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Greetings to you, too :)

1)Tim Worker expects Social Security benefits of $937 per month. His wife, Mary, is eligible for a wife's benefit of 50 percent of Tim's benefit. How much will Mary receive?

50%  of  937  =  .50 * 937  = $468.50

2) The Brown family's dinner bill was 75.89 and they left 10.00 as a tip. What percent was the tip?

Assuming that 75.89 was the total bill.....the percent of the tip is given  by :

10.00 / 75.89  ≈ 13.2 %

As for the third question....I don't know.....!!!

Hello  \(X^2\)

Neither of us is wrong.
Right, who puts brackets to represent the meaning of his term.
greetings :)

If your calculator evaluates the expression \(6/2(2+1)\) as 1, you need to use a different calculator. 

LOL!!!!!!......we'll see......don't get too cocky....!!!!

I think we've worn this one out.....

By the order of operations......multiplication and  division arre carried out at the same level of precedence......that is....we evalutate them from left to right as we encounter them....so...

6/2 (2 + 1) .......we encounter the division of 6/2, first  ...and this =  3

3 ( 2 + 1)      simplifying inside the parentheses

3 (3)  = 9

Solve the separable equation ( dy(x))/( dx) = (y(x) + 1) cos(x):
Divide both sides by y(x) + 1:
(( dy(x))/( dx))/(y(x) + 1) = cos(x)
Integrate both sides with respect to x:
 integral(( dy(x))/( dx))/(y(x) + 1) dx = integral cos(x) dx
Evaluate the integrals:
log(y(x) + 1) = sin(x) + c_1, where c_1 is an arbitrary constant.
Solve for y(x):
y(x) = e^(sin(x) + c_1) - 1
Simplify the arbitrary constants:
Answer: | y(x) = c_1 e^(sin(x)) - 1

y = e^x cos (x)

y '  = e^x cos(x)  - e^x sin(x)

y "  = e^x cos (x)  - e^x sin (x)  - e^x sin (x)  - e^x cos(x)  =  -2 e^x sin (x)

So

y"   -  2y' + 2y  =

 -2 e^x sin (x)  - 2 [  e^x cos(x)  - e^x sin(x) ] + 2 [ e^x cos(x) ]   =  0

f " (x)  = 2x   f ' (1)   = 2, f (0)  = -1

f "(x)   = 2x        integrate to find the first derivative

f ' (x) =  x^2  + C₁      applying the initial condition to find C₁ ,  we have that

f '( 1)  = 2

2  =    (1)^2 + C₁

2  = 1 + C₁

1  = C₁

So....  f'(x) =  x^2 + 1

Integrate this again to find y(x)

f(x)  = (1/3)x^2 + x  + C₂

And applying  the  second initial condition to find C₂, we have that

f(0) = -1

-1 = (1/3)(0)^3 + 0 + C₂

C₂  = -1

So.....   f(x)  =  (1/3)x^3 +  x   - 1

Thanks CPhill! I'll have to remember that a line like that always splits those sides into equal ratios...

BTW....What does LSU have in common with a sand castle ????

...They both look good until the Tide rolls in !!!!!!!!!!!!!!    

x + y = 9      →       x  =  9 - y      (1)       note......this isolates x

-10x + 6y = 6     (2)

Sub (1)  into (2)

-10 ( 9 - y) + 6y  = 6     simplify

-90 + 10y + 6y  = 6

-90 + 16y  = 6      add 90 to both sides

16y = 96     divide both sides by  16

y  = 6          and using (1),    x =  9 - 6   =   3

y(1+x^2)y'-x(1+y^2) = 0     this is a separable equation

y ( 1 + x^2)y'  =  x ( 1 + y^2)   rearrange as

[ y] / [ 1 + y^2 ] y'  =  [x] / [ 1 + x^2]   integrate both sides

∫ y / [ 1 + y^2] dy  = ∫ x / [ 1 + x^2]  dx

(1/2) ln ( 1 + y^2)  =   (1/2) ln ( 1 + x^2) + C

ln (1 + y^2)  =  ln ( 1 + x^2 ) + C    exponentiate both sides

e^[ln (1 + y^2) ] =  e ^ [  ln ( 1 + x^2 ) + C]

1 + y^2  =  e^[ ln ( 1 + x^2)] * e^C       →     let e^C   =  C₂

1 + y^2  =  (1 + x^2) * C₂

y^2  = ( 1 + x^2) *C₂  - 1

y = ± √  [   ( 1 + x^2) *C₂  - 1 ]

y (x) =   ± √  [   C₂x^2 + C₂  - 1 ]

Now....since the intial condition is that   y(0)  = √3  .....the positive root must be correct....solving for C₂, we have that

√3  =   √  [   C₂(0)^2 + C₂  - 1 ]

√3 = √ [ C₂ - 1]

3  = C₂ - 1

C₂ = 4

So....the particular solution is   y(x)  = √  [   4x^2 + 4  - 1 ]  =   √  [  4x^2 + 3 ]

Here's another way to determine t  with polynomial long division

                                 4x^2  -  6x  + 10

5x^2 - 6x + 7     [     20x^4 -  54x^3  + 114x^2  - 102x  + 70  ] 

                                20x^4 - 24x^3  +    28x^2

                                ________________________________

                                             - 30x^3 +  86x^2  - 102x   

                                            - 30x^3  + 36x^2  -    42x

                                            _____________________  

                                                            50x^2  -  60x  + 70

                                                            50x^2 -  60x   + 70

                                                           ___________________

So...it appears that  t =  -6

Assuming that the side lengths correspond.....note that the ratio of the first two sides in each triangle = 18 / 12  = 3 / 2

So.....x   will  =  (3/2) 14 =   21 cm

I hope you meant solve for the variable t, because this took me too long for me to get mad if that's not what you meant

(5x2−6x+7)(4x2+tx+10)=20x4−54x3+114x2−102x+70

Step 1: Add -70 to both sides.

5tx3+20x4−6tx2−24x3+7tx+78x2−60x+70+−70=20x4−54x3+114x2−102x+70+−70

5tx3+20x4−6tx2−24x3+7tx+78x2−60x=20x4−54x3+114x2−102x

Step 2: Add -78x^2 to both sides.

5tx3+20x4−6tx2−24x3+7tx+78x2−60x+−78x2=20x4−54x3+114x2−102x+−78x2

5tx3+20x4−6tx2−24x3+7tx−60x=20x4−54x3+36x2−102x

Step 3: Add -20x^4 to both sides.

5tx3+20x4−6tx2−24x3+7tx−60x+−20x4=20x4−54x3+36x2−102x+−20x4

5tx3−6tx2−24x3+7tx−60x=−54x3+36x2−102x

Step 4: Add 60x to both sides.

5tx3−6tx2−24x3+7tx−60x+60x=−54x3+36x2−102x+60x

5tx3−6tx2−24x3+7tx=−54x3+36x2−42x

Step 5: Add 24x^3 to both sides.

5tx3−6tx2−24x3+7tx+24x3=−54x3+36x2−42x+24x3

5tx3−6tx2+7tx=−30x3+36x2−42x

Step 6: Factor out variable t.

t(5x3−6x2+7x)=−30x3+36x2−42x

Step 7: Divide both sides by 5x^3-6x^2+7x.

t(5x3−6x2+7x)5x3−6x2+7x=−30x3+36x2−42x5x3−6x2+7x

t=−30x2+36x−425x2−6x+7

Answer:

t=−30x2+36x−425x2−6x+7

If they from an arithmetic progression...let the smallest angle  = x    and the largest angle  = x + 44

So...we have that

x +  4d  =  x + 44      where d is the common differnce between consecutive angles

4d  = 44

d  = 11

And the interior angles of a pentagon will sum to 540°....therefore

x + (x + 11) + (x + 22) + ( x + 33) + (x + 44)  = 540    simplify

5x + 110  = 540    subtract 110 from each side

5x = 430      divide both sides by 5

x = 86°   and this is the smallest angle

The other angles are   97°, 108°, 119° and 130°

And [ 86 + 97 + 108 + 119 + 130 ] =   540

Here's another way without using vectors.....let B  = (x, 0)

If the right angle is at A.....the hypotenuse is BC....and this distance  is  just  

sqrt [ (7 - x)^2 + (3 - 0)^2 ] =

sqrt [ 49 - 14x + x^2 + 9 ]   =  sqt [ 58 - 14x + x^2 ]

And AB  forms one of the legs...and its length is just    sqrt [ (-2 - x)^2 + (6 -0)^2 ] =  

sqrt [ ( 4 + 4x + x^2 + 36 ] =

sqrt [ 40 + 4x + x^2 ]

And AC  is the other leg....and its  length is just  sqrt [ (-2 - 7)^2 + (6 -3)^2 ] =

sqrt [ 81 + 9 ]  = sqrt [90]

And by the Pytahgorean Theorem.........AB^2 + AC^2  = BC^2   ....so...

[ 40 + 4x + x^2 ]  +  90    =  [ 58 - 14x + x^2 ]   simplify

130 + 4x  =  58 - 14x          subtract 4x, 58 from both sides

72 = -18x             divide both sides by -18

-4  = x       so ....B  = (-4, 0)

Obviously.....vectors make the process easier.....!!!!

Area of a sector  =  theta * pi * r^2 / 360   .....so...we have

30  =  theta * pi * (225) /360  simplify

30  = theta  * pi  * (5/8)      divide both sides by   pi * (5/8)

30 / [ pi * 5/8]  = theta  ≈  15.3° 

Yep....that makes it way easier.....!!!!

Excellent thinking, hectictar....!!!!!....this one was a little tricky, for sure....!!!!!

Here's one more thought......dropping a perpendicular from Z to H on  AC means that  ZH is parallel to BA....and whenevever   a segment is drawn parallel to a base, it splits the sides of the  triangle into  equal ratios....that is.....CZ / BZ  =  CH / AH.....but since BZ  = CZ  [ because Z is the midpoint of BC ], then  AH  = CH

Then by SAS, triangle CHZ   is congruent to triangle AHZ.....and since you found that BCA  = 32°  = ZCH  =  ZAH  = ZAC

Obviously.....dropping that perpendicular as you did was the key to the whole thing.....not bad for a 'Bama fan....LOL!!!!

Wow sir you are a genius

Thank you for your response

May you wear your sunglasses with pride 

Sonji has $5.25 in nickels and dimes.The coin value can be modeled by the equation 0.10d + 0.05n = 5.25, where d represents the number of dimes and n represents the number of nickels. If Sonji has 21 dimes, how many nickels does she have?

.10 (21)  + .05 n  = 5.25

2.10  + .05n  = 5.25   subtract  2.10 from both sides

.05n  = 3.15           divide both sides by .05

n  = 3.15 / .05    =    315 / 5  =  63  nickels

Here's another way :

The first and last terms sum to 26....so....the sum is given by

[ Sum of first and last term] [ Indices of last term  - indices of first term + 1 ]  / 2  =

[26 ] [7 - 3 + 1 ] / 2  =

[26] [ 5] / 2  =

[26 / 2 ] * 5   =

13 * 5  =

65