hectictar

The third question says, "Find the area of the parallelogram."

So we know that the shape is a parallelogram, and we know that the opposite sides are congruent.

That means the unlabeled side lengths would be labeled like this:

And...

the perimeter, in ft   =   26 + 23 + 26 + 23   =   98

Remember, the perimeter of any polygon is the sum of the lengths of its sides.

So to find the perimeter of  (a)  and  (b) ,  add up the lengths of the sides.

I'll do  (b) :

perimeter of shape  b  , in meters   =   3.4 + 3 + 2 + 2.8 + 3.8

perimeter of shape  b  , in meters   =   15

The answer to part  (b)  is  15 .

Can you answer part (a)? To get the answer, add up the lengths of the four sides.

30y² + x    This is a polynomial.

30y² + x^1/2    If this your expression, it is not a polynomial because the exponent of x is a fraction.

(30y²)/x    This is not a polynomial because  x  is in the denominator.

30y²    This is a polynomial.

550 dollars per year /  12 months per year

=   550/12 dollars per month

≈   45.83 dollars per month

4x + 5a + 4y + 5b

                                   Numbers can be added in any order, so we can rearrange the expression.

=  4x + 4y + 5a + 5b

                                   Now we can factor  4  out of the first two terms and  5  out of the last two terms.

=  4(x + y) + 5(a + b)   Notice that if we distributed the  4  and the  5 , we would get the last expression.

                                   Since  x + y  =  -2 ,  we can substitute  -2  in for  x + y .

=  4( -2 ) + 5(a + b)

                                   Since  a + b  =  -2 ,  we can substitute  -2  in for  a + b .

=  4( -2 ) + 5( -2 )

=  -8  +  -10

=  -18

Find   $\left(\frac{1+i}{\sqrt{2}}\right)^{46}$ .

Let's convert  1 + i  to polar form so we can use de Moivre's Theorem.

$1+i=\sqrt2(\cos\frac{\pi}4+i\sin\frac{\pi}4)$

Now lets raise both sides of that equation to the power of  46 .

$(1+i)^{46}=[\sqrt2(\cos\frac{\pi}4+i\sin\frac{\pi}4)]^{46}\\~\\ (1+i)^{46}=(\sqrt2\,)^{46}(\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}$

And divide both sides by  $(\sqrt2\,)^{46}$

$\frac{(1+i)^{46}}{(\sqrt2\,)^{46}}=(\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}\\~\\ \left(\frac{1+i}{\sqrt{2}}\right)^{46}=(\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}$

De Moivre's Theorem says....

$(\cos(x)+i\sin(x))^n=\cos(nx)+i\sin(nx)$    So....

$(\cos\frac{\pi}4+i\sin\frac{\pi}4)^{46}=\cos(\frac{46\pi}{4})+i\sin(\frac{46\pi}{4})\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=\cos(\frac{46\pi}{4})+i\sin(\frac{46\pi}{4})\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=\cos(\frac{23\pi}{2})+i\sin(\frac{23\pi}{2})$

Now let's find a reference angle that is coterminal with  $\frac{23\pi}{2}$  by subtracting  $2\pi$  five times.

$\frac{23\pi}{2}-2\pi-2\pi-2\pi-2\pi-2\pi\,=\,\frac{23\pi}{2}-10\pi\,=\,\frac{23\pi}{2}-\frac{20\pi}{2}\,=\,\frac{3\pi}{2}$

So....

$\left(\frac{1+i}{\sqrt2}\right)^{46}=\cos(\frac{3\pi}{2})+i\sin(\frac{3\pi}{2})\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=0+i(-1)\\~\\ \left(\frac{1+i}{\sqrt2}\right)^{46}=-i$

To make your fractions what you mean, you need to put parenthesees around each numerator and denominator.

Every fraction should be in this format:  ( stuff in numerator )/(  stuff in denominator  )

For example,     $\frac{x^2-x-2}{x-3}$    must be written with parenthesees like this:     (x^2 - x - 2)/(x - 3)

I can probably guess what your fractions are supposed to be, but since I already started making a point about this.....I don't want to just give it up.  

-3(y - 1)  =  9

                          First let's solve this equation for  y . Divide both sides by  -3 .

y - 1  =  9 / -3

y - 1  =  -3

                          Add  1  to both sides of the equation.

y  =  -3 + 1

y  =  -2

Now that we know  y  =  -2  ,   we can find the value of  $\frac12$y .

$\frac12$y  =  $\frac12$(-2)

$\frac12$y  =  -1

5280 ft / (2$\frac12$ ft per stride)   =   5280 ft / (2.5 ft per stride)   =   2112 strides

So Jim must take  2112  strides to go exactly one mile.

num of strides Jim takes  =  2112

5280 ft / (2$\frac23$ ft per stride)  =   5280 ft / ($\frac83$ ft per stride)   =   1983.75 strides

So Jeffrey must take 1983.75 strides to go exactly one mile.

num of strides Jeffrey takes  =  1983.75

ratio of the num of strides Jim takes to the num of strides Jeffrey takes

=   num of strides Jim takes / num of strides Jeffrey takes

=   2112 / 1983.75

=   211200 / 198375

=   2816 / 2645

However...

if Jeffrey cannot take a fraction of a stride, he must take 1984 strides to reach a mile. If so...

num of strides Jeffrey takes  =  1984

And...

ratio of the num of strides Jim takes to the num of strides Jeffrey takes

=   num of strides Jim takes / num of strides Jeffrey takes

=   2112 / 1984

=   33 / 31

Okay, thanks for editing it! But just to make sure.....

(3/2y) + (5/3y^2)   means   $(\frac32y)+(\frac53y^2)$

(1/x-3) - (4/x-5)   means   $(\frac1x-3)-(\frac4x-5)$

(x^2-x-2/x-3) times (x^2-9/x+1)   means   $(x^2-x-\frac2x-3)(x^2-\frac9x+1)$

(3/p + q/6) / (2q/3p)   means   $\frac{\frac3p+\frac{q}6}{2\frac{q}3p}$

....Are these what you meant?