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Sandy has two 64-ounce pitchers,  and she has four 48-ounce pitchers.

So the total number of ounces that Sandy can carry to the bake sale...

=  64 + 64 + 48 + 48 + 48 + 48

= 2(64) + 4(48)

= 320

thank you! I have been trying so trying to figure this out for a while!

To find the vertical asymptotes, we must figure out when the denominator of \(\frac{5x^2-9}{3x^2+5x+2}\) equals 0. This will give us the vertical asymptotes. Let's do that!

x=-(2/3) and x=-1 are the location of both vertical asymptotes on the graph. 

Now, add both answers together to get the result of \(a+b\). 

\(-1-\frac{2}{3}=\frac{-3}{3}-\frac{2}{3}=-\frac{5}{3}=-1.\overline{66}\)

Circumference  =  2 * pi * radius

49.6  = 2 * pi * radius          

Divide both sides by 2 pi

[ 49.6 ]  / [ 2 pi ]  = radius  ≈  7.89 inches

16225

Coincidentally, trigonometry was unnecessary in this problem (although use whatever solving method you'd like!). 

The height of the rectangle (we know it is a rectangle because it a quadrilateral with 4 right angles) \(FLAX\) is of equal length to the height of\(\triangle ZAX\). 

Let's draw in \(\overline{ZB}\) such that the segment is a perpendicular bisector of \(\overline{FL}\) and \(B\) lies on the midpoint of \(\overline{XA}\).

Inserting the altitude of an isosceles (or equilateral, in this case) triangle has 2 lesser-known properties.

1) The altitude bisects the intersected angle.

2)  The altitude bisects the intersected segment.

Put all of this information together \(m\angle AZB=30^{\circ}\hspace{1mm}\text{and}\hspace{1mm}m\angle ZBA=90^{\circ}\hspace{1mm}\text{and}\hspace{1mm} m\angle BAZ=60^{\circ}\). \(AB=5\hspace{1mm}\text{and}\hspace{1mm} AZ=10\)

It's a 30-60-90 triangle. Yay! By definition, the ratio of the side lengths is \(1:\sqrt{3}:2\). Knowing this, we can calculate the length of the altitude. 

It has already been established that ZB is the height of the rectangle. Therefore, multiply the length and the width together to get the total area.

\(A=10*5\sqrt{3}=50\sqrt{3}units^2\)

Since ZAX  is equilateral.....it must have a side  = 10

And the height of the quadrilateral, h, is given by : 

h =  10 * sin (60 degrees)  =  10√3/2  = 5/√3  units

And the area of the quadrilateral   =   side of ZAX * height of quadrilateral  = 10 *  5√3 =

50 √3   square units

The area of quadrilateral FLAX is 10 x sqrt(75). 

Triangle ZAX is equilateral and has a perimeter of 30. Each side is equal and there are 3 sides (), so 30/3=10. Therefore, line XA and FL are 10 units long. Since triangle ZAX is equilateral, point Z is in the middle of line FL. Line FL is 10, so lines FZ is 5 units long. Line ZX (part of the triangle) is 10 units long. We can use the Pythagorean theorem (a² + b² = c² ) to find the length of the last line. A = 5 and C = 10. 5² = 25 and 10² = 100. Therefore, 25 + b² = 100. 100 - 25 = 75, so b² = 75. This means that b = the square root of 75. The area would be 10*sqrt(75), or about 86.6025.

Hope this helps!! 

The largest shape I could conceive is a semicircle. 

If the corridor is x units long, then the semicircle has the area of \(\frac{\pi x^2}{2}\). This is better than a square.

Subtracting the value of the grants from the projected cost gives the cost that must be paid by the residents:

          $20,066,340 - $5,216,340  =  $14,850,000

If there were 'x' residents, the cost per resident would be:  14 850 000 / x

However, since there are 2200 more residents, the cost per resident would be:  14 850 000 / (x + 2200)

The difference between the two is $75; therefore, we get this equation:

     14 850 000 / x  -  14 850 000 / (x + 2200)  =  75

To get rid of all denominators, multiply each term by  (x)(x + 2200)

     (14 850 000)(x + 2200)  -  (14 850 000)(x)  =  (75)(x)(x + 2200)

Multiplying out:

     14 850 000x + 32 670 000 000  -  14 850 000x  =  75x² + 165 000x

Simplifying:

     32 670 000 000  =  75x² + 165 000x

Dividing all terms by 75:

     435 600 000  =  x² + 2200x

Subtracting  435 600 000  from both sides:

      0  =  x² + 2200x - 435 600 000

Factoring:  (Actually, I would either graph this or use the quadratic formula.)

     0  =  (x - 19 800)(x + 22 000)

So,  x  is either  19,800  or  - 22,000

Since  -22,000 is impossible, x = 19,800

So, the current population is  19,800 + 2,200  =  22,000

   (3x² + 9x - 1) + (-5x² + 6x - 10)

We can get rid of the parenthesees since there is only addition between them.

=  3x² + 9x - 1 + -5x² + 6x - 10

Now we can rearrange the terms so like terms are together.

=  3x² - 5x² + 9x + 6x - 1 - 10

Combine like terms. Just like  3 apples - 5 apples = -2 apples ....so does  3x² - 5x² = -2x²

=      -2x²     +  15x        - 11

(1)

Note that  Distance / Rate  = Time

Let Emily's rate  = r  mph

Let  Ashley's rate  = [r + 1] mph

And let the current rate  = c

So...Emily travels with the current and we have that

6 / [ r + c ] =   1 hour

6 = [r + c]

c = 6 - r

And  Ashley rows against the current .....so

6 / [ r + 1 - c]  = 2 hours

6  = 2 [ r + 1 - c]

6 = 2 [ r + 1 - (6 -r) ]

6  = 2 [ 2r - 5]

6  = 4r - 10

16  = 4r

r   = 4 mph   .....this is Emily's rate

So Ashley's rate  = 5 mph

And the rate of the current  =  6 - r  =  6 - 4  = 2 mph 

(2)  

For a round trip

Emily would take     6 / [ 4 + 2]  + 6 / [ 4 - 2 ]  =  1 + 3 =  4 hrs 

Ashley would take  6 / [ 5 + 2]  + 6 / [5 - 2]  =  6/7 + 6/3  = 6/7 + 2 = 20/7  hrs  ≈ 2.86  hrs

cups of vinaigrette  =  cups of vinegar + cups of oil

Let's call the amount of additional cups of oil  " x " .

There are   3  cups of vinegar   , and there are   1 + x  cups of oil   in the batch so far.

cups of vinaigrette  =  (3) + (1 + x)

cups of vinaigrette  =  4 + x

We want there to be  25  cups of vinegar for every  100  cups of vinaigrette.   That is....

\( \frac{\text{number of cups of vinegar}}{\text{number of cups of vinaigrette}}=\frac{25}{100} \)       Plug in  3  for cups of vinegar and  4 + x  for cups of vinaigrette.

\(\frac{3}{4+x}\,=\,\frac{25}{100} \\~\\ (4+x)(25)\,=\,(3)(100) \\ 100+25x\,=\,300 \\ 25x\,=\,200 \\ x\,=\,8\)                 Now solve this for  x  . First, let's cross-multiply.

So....she needs to add  8  cups of oil .

--------------------------------------------------

The final recipe will have  9  cups of oil. To find the price, we need to know how many ounces this is.

1 cup    =  8 ounces          Multiply both sides by  9  .

9 cups  =  72 ounces

The final recipe will have  3  cups of vinegar.

1 cup    =  8 ounces

3 cups  =  24 ounces

the total price, in dollars   =   0.50 * (ounces of olive oil)  +  0.35 * (ounces of vinegar)

                                         =   0.50 * 72   +   0.35 * 24

                                         =   36   +   8.4

                                         =   44.40

You know you're on a calculator website, right? 295*55 = 16225

There are 4 weeks in every month...so....the first one becomes

332 + 4 * x * 11 =

332 + 44x

For the second one....the additional amount that each player must raise  = x  = 

[ amount left to be raised] /  number of players  = [ 2000 - 1456 ] / 11   = 544 / 11  ≈  $49.45

The product of a number, k, and four is seven more than the quotient of the number and two-thirds. Write an equation to represent this situation.

Product of k  and 4  = 4k

Quotient of  k and 2/3  =  k / [2/3]  =  (3/2)k

So......7 plus the second thing  =  the frist thing....or....

4k  = (3/2)k + 7

A local basketball court dimensions are 50 ft. long, 20 ft. wide, the court is lost square footage by 1/4. What are the new dimensions of the court? Write the equation out then solve. 

If  1/4 of the square footage is lost....then 3/4 must remain

And  L  x W  = Area

So.....the new area  =  [L x W] * (3/4)  =   [50 x  20] x (3/4)   =  1000 x 3/4  =  750 sqare feet

However.....the new dimensions could be variable

If  we assume  that  each dimension was √3/2  that of the old...we have that

(50 * √3/2) * (20 * √3/2)  =

(25 * √3) *  ( 10 √3)  =

(≈ 43.3 ft )  * (≈ 17.32 ft)   =  (new length ) * (new width)  ≈   750 square feet

However....if the problem is claiming that each dimension was reduced by 1/4....the new dimensions are (50 * 3/4) * (20 * 3/4)  =  (37.5 ft)  * (15 ft)  and the new area  is 562.5 square feet

This one was unclear....!!!

The school district has $1045 to throw a company event downtown for 50 people. If the venue cost $320 to rent out, as well as dinner cost $290. The party director wants to give bonuses to everyone with the remainder of the budget.  How much would each employee get as a bonus? Write out the equation, let x stand for the bonus.

Let's first figure out how much remains......this is given by [ 1045 - 320 - 290]

So....this amount divide by the number of people is each person's bonus.....so we have...

[ 1045 - 320 - 290 ] /  50    = x      solving, we have

[ 435 ] / 50   = x  = $8.70  per person

Some big "bonus," huh???

Try typing this into the bar:       asin(0.591)         and press enter.

You should get approximately 36.228º  .

THANK YOU!

\( \[f(x) = \begin{cases} x^2+9 &\text{if }x<-5, \\ 3x-8&\text{if }x\ge-5. \end{cases} \]If $f(x)=10$\)

Find the sum of all possible values of x

If   x^2 +  9  = 10

Then x^2  = 1

And x  =  ± 1     .....   but for this function,  x < -5,  so neither of these are good

If  3x  - 8  = 10

3x  = 18

x  = 6  .......   and  x  ≥ -5

So....the sum of all possible values of x  =  6

 3y = 5x-1     divide both sides by 3

y = (5/3)x  - 1/3

The slope of the given line is in red.....a perpendicular line to this will have a negative reciprocal slope =  -3/5

"D"  looks to be the correct answer....!!!

 -6 + (14 + 2[60 - 9(1 + 3)]) =

-6 + (14 + 2 [60  - 9(4) ] ) =

-6 + (14 + 2 [ 60 - 36 ] )  =

-6  + ( 14  + 2 [ 24] )  =

-6 + ( 14 + 48 )  =

-6 + 62  =

56

(6^2 - 2^2) / 8  =

(36 - 4)  / 8  =

32 / 8 =

4

7/3 + 3 (3/4 - 2/3)^2  =

7/3 + 3 ( 9/12 - 8/12)^2 =

7/3 + 3 (1/12)^2  =

7/3 + 3 (1/44)  =

7/3 + 3/144  =

7/3 + 1/48   =   

[7 * 16 ] / [3 * 16]  +  1/48  =

[ 112] /[48] +  1/48 =

113 / 48

6/7 + 6(1/2 - 1/3)^2  =

6/7 + 6 ( 3/6 - 2/6)^2 =

6/7 + 6 (1/6)^2 =

6/7 + 6 (1/36) =

6/7 + 6/36 =

6/7 + 1/6 =

[ 6 * 6] / [ 6 * 7]  +  [ 1 * 7] / [ 6 * 7]  =

36 /42  + 7 /42  =

43 /42

Let L  be the weight of the large box and S  be the weight of a small box

So.....we have this system

12L +  2S  = 253

 3L  +  5S  = 133

Looking at the second equation  and multiplying both sides by 4, we have that

12L + 20S  = 532   

Which implies that   12L  = 532 - 20S

Sub  this expression for 12L  into the first equation and we have

532 - 20S + 2S  = 253     simplify

532 - 18S  = 253           subtract  532 from each side

-18S  = -279     divide both sides by  -18

S =  279/18   = 15.5 kg

And the large box weighs   

12L  = 532 - 20 (15.5)

L =   [ 532 - 20(15.5) ] / 12  = 18.5 kg

2(2r-1)/6+3=3(r+7)/18

I'm taking literally what you wrote which is

2(2r -1)                     3(r + 7 )

______    +   3   =    _______                multiply through by 18  to clear the fractions

     6                             18

3 [ 2(2r -1) ]  +  54  =  3(r + 7 )          simplify

12r - 6   +  54  =  3r +  21

12r + 48  =  3r + 21          subtract   3r, 48 from  both sides

9r  = -27            divide both sides by 9

r  = -3

We can break this down in parts

The diagonal distance [in the same plane]  across the bottom of the box will be  

f  =sqrt [ (d^2 -a^2)  +  (e^2 - a^2) ]  =  sqrt [ b^2 + c^2] 

And  f^2 =  (b^2 + c^2)

Then....the height of the box  = a =  sqrt [ e^2 - c^2 ]  

And  a^2  = (e^2  - c^2)

So....by the Pythagorean Theorem....g  =  

sqrt [ f^2 + a^2] =  

sqrt [ (b^2 + c^2)  + (e^2 - c^2) ] =  

sqrt [ b^2  + e^2]

Yes thank you, how did you get these answers ?