Dimensional Change Questions II

Dimensional change questions II:   Answer key below.
If you've found you are not solid yet with these problems,
slow down and start with Dimensional change questions I.

1a. There is a regular cylinder, which has a height equal to its radius. If the radius and height are both increased by 20%, by what % does the total volume of the cylinder increase?

1b. If the radius and height are both decreased by 20%, by what % does the total volume of the cylinder decrease?

1c. If the radius is increased by 50% and the height is decreased by 25%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

1d. If the radius is increased by 25% and the height is decreased by 50%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

1e. If the height is increased by 300%, what % does the radius need to be decreased by for the volume to remain the same?

2. If the side of a cube is increased by 30%, by what % does the total surface area of the cube increase? By what % does the volume increase?

3a. If the volume of a cube increases by 174.4%, by what % does the total surface area of the cube increase?

3b. By what % did the side length of the cube increase?

















 

Answer key to dimensional change questions II: 

1a. There is a regular cylinder, which has a height equal to its radius. If the radius and height are both increased by 20%, by what % does the total volume of the cylinder increase?

72.8%

1b. If the radius and height are both decreased by 20%, by what % does the total volume of the cylinder decrease?

48.8% (Only 0.8³ = 0.512 = 51.2% of the original percentage left and 100% - 51.2% = 48.8%.)

1c. If the radius is increased by 50% and the height is decreased by 25%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

168.75%

1d. If the radius is increased by 25% and the height is decreased by 50%, what % of the volume of the original cylinder does the volume of the new cylinder represent?

78.125%

1e. If the height is increased by 300%, what % does the radius need to be decreased by for the volume to remain the same?

50%

2. If the side of a cube is increased by 30%, by what % does the total surface area of the cube increase? By what % does the volume increase?

The surface area will increase 69% and the volume will increase 119.7%

3a. If the volume of a cube increases by 174.4%, by what % does the total surface area of the cube increase?

96%

3b. By what % did the side length of the cube increase?

40%

Analytical Geometry : Circle Equations

Circle Equations from Math is Fun 

How to Find Equation of a Circle Passing 3 Given Points 

7 methods included ; Amazing !!

Practice finding the equation of a Circle given 3 points -- 

Q #1 : (1, 3), (7, 3) and (1, -3)

Answer : (x -4)² + y² = 18

Q #2  : (3, 4), (3, -4), (0, 5)

Answer : x² + y² = 25

Q #3 : A (1, 1), B (2, 4), C (5, 3)

Answer : (x-3)² + (y -2)² = 5

Solution : 

The midpoint of line AB on the Cartesian plane is \((\frac{3}{2}, \frac{5}{2})\) and the slope is \((\frac{3}{1})\) so the slope of the perpendicular bisector of line AB is \((\frac{-1}{3})\).

The equation of the line bisect line AB and perpendicular to line AB is thus :  

y - \((\frac{5}{2}\)) =\(\frac{-1}{3}\) [x - \((\frac{3}{2})\)] --- equation 1

The midpoint of line BC on the Cartesian plane is \((\frac{7}{2}, \frac{7}{2})\) and the slope is \((\frac{-1}{3})\) so the slope of the perpendicular bisector of line BC is 3.

And the equation of the line bisect line BC and perpendicular to line BC is 

y - \((\frac{7}{2})\) = 3 [x - \((\frac{7}{2})\)] --- equation 2

Solve the two equations for x and y and you have the center of the circle being (3, 2)

Use distance formula from the center circle to any point to get the radius = 

\(\sqrt{5}\).

The answer is : (x - 3)² + (x - 2)² = 5

More practices on similar questions :  (Answers below for self check)

Q #1 : A (2, 5) , B (2, 13) ,  and C (-6, 5 )

Q #2 : A (0, 7), B ( 6, 5 ), and C (-6, -11 )

Q #3 : A (3, -5) , B (-4, 2) and C (1, 7 )








Answer key :

#1 :  (x - 2) ² + ( y - 9 ) ² = 32

#2 :   x² + (y + 3)² = 100

#3 :  (x - 2) ² + ( y -1) ² = 37

This Week's Work : Week 53 for Inquisitive Young Mathletes

For this week's work, review 1991 Mathcounts National sprint and target round questions you got wrong or not fast enough. Please try once more to see if you now can solve them at ease.

For this week's review, try the following Mathcounts Mini :

Similar Triangles and Proportional Reasoning

Try the follow-up problems

Detailed solutions

I'll send you notes/solutions/links to some of the hardest problems later.

Take care and happy problem solving !!

This Week's Work : Week 52 for Inquisitive Young Mathletes

Finish 2000 Mathcounts National sprint and target rounds according to the rules.
Check your e-mail.

E-mail me your scores + what questions you skip, couldn't solve fast, preferably with
the wrong answers put down so I can tell if you have some ideas.

Constructive Counting from Mathcounts Mini

Try some questions from the activity sheet till you fully understand the concepts.

Activity sheet solutions 

This Week's Work : Week 51 for Inquisitive Young Mathletes

For this week's work :

Try 2013 Mathworks Math Contest problems from Texas State University

Answer key and statistics can be viewed here. 

I'll send you detailed solutions once we finish all the problems.

Practice more "at least" problems :

2014 AMC-10 B problem 16 

Solution

From Mathcounts Mini : Video tutorials on counting and probability for Mathcounts state/national prep concepts are in order the of difficulty.

Counting the Number of Subsets of a Set

Constructive Counting

More Constructive Counting 

Probability and Counting