Bosco

A cylinder has a height of 6 centimeters and a diameter of 20 centimeters. What is its volume? Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.  

The volume of a cylinder is the area of the base times the height.  

Area of base:                 π r²  =  3.14 • 10²  =  314  

Height:                                                                  6  

Volume:                                                           1884 cm³    

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There are four cities: Capital City, Little Village, Mytown, and Yourtown.  The distance from Capital City to Little Village is $5$ miles.  The distance from Capital City to Mytown is $2$ miles.  The distance from Mytown to Yourtown is $3$ miles.  The distance from Little Village to Yourtown is $4$ miles.  Find the distance from Capital City to Yourtown, in miles.  

          It could work like this.  Let each x represent one mile.    

           x          x          x          x          x          x          x          x         

          MT                  CC       YT                                          LV      

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All the sides of a triangle are integers.  If the perimeter of the triangle is $3,$ then how many different possible triangles are there?  (Assume that the triangle is non-degenerate.  Two triangles are considered the same if they are congruent.)  

A triangle has 3 sides.  

All the way around is 3.  

Every side is an integer.  

Each side can equal only 1.  There is one triangle.  

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 Fill in the blanks to make a quadratic whose roots are 5 and -5.   x^2 + ___ x + ___  

Since the roots are +5 and –5, the quadratic factors to (x – 5)(x + 5)   

Multplied out, (x – 5)(x + 5) is x² – 25.  

Fill in the blanks with 0 and –25 ............. x²  +  0x  +  –25    

^.

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?  

In order not to have two yellows next to each other,  

the three yellows must occupy squares 1, 3, and 5. 

That leaves squares 2 and 4 to fill, and two colors  

to do it with. There are only four ways. 

Blue Blue     Red Red     Blue Red     Red Blue  

So, the answer is four ways.  

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The perimeter of the triangle below is an integer.  What is the smallest possible value of the perimeter?  (Assume that the triangle is non-degenerate.)     

When the unlabeled side is 26, then the sum of it and the 24 side will be 50,  

which is the minimum integer value to satisfy the triangle inequality rule.   

Had to edit.  The problem asks for perimeter, not third side.  The perimeter is 99   

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The diagram below consists of a small square, four equilateral triangles, and a large square.  Find the area of the large square.   

At the base of the triangles, extend the line on over to the right side of the square.  

That line is 2 & ¹/₂ triangle bases long.  The base of each triangle is one unit.  

So the width of the large square is 2 & ¹/₂ units.  

Two and a half squared is six and a quarter.  The area of the large square is 6.25 units².   

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In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?  

In order not to have two yellows next to each other,  

the three yellows must occupy squares 1, 3, and 5. 

That leaves squares 2 and 4 to fill, and two colors  

to do it with. There are only four ways. 

Blue Blue     Red Red     Blue Red     Red Blue  

So, the answer is four ways.  

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Get the measure of the angles on either side of PAQ   

then subtract them from 90^o to leave the size of PAQ.  

tangent of BAP is 5 / 7  =  0.7143   

tangent of DAQ is 5 / 7  =  0.7143  

note that tan^–1(0.7143)  =  35.54^o  

BAP + DAQ  =  35.54^o + 35.54^o  =  71.08^o  

PAQ  =  90^o – 71.08^o  =  18.92^o  

sin(18.92^o)  =  0.324  

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A baking club wants to form an executive committee. There are $20$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $7$ people?  

This is a Combinations problem, of 20 items taken 7 at a time. 

One way this is abbreviated is             ₂₀C₇    

it's calculated as                                     20!   

                                                       –––––––––––   

                                                        (7! • (20–7)!)   

I can't write down the numbers.  

20! is too big, something times e18.  

My calculator screen won't display it all.  

The final answer works out to               77,520   

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