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Look for a coupleof definite points    (2,-2)   and   (4,-8)  are on the graph and easily defined

a² / b  = -2    and    re arrange   a²/-2 = b    (sub this into the next equation)

a⁴ / b = -8        

                          a^4/  (a^2/-2)  = -8

                          -2a^2 = -8

                              a^2 = 4        a = 2     or -2   (we'll just take the positive root)

then b = -2

a+b = 2-2 = 0

Well the 14.2  and 19.3 cm dimensions will definitely  fit in the box

   so the only dimension we are concerned about is the 1.5 cm thickness.....will 17 of them be less than or equal to the 28 cm dimension?

        1.5 cm  x 17 = 25.5 cm     which is definitely less than the 28 cm box dimension....so yes  they will fit .

Hi geno3141.  I was working on my answer (just below) when you posted yours.  It takes me longer because I explain why I did each step, as if the student were here and I was talking to him/her.  I figure that since they need help with a relatively simple problem like this, they aren't familiar with the process.  Do you think that my "talking" to the student is overdoing it?  Is it too much?  I've wondered this before but this is the first time I've asked.  I'd appreciate your opinion, also from anybody else who would like to offer one.  Thanks. 

_..

Second problem: 

First we'll solve for AC

The formula for the area of a triangle is              A = (1/2)(base)(height) 

The area of triangle ABC is 42 

BC is 6 and we'll use it as the base

AC is the height of triangle ABC so                     42 = (1/2)(6)(AC) 

Do a little arithmetic and                                      42 = 3 • AC

Divide both sides by 3 and                                  14 = AC

Now we have learned that                                    AC = 14

and because (22 – 6) = 16                                  CD = 16

We've got the hard part done, so now

it's time for a lilttle basic trigonometry 

The tangent of angle ADC is

the opposite side over the adjacent side              Tan(x) = AC/CD = 14/16 

We could do the division by hand but 

since we'll use the calculator to get the

arctan aka tan^–1 let's just do the division

on the calculator too, for the convenience            Tan(x) = 14/16 = 0.875 

We see that the tangent is "close to" 1 

so we know the angle will be "close to" 45^o 

This is called "forecasting" or "guesstimating" 

We use the arctan aka tan–1 function of  

the calculator and find that the angle is                  Tan^–1 (0.875) = 41.1859 

ta da, and there you go (round it down a little)         angle x  =  41.2^o 

_..

1) You have to COMBINE!!! That was my mistake!
 

a)320 grams: greatest: 319.999 grams

                       least: 319.49

Really? I will try again, then. Sorry about that

it says the answer is wrong

1) The greatest weight for 32 grams is: 31.99 grams

    The least weight for 32 grams is 32.5

1) Check my answer here:

Second problem:

Find AC:

Area( triangle(ABC) )  =  42  =  ½·base·height  =  ½·BC·AC  =  ½·6·h

  --->   42  =  ½·6·h

  --->   42  =  3h

  --->   h  =  14

Since  BC = 6  and  BD = 22, DC  =  22 - 6  =  16

In triangle(ADC):  AC = 14 and DC = 16   --->   tan( D )  =  14/16   --->   D  =  tan^-1(14/16)

  --->   D  =  41.2^o  

Here's another.

\(\displaystyle S = 1_{6}+2_{6}+3_{6}+\dots+45_{6}.\)

\(\displaystyle S=45_{6}+44_{6}+43_{6}+\dots+1_{6}.\)

Add

\(\displaystyle 2S=50_{6}+50_{6}+50_{6}+\dots +50_{6}=50_{6}\times45_{6}=4010_{6},\\S=4010_{6}/2=2003_{6}.\)

First problem:

Call the point where the dashed line segment hits the base "X".

In triangle(BXA):  (AX)² + (XB)²  =  (AB)²     --->     (AX)² + (6)²  =  (6.5)²

   --->   A  =  2.5

Drop a dashed line from C perpendicular to AB. Label the point where it intersects the base "Y",

Since BC = XY,  AX + SY + YD  =  2.5 + 9 + YD  =  23     --->   YD  =  11.5

In triangle(CDY):  CY = 6  and  YD = 11.5   --->   tan(D)  =  6/11.5   --->  D  =  tan^-1(6/11.5)

   --->   D  =  27.6^o

Green to total    =    2:9

red to total = 1:3  =  3:9      Which do you think is the greater ....red or green?

I agree with guest because for a number to be divisible by 9, the sum of its digits must be divisible by 9.

All numbers smaller than 10,000 has either 1, 2, 3, or 4 digits.

The only 1-digit number that contains only 1s and 8s are 1 and 8; both obviously fail.

The only 2-digit numbers are 11, 18, 81, and 88, giving us 18 and 81.

If they have more than two digits, we may want to use the "divisble by 9" rule.

The three-digit numbers have either:

  3 1s and no 8s:  1 + 1 + 1 = 3, not divisble by 9

  2 1's and 1 8:     1 + 1 + 8 = 10, not divisible by 9

  1 1 and 2 8's:     1 + 8 + 8 = 17, not divisible by 9

  no 1s and 3 8s:  8 + 8 + 8  = 24, not divisible by 9

The four-digit number have either:

  4 1s and no 8s:  1 + 1 + 1 + 1 = 4, not divisble by 9

  3 1s and 1 8:      1 + 1 + 1 + 8 = 11, not divisible by 9

  2 1s and 2 8s:    1 + 1 + 8 + 8 = 18, divisible by 9:  1188, 1818, 1881, 8118, 8181, 8811

  1 1 and 3 8s:      1 + 8 + 8 + 8 = 25, not divisible by 9

  no 1s and 4 8s:  8 + 8 + 8 + 8 =  32, not divisible by 9 

Here is one way...convert to base 10....find answer ....then convert to base 6

1+2+3+4........+29 = \(\sum_{1}^{29}\) n   = 435      =     2003₆

let x = .4888888

then 10 x = 4.888888....

and 100x = 48.8888888....

100x - 10x   =      48.88888  ...    - 4.88888

             90x  =     44

                  x  = 44/90    is the first fraction      can you do the second one....and then do the subtraction?

This is a binomial probability problem.  The relevant formula is:  prob  =  _nC_r·p^r·q^{n - r} 

where:  n  =  total number of trials          n = 5

             r  =  number of successes         r = 0, 1, 2

             p  =  probability of success        p = 0.18

             q  =  probability of failure           q = (1.00 - 0.18)  =  0.82

For this problem, to find the total probability, we need to find the probability for 0 successes, then 1 success, and

finally 2 successes, and add them together.

Total probability  =  ₅C₀·(0.18)⁰·(0.82)⁵  +  ₅C₁·(0.18)¹·(0.82)⁴  +  ₅C₂·(0.18)²·(0.82)⁴ 

Now, it's between you and your calculator!

sin of angle  =  opposite / hypotenuse      (in right triangles such as this)

    sin(yzx)    =  7.5/22.3

xyz =   arcsin (7.5/22.3)  = ........

Total of angles in triangle  180^o

8+3+4 = 15      one angle is    8/15 of 180   = 96^o

                             one angle is   3/15 of 180^o =  .......

                         one angle  is  4/15              = ......

(41.5-25) / 41.5   x 100% = 39.76%

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Here is Venn diagram

ginger ale hanging with kids