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thank you alan

100/2=50

lower bound of 6400kg = 6350kg

10/2=5 

upper bound of 90kg = 95kg

6350kg/95kg=66kg

The angle between them (call it c) is 125°

At 12 knots, after 2 hours ship A will have travelled A = 24 nautical miles.

At 16 knots, sfter 2 hours ship B will have travelled B = 32 nautical miles.

Cosine rule:   C² = A² + B² - 2ABcos(c)

so distance between ships is C = sqrt(24² + 32² -2*24*32*cos(125°))  nautical miles.

I'll leave you to crunch the numbers.

The four points  A(-4,0) ,  B(0,-4) ,  X(0,8) ,  and  Y(14,k)  are grouped on the Cartesian plane.

If segment AB is parallel to segment XY, then...

the slope of AB   =   the slope of XY

\(\frac{-4-0}{0--4}\,=\,\frac{k-8}{14-0}\\~\\ \frac{-4}{4}\,=\,\frac{k-8}{14}\\~\\ -1\,=\,\frac{k-8}{14}\\~\\ -1\cdot14\,=\,k-8\\~\\ -14\,=\,k-8\\~\\ -14+8\,=\,k\\~\\ -6\,=\,k\)

The answer is D, not A-

If f(x)=x³, f'(0)=0 but f is strictly increasing.

4.   Let  f  be a function such that   f(x + y)  =  x + f(y)   for any two real numbers  x  and  y .

      If  f(0)  =  2  ,  then what is  f(2012)  ?

f(x + y)  =  x + f(y)

f( -2012 + 2012 )   =   -2012 + f(2012)

f(0)   =   -2012 + f(2012)

2   =   -2012 + f(2012)

2 + 2012   =   f(2012)

2014   =   f(2012)

I don't know....you're always pretty good with these function problems.....!!!

Ah...thanks..but....my final answer happened to be right but my working out was all messed up!!  I had to fix it!!

Very nice, hectictar  !!!!

2.   Let   f(x)  =  3x² - 4x  .  Find the constant  k  such that   f(x)  =  f(k - x)   for all real numbers  x .

f(x)   =   f(k - x)

                                                 And   f(x)  =  3x² - 4x

3x² - 4x   =   f(k - x)

                                                 And   f(k - x)  =  3(k - x)² - 4(k - x)

3x² - 4x   =   3(k - x)² - 4(k - x)

3x² - 3(k - x)² - 4x + 4(k - x)   =   0

3[x² - (k - x)²] - 4[x - (k - x)]   =   0

3[x + (k - x)][x - (k - x)] - 4[x - (k - x)]   =   0

3[x + k - x][x - k + x] - 4[x - k + x]   =   0

3[ k ][ 2x - k ] - 4[ 2x - k ]   =   0

( 2x - k )( 3k - 4 )   =   0

2x - k  =  0      or      3k - 4  =  0

k  =  2x                      k  =  4/3

The constant value that works is   k  =  4/3

1.   The function  \(f\)  satisfies   \(f(\sqrt{x+1})=\frac1x\)   for all   \(x\geq-1\,,\quad x\neq0\) .   Find  \(f(2)\) .

We want to find an  x  value such that...

\(\sqrt{x+1}=2\\~\\ x+1=4\\~\\ x=3\)

So......

\(f(\sqrt{x+1})=\frac1x\\~\\ f(\sqrt{3+1})=\frac13\\~\\ f(\sqrt{4})=\frac13\\~\\ f(2)=\frac13\)

OK....good job  !!!

Thank you so much! I actually managed to work this out before your reply and realized I was just overthinking it!

You should get more than 1 point here, hectictar....!!!

The coefficient is the number in front of the variable  =  -13.1

x^2  - k^2   =  (x + a) (x - b)

(x + k) ( x - k)  =  (x + a) ( x - b)

So   a  =   k     and   b  = k

So

a - b   =  k - k   =    0

47.32 / 3.79   ≈  12.485   gal 

2.

m∠A  +  m∠C   =   180°

(2x + 9)°  +  (3x + 1)°   =   180°

2x + 9 + 3x + 1   =   180

5x + 10   =   180

5x   =   170

x   =   34

m∠A   =   (2x + 9)°

m∠A   =   (2(34) + 9)°

m∠A   =   (68 + 9)°

m∠A   =   77°

3.

m∠B  +  m∠D   =   180°

(4x - 20)°  +  x°   =   180°

4x - 20 + x   =   180

5x - 20   =   180

5x   =   200

x   =   40

m∠B +  m∠D   =   180°

m∠B  +  40°   =   180°

m∠B   =   140°

4.

m∠D  +  m∠B   =   180°

x°  +  148°   =   180°

x°   =   180°  -  148°

x°   =   32°

x   =   32

m∠A   =   (2x + 1)°

m∠A   =   (2(32) + 1)°

m∠A   =   65°

5.

m∠B + m∠D   =   180°

(x + 10)° + (x + 24)°   =   180°

x + 10 + x + 24   =   180

2x + 34   =   180

2x   =   146

x   =   73

m∠A  +  m∠C   =   180°

(x + 15)° + m∠C   =   180°

(73 + 15)° + m∠C   =   180°

88° + m∠C   =   180°

m∠C   =   180° - 88°

m∠C   =   92°

4 terms

3 coefficients   { 5, 2 and  -3 }

Remember that 6%  = .06

Let the total  cost be  T...so...

T  =  C +  .06C        we can simplify this as

T  =  C (1 + .06)

T  = C (1.06)

Let  x  be the first integer

x + 2  the second

x + 4 the third

x + 6  the fourth

So  we have

x  + (x + 2)  + ( x + 4) + ( x + 6)  < 192  simplify

4x + 12 <   192     subtract  12  from  both sides

4x < 180     divide both sides by  4

x < 45

So......the greatest even integer < 45  is 44

So...the four consecutive even integers   are  44  46   48   and 50

And their sum is 188

I think these might all be the same type of problem

2. Area  =  (1/2)(13) (7.5)sin(94)  =  48.6 in^2

3.  Area  = (1/2) (4) (8.5)sin (25)  = 7.2 m^2

4.  Area   =   (1/2)(4) (5) sin (79)   =  9.8 ft^2

5. Area   = (1/2) (40) (25) sin (80)  = 492.4 m^2

What do you need to calculate, Sam???...area or volume    ???