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"5 pencils and 4 crayons are 26.3 inches. 6 pencils and 5 crayons are 32.0 inches. Each pencil is the same length as the other pencils, and each crayon is the same length as the other crayons. What is the length of 1 pencil? What is the length of 1 crayon?"

5p + 4c = 26.3           ...(1)

6p + 5c = 32              ...(2)

Multiply (1) by 5 and (2) by 4

25p + 20c = 5*26.3     ...(3)

24p + 20c = 4*32        ...(4)

subtract (4) from (3)

p = 5*26.3 - 4*32 =  ...

Substitute the result into (1) or (2) and rearrange to find c.

I'll leave the number crunching to you. 

You mean "what is the probability of rolling 2 "ones" and 2 "sixes"?

The probability is: 4C2 * 2C2 =6 / 6^4 = 1 / 216

11C2 = 55 combinations.

[(1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (1, 7), (1, 8), (1, 9), (1, 10), (1, 11), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8), (2, 9), (2, 10), (2, 11), (3, 4), (3, 5), (3, 6), (3, 7), (3, 8), (3, 9), (3, 10), (3, 11), (4, 5), (4, 6), (4, 7), (4, 8), (4, 9), (4, 10), (4, 11), (5, 6), (5, 7), (5, 8), (5, 9), (5, 10), (5, 11), (6, 7), (6, 8), (6, 9), (6, 10), (6, 11), (7, 8), (7, 9), (7, 10), (7, 11), (8, 9), (8, 10), (8, 11), (9, 10), (9, 11), (10, 11)],  >Total distinct combinations = 55

(3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 5, 6, 7, 8, 9, 10, 11, 12, 13, 7, 8, 9, 10, 11, 12, 13, 14, 9, 10, 11, 12, 13, 14, 15, 11, 12, 13, 14, 15, 16, 13, 14, 15, 16, 17, 15, 16, 17, 18, 17, 18, 19, 19, 20, 21)>Total = 55 sums

(3, 5, 5, 7, 7, 7, 11, 11, 11, 11, 11, 13, 13, 13, 13, 13, 17, 17, 17, 19, 19)>>Total = 21 pairs are prime numbers

The 6 numbers 1 to 6 have: 6! =720 permutations . Each permutation begins with one of the 6 numbers this many times: 720 / 6 =120 times with each number. So, the first 120 permutations will begin with 1. The next 120 permutations will begin with 2. So, that is: 120 + 120 =240 permutations. The next 120 will begin with 3 as follows:  312456 , 312465 , 312546 , 312564 , 312645 , 312654 , 314256 , 314265.......etc. So, the 8th permutation beginning with 3 =314265, and that is the 240 + 8 =248th permutation.

Sorry....can't  read this.....

Find the average speed of a mobile that during its journey north had the following speeds

We will need to know HOW MUCH TIME the object traveled at each of the given velocities.....

Plug in the answer choices given....find which one 'works'......

ElectricPavlov,

thank you very much!..much appreciated!

Melody,

yes, I understand..brrr...okay, it's a lot right now, I wil have a look at all the replies a bit later on...my brain is smoking!....but..I will check it and I hereby also say a BIIIG thank you!..

Hi Omi67,

I see your answers are different to ElectricPavlov's?...Thanx for your time..

ALWAYS REMEMBER:

The first derivative at any given point gives the GRADIENT of the tangent to the curve at that point.

It is essential that you understand and memorize this fact.

  You can search and find this Q was aked ...and answered recently......but you did not include the number of bounces....as I recall, the original Q asked how far the ball had travelled as it hit for the 6th time....     ~EP

We can let the area of the entire octagon be 1. Then..

[ABCDE]   =   1/2

area of △AOB   =   1/8

area of △BOM   =   1/2 of 1/8   =   1/16

[ABMO]   =   area of △AOB + area of △BOM   =   1/8 + 1/16   =   3/16

[ABMO] / [ABCDE]   =   ( 3/16 )/( 1/2 )   =   3/8

So long as you are learning I am pleased to give my time. :)

Thanks guest.

I think your solution is similar to Hectictar's.

There are many ways to do this problem :)

No that is not what it means at all.

It just means that when x=1 the gradient of the tangent will be 12

And when x=1 the concavity will be -24.   Only the negative sign needs to mean anything to you.

the concavity at x=1 is negative so at that point will be on a  concave down part of the graph.

I can see that EP is composing an answer as well.

He will probably show you the algebra of what you have to do.

I will come back later if he does not fill in the holes to your satisfaction.  Let us know how you get on.  :)

f' = first derivative         = 9x^2 + 2bx + c

f" = second derivative  = 18x +2b

f" (1) =- 24                18(1)+ 2b = -24     then b = -21

Use this valueof b in the first equation

f'(1) = 12 = 9(1^2) + 2(-21)(1) + c = 12    

                    9    +  -42     + c = 12       c = 45

From a mathematics point of view, I don't like jugoslav's answer.

It's what I would regard as an ' engineer's answer ', "  close enough for all practical purposes ".

Close enough, but you can't be sure that it's exact.

It's just correct to within the accuracy limits of the calculator.

The problem arises with the use of a calculator to calculate the angle DAQ.

You might have this to however many decimal places, but evenso, all that you have is an approximation to the angle.

Follow this through, and you have to say that the final result need not be exactly 0.6 .

Melody's answer using the cosine rule is what I would regard as being mathematical.

At the end of that, I could take it as being definate that the sine of the angle PAQ is 0.6 (exactly).

Here's an alternative derivation.

Making use of Melody's diagram, call the (equal) angles DAQ and PAB, phi.

Then

\(\displaystyle \theta + 2\phi = 90^{\circ}, \\ \sin \theta = \sin(90^{\circ}-2\phi)\\ =\cos2\phi=2\cos^{2}\phi-1=2(4/5)-1= 3/5.\\ \)

Here is one way:

tan a = opp/ adj    for  Brett's pictured triangle ...where 'a' is the end of his shadow

         = 1.85 / .6

       arctan(1.85/.6) = 72 degrees

Now use this angle in the tower diagram

   cos a = adj/hyp = 9.00/guywire

    cos 72 = 9/guywire

    guywire = 9 / (cos 72) = 29.17 m

A=4.5

Calculate b and c.

ElectricPawlow's answer is right.

It's like a flower blooming...it really is not that difficult after all. In a certain sense I feel I have wasted your time on this...I should have been able to see this..anyways, as always, you are appreciated!!..Thank you..

You can also do it mathematically as follows:

[6C2] * [4C3] * 4 =15 * 4 * 4 = 240 / 6^6 = 5 / 972 - which is same probability.

It is much easier and simpler to list the "combinations" first and then calculate the "permutations" as follows:

112223 , 112224 , 112225 , 112226 , Total =  4 combinations.

Each of the above combinations has: 6! / [2! * 3!] =60 permutations

And the probability is: [4 x 60] = 240 / 6^6 =5 / 972

Well in that case it is concave up and the minimum is (-2,-2)

\(y=ax(x+4) \qquad \text{Where a is any real number.}\\ \text{passes though }(-2,-2)\\~\\ \text{Subthe point in}\\ -2=a*-2(-2+4)\\ -2=a*-4\\ a=\frac{1}{2}\\ \text{So the equation is}\\ y=0.5x(x+4)\\ y=\frac{x(x+4)}{2}\\ \)

LaTex

y=ax(x+4) \qquad \text{Where a is any real number.}\\
\text{passes though }(-2,-2)\\~\\
\text{Subthe point in}\\
-2=a*-2(-2+4)\\
-2=a*-4\\
a=\frac{1}{2}\\
\text{So the equation is}\\
y=0.5x(x+4)\\
y=\frac{x(x+4)}{2}\\