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Your answer is correct too  

also it is probably obvious but I accidentally replaced h with f  (I didn't use it anyway)

Hi Blank, 

Here is the pic.  (maybe next time you could display some of your own working for us)

Note that a=a'    and   k_1=k   (because it is an isosceles triangle)

I added g, it is g that you need to find in terms of the original letters.

Width of alley= k+g

\(cos75= \frac{g}{a}\\ g=a*cos75\)

So width of alley = k + a*cos75

At this point I have answered the question but it is not very difficult to go further.

Using pythagoras a=sqrt(2k^2)

And cos75 can be estimated on the calculator (which is sensible since it is a real life situation)

But the exact value of cos 75 can also be found using the expansion of cos(45+30)

Yes that is right :)

The positive difference between two consecutive even perfect squares is 228. Compute the larger of the two squares.

Hello Guest!

\({\color{blue}22^2}-16^2=228\)      

The larger of the two squares is   \(22^2=484.\)

  !

Write down the quadratic equation whose roots are x = 1 and x = -7 and the coefficient of x is 1.     

Since the roots are +1 and –7,  

  the factors are (x – 1)(x + 7)  

     which multiplies out to  .........  x² + 6x – 7   

_.

There are 9 zeros between 1 and 99 ==9

There are 20 zeros between 100 and 199==20 x 9 ==180

There are 3 zeros in 1000 ==3

Total ==9 + 180 + 3==192 zeros between 1 and 1000 inclusive.

The quadratic is x^2 + x - 7.

s(1) = 26.

The cards are ordered, so there are 52*51*2 = 5304 ways.

(400,000)^2 == 16  x  10^10

The digit 0 appears 223 times.

The remainder when k is divided by 13 is 10.

You don't say.  Prove it. 

[10! x 11!] ==144850083840000 = 2^16 * 3^8 * 5^4 * 7^2 * 11

x==36 possible integer values as follows:

x ==1 , 2 , 3 , 4 , 5 , 6 , 8 , 9 , 10 , 12 , 15 , 16 , 18 , 20 , 24 , 30 , 32 , 36 , 40 , 45 , 48 , 60 , 72 , 80 , 90 , 96 , 120 , 144 , 160 , 180 , 240 , 288 , 360 , 480 , 720 , 1440 , Total == 36 such integers.

sqrt(x^2 - 5x + 6)   <===== this part cannot be negative

x^2 -5x +6     is a bowl shaped parabola.....   the negative area is between the zeroes and is not part of the domain of the given expression....all othe 'x' values are valid......    find the zeroes  of   x *^2 -5x+6

(x-6)(x+1)  = 0         so zeroes are  6 and -1       between 6 and -1 are not part of the domain (BUT 6 and 0 ARE part of the domain)

can you finish?

You have several solutions to your equation as follows:

       a      b       c

1  = 23   22    10
2  = 0     38   22
3  = 23   3     34
4  = 0    19    46
5  = 0    0      70

Quadratics are usually written as    a x^2 + b x     + c

So for this one      a = 3    b = 5   

This is not correct. If the red group is (1,2,3) and the blue group is (4,5,6,7,8,9,10), then the desired condition is not possible.

Yaneev thinks of a number  He subtracts three from the number, then doubles it. Then he subtracts three again, and doubles the number again. The result is his original number  What is ?     

Let X represent the original number.  

X  =  {[(X – 3) • 2] – 3} • 2  

Let me see if I can multiply that out without getting lost.  One step at a time.  

X  =  [(2X – 6) – 3] • 2  

X  =  4X – 12 – 6  

3X  =  18  

X  =  6   

check answer:  

6 minus 3 is three.  3 doubled is 6.  6 minus 3 is 3.  3 doubled is 6.  Looks correct.     

_.

Are we looking at the same problem?  The problem I'm looking at says "roll sum of 11" not sum of 6.   

When rolling two standard cubes (6-sided die), what is the probability that you roll sum of 11? Convert your answer to decimal form, then round to 3 decimal places    

Here are all the possible rolls.  

1,1   1,2   1,3   1,4   1,5   1,6  

2,1   2,2   2,3   2,4   2,5   2,6  

3,1   3,2   3,3   3,4   3,5   3,6  

4,1   4,2   4,3   4,4   4,5   4,6  

5,1   5,2   5,3   5,4   5,5   5,6  

6,1   6,2   6,3   6,4   6,5   6,6

How many ways to total 11 ....  count them, there are 2  

How many possible rolls ........  count them, there are 36  

The likelihood of rolling an 11 is 2 out of 36, or in decimal form to three places, 0.056    

_.

There is an equation to find this sort of thing......but most people do not remember it....here is another way:

Find the equation of the line that is perpindicular to y = 2x+8    that contains the point   (6,0)

    (HINT:   Perindicular slope = - 1/2 )

then equate the two line equations to find the x coordinate of the point of intersection

   then use that x coordinate in either of the line equations to find the y coordinate of the point of intersection

THEN, use the distance formula to find the distance between this point of intersection and the point (6,0) ...

    See what you can do !  

Here is how to find the 'x' coordinate...... you see if you can do the same for the 'y'

From A to B      x goes from   2 to 14      a distance of 12....   we need to go  1/3  more for BC = 1/3 AC

                                              so x   will go 4 more  ( which is 1/3 of 12)   from B   to   x = 18

The probability is 3/50.