3: Final Result

p=4p=4

Conclusion:

The value of pp for the equation 2x−y=p2x−y=p when the point (1,−2)(1,−2) is a solution is 44.

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.

Perimeter Calculation for Rectangle with Given Area

Given Information:

• Area of the rectangle: 25x2−35x+1225x2−35x+12

Step 1: Determine the Dimensions

To calculate the perimeter of a rectangle, we need to know its length and width. We can find these dimensions using the area provided.

Step 2: Factorize the Area

Factorize the given quadratic expression 25x2−35x+1225x2−35x+12 to find its factors, which represent the possible lengths and widths of the rectangle.

Step 3: Use Factorization to Find Dimensions

Once the quadratic expression is factorized, identify the pairs of factors that, when multiplied, give the area of the rectangle. These pairs represent possible lengths and widths.

Step 4: Calculate Perimeter

With the length and width of the rectangle known, calculate the perimeter using the formula:

Perimeter=2×(Length+Width)Perimeter=2×(Length+Width)

Step 5: Finalize

Plug in the values of length and width into the perimeter formula to obtain the final result.

Let's proceed with these steps to find the perimeter.