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Answered on 18 Apr Learn 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:
Solution Steps:
Conclusion:
Answered on 18 Apr Learn Sphere
Nazia Khanum
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:
Step 3: Determine the Ratios
Let's denote:
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.
Answered on 18 Apr Learn Sphere
Nazia Khanum
Problem Solving: Finding Height and Total Surface Area of a Cylinder
Given Information:
Step 1: Finding the Height of the Cylinder
The formula for the volume of a cylinder is given by: V=πr2hV=πr2h
Where:
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:
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:
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Answered on 18 Apr Learn Sphere
Nazia Khanum
Problem Analysis:
Solution:
Determine Area Covered by Each Revolution:
Calculate Total Area Covered:
Convert Area to Square Meters:
Determine Cost of Levelling:
Final Calculation:
Detailed Calculation:
Final Answer:
The cost of levelling the playground at Rs. 2 per square meter is Rs. [insert calculated value].
Answered on 18 Apr Learn Real Numbers
Nazia Khanum
Visualizing numbers on a number line can be a helpful technique to understand their placement and relationship to other numbers. Let's explore how we can visualize the number 3.765 using successive magnification.
Identify the Initial Position:
First Magnification:
Second Magnification:
Final Visualization:
Visualizing numbers on the number line using successive magnification helps in understanding their precise location and relationship to other numbers. By breaking down the intervals into smaller parts, we can accurately locate decimal numbers like 3.765 on the number line.
Answered on 18 Apr Learn Real Numbers
Nazia Khanum
Adding Radical Expressions
Introduction: In mathematics, adding radical expressions involves combining like terms to simplify the expression. Radical expressions contain radicals, which are expressions that include square roots, cube roots, etc.
Problem Statement: Add 22+5322
+53 and 2−332−33
.
Solution: To add radical expressions, follow these steps:
Identify Like Terms:
and 22
and −33−33
Combine Like Terms:
Write the Result:
+53 and 2−332−33 is:
+23
Conclusion: The addition of 22+5322
+53 and 2−332−33 simplifies to 32+2332+23
.
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Answered on 18 Apr Learn Linear equations in 2 variables
Nazia Khanum
Understanding Linear Equations: Linear equations are fundamental in mathematics, representing straight lines on a coordinate plane. They're expressed in the form of ax+b=0ax+b=0, where aa and bb are constants.
Identifying Axis: In the context of linear equations, the term "axis" typically refers to either the x-axis or the y-axis on a Cartesian plane.
Analyzing the Equation: The linear equation provided is x−2=0x−2=0.
Finding the Axis: To determine which axis the given linear equation is parallel to, let's analyze the equation:
Equation Form:
Solving for x:
Interpretation:
Conclusion: The linear equation x−2=0x−2=0 is parallel to the y-axis.
Answered on 18 Apr Learn Polynomials
Nazia Khanum
Problem Statement: Find the value of x2+y2x2+y2, given x+y=12x+y=12 and xy=32xy=32.
Solution:
Step 1: Understanding the problem
Step 2: Solving the equations
Step 3: Finding the values of xx and yy
Step 4: Finding corresponding values of yy
Step 5: Calculating x2+y2x2+y2
Step 6: Presenting the solution
Final Answer:
This structured approach helps in solving the problem systematically, ensuring accuracy and clarity.
Answered on 18 Apr Learn Polynomials
Nazia Khanum
Determining the Value of k
Introduction: To find the value of k when (x – 1) is a factor of the polynomial 4x^3 + 3x^2 – 4x + k, we'll utilize the Factor Theorem.
Factor Theorem: If (x – c) is a factor of a polynomial, then substituting c into the polynomial should result in zero.
Procedure:
Step-by-Step Solution:
Substitute x=1x=1:
Solve for k:
Conclusion: The value of k when (x – 1) is a factor of the given polynomial is k=−3k=−3.
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Answered on 18 Apr Learn Polynomials
Nazia Khanum
Solution: Finding Values of a and b
Given Problem: If x3+ax2–bx+10x3+ax2–bx+10 is divisible by x2–3x+2x2–3x+2, we need to find the values of aa and bb.
Solution Steps:
Step 1: Determine the factors of the divisor
Given divisor: x2–3x+2x2–3x+2
We need to find two numbers that multiply to 22 and add up to −3−3.
The factors of 22 are 11 and 22.
So, the factors that add up to −3−3 are −2−2 and −1−1.
Hence, the divisor factors are (x–2)(x–2) and (x–1)(x–1).
So, the divisor can be written as (x–2)(x–1)(x–2)(x–1).
Step 2: Use Remainder Theorem
If f(x)=x3+ax2–bx+10f(x)=x3+ax2–bx+10 is divisible by (x–2)(x–1)(x–2)(x–1), then the remainder when f(x)f(x) is divided by x2–3x+2x2–3x+2 is zero.
According to Remainder Theorem, if f(x)f(x) is divided by x2–3x+2x2–3x+2, then the remainder is given by f(2)f(2) and f(1)f(1) respectively.
Step 3: Find the value of aa
Substitute x=2x=2 into f(x)f(x) and equate it to 00 to find the value of aa.
f(2)=23+a(2)2–b(2)+10f(2)=23+a(2)2–b(2)+10
0=8+4a–2b+100=8+4a–2b+10
18=4a–2b18=4a–2b
4a–2b=184a–2b=18
Step 4: Find the value of bb
Substitute x=1x=1 into f(x)f(x) and equate it to 00 to find the value of bb.
f(1)=13+a(1)2–b(1)+10f(1)=13+a(1)2–b(1)+10
0=1+a–b+100=1+a–b+10
11=a–b11=a–b
a–b=11a–b=11
Step 5: Solve the equations
Now we have two equations:
We can solve these equations simultaneously to find the values of aa and bb.
Step 6: Solve the equations
Equation 1: 4a–2b=184a–2b=18
Divide by 2: 2a–b=92a–b=9
Equation 2: a–b=11a–b=11
Step 7: Solve the system of equations
Adding equation 2 to equation 1: (2a–b)+(a–b)=9+11(2a–b)+(a–b)=9+11
3a=203a=20
a=203a=320
Substitute a=203a=320 into equation 2: 203–b=11320–b=11
b=203–11b=320–11
b=20–333b=320–33
b=−133b=3−13
Step 8: Final values of aa and bb
a=203a=320
b=−133b=3−13
So, the values of aa and bb are a=203a=320 and b=−133b=3−13 respectively.
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