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(a)  Call x the additional consecutive number of FTs she needs to make

We have this equation

[12 + x ] / [ 15 +  x ] = .85

(b)   Multiply both sides by  15 + x

12 + x  = .85 [ 15 + x]

12 + x  =  12.75  + .85x      subtract  .85x, 12 from both sides

.15x  = .75     divide both sides by .15

x  = 5  additional consecutive FTs

Let  the total number of skateboards = X

Let the total number of scooters  = Y

And we know that

(1/4)X  = (2/3)Y       multiply both sides by 3/2

(3/8)X  =  Y

So fraction  of  red skateboards and scooters to the total  =

[ 1/4X + (2/3)Y ]  /  [  X + Y  ]  =

[ 1/4X  + 1/4X  ]  /  [ X + (3/8)X ] =

[ (1/2) X ]  / [ (8/8)X + (3/8)X ] =

[   (1/2)X ]  / [ (11/8) X ] =  (cancel the  X's )

(1/2)  / (11/8)  =  (invert the second fraction and multiply )

(1/2) * (8 /11)  =

8 / 22  =

4 /11

The sides of the square must be √6 cm  each

Let x be the distance between the legs of the black triangle and the side of the square

The white area is composed of a square of sides of  x  and two rectangles with dimensions of x and [ √6 - x ]   

So.....the white area  =    x^2 + 2x ( √6 - x )

And the area of the  black triangle  =  ( √6 - x )^2 / 2  = [ 6 - [2√6]x  + x^2] / 2

Since these areas are equal  we have that

x^2 + 2x ( √6 - x )  = [ 6  - [2√6]x + x^2] / 2

2 [ x^2  + 2x (√6 - x) ] = 6 - [2√6]x + x^2 

2x^2 + 4x(√6 - x ]   = 6 - [2√6]x  + x^2

x^2 + [4√6] x - 4x^2  = 6 - [2√6]x

-3x^2 + [ 6√6]x   = 6

3x^2  - (6√6]x  = -6

x^2 - [2√6]x = -2     complete the square

x^2  - [2√6]x  +  6  =  -2 + 6

( x - √6)^2  = 4    take both roots

x - √6  = 2        or     x - √6  = -2

x = √6 + 2       or     x  =  √6  - 2

(reject)                           (accept)

So  ...let the original position of A   = (0, 0)

And the new position  of A  =  ( √6  - ( √6  - 2) ,  √6 - ( √6 - 2) )  = ( 2 , 2)

And its distance from its original position is

√ [ 2^2  +  2^2  ]  =

√8  =

2√2

I will answer this one and you answer the rest of them because they are all similar:

7. A bank offers the following loans: 
 

Bank A

$235,000 
20-year fixed 
4.375% 
1.75 discount points 
M = $1,470.92 

Bank B

$235,000 
20-year fixed 
4.7% 
0 discount points 
M = $1,512.22 

What is the break-even point for the loans? (4 points)

A. about 5 years

B. about 8 years

C. about 6 years

ANSWER:

1.75 discount points is the percentage of fees you pay the lending Bank A to get a reduced interest rate and thereby a smaller monthly payment.

1.75 / 100 =0.0175

0.0175 x $235,000 =$4,112.50 - this is the fee you have to pay to Bank A.

$1,512.22 - $1,470.92 =$41.30 - this is the monthly savings between Bank A and Bank B.

$4,112.50 / $41.30 =99.57 months to get to break-even point.

99.57 / 12 months =~ 8 years to get to the break-even point or B on your list.

This is clearly not what the guest asked, his question is missing something. Why did you mark this with a checkmark?

Why was this question marked with a checkmark?

well there is just one integer between 1 and 3  that is the number 2

there are 5 integers between 1 and 7 they are  2,3,4,5,6

So there must be  1000-1-1 = 998 integers BETWEEN 1 and 1000.

Yes what he said! 

Find the smallest value.

A diagram please ?, or at least a less garbled description of it. 

The following graph should help:

I think something in your question is missing

You must not be asking this correctly.

The decimal representation for any integer terminates.  They have no non-zero digits to the right of the decimal point at all.

\(i \text{ is the value such that }i^2 = -1 \\ \text{you'll often see it written that }i=\sqrt{-1} \\ \text{but this is a bit of abuse of notation}\)

\((7-3i)(5i+4)-(3-i) = \\ 35i+28-15i^2-12i - (3-i) = \\ 28+15+(35-12)i - (3-i) = \\ 43+23i -3 + i = \\ 40+24i\)

\(P=b-1\\ (b-1)^2 = 31_b = 3b+1\\ b^2 -2b+1 = 3b+1\\ b^2 = 5b \\ b=5 \text{ (or 0, but 0 is not a valid base)}\\ b=5, ~P=4\\ (4_5)^2 = 16_{10} = 31_5\)

Correct !!!!

\(\text{Assuming }f(x) \text{ is invertible }\forall x \in \mathbb{R} \\ \text{then }\forall k\neq 0, \exists y = k x \ni f^{-1}(y) = k x \\ \text{i.e. }f(kx) \text{ is invertible.}\\ \text{Thus it must be the only value of k that makes the original statement is valid is }\\ k=0 \\ \text{and the sum of this is just 0} \\ \text{It's true that }\exists f(0) \ni f^{-1}(f(0))=0 \text{ but }f(0\cdot x) \text{ is not invertible}\)

Let's simplify the denominator, first

1/6  +  1 / [x + 1]  =

[ x + 1  + 6 ]  / [ 6(x + 1)] =

[ x + 7]  / [ 6(x + 1) ]

So we have

    5  / 1

_________________       [flip the fraction in the denominator and multiply ]

  [x + 7 ] / [ 6 (x + 1)

5/1  *  [ 6 ( x + 1) ] /  ( x + 7 ]  =

30(x + 1)              30x + 30

_______  =         ________

  x + 7                    x +  7

a)What is the area between the graph of y=6f(x-6) and the x-axis?

The (x - 6) part just shifts the graph to the right by 6 units.....this does not change the area

The  "6"  part  makes every point on the graph 6 times as high..so the area  is

6 * 10   = 60 units^2

b)What is the area between the graph of y=1/2f(-x) and the x-axis?

The (-x)  part rotates the graph about the y axis....the area is unchanged

The (1/2) part makes every point on the graph 1/2 as high....so the area is

(1/2) * 10   =  5 units^2

c)What is the area between the graph of y=f(x+2) and the x-axis?

This just shifts the graph 2 uits to the left....the area is unchanged

Yes...the second answer.  

The slant height of the cone  is  sqrt [ 3^2 + 4^2]  = sqrt [25]  =5

And its lateral surface area is given by pi * radius * slant height  =

pi * 4 * 5  =

20 pi   [units^2]   = A

The lateral surface are of the cylinder is given by

2pi * radius * height  =

2pi * 4 * 3  =

24pi  [units^2 ]  = B

So

A/B  =   20 pi / [24 pi]  =  20 / 24  =   5/6

[ x^2 + 4x + 3 ] / [ 2x - 1 ]

_____________________  =   [ invert the bottom fraction and multiply ]

[ x^2 + x ] / [ 2x^2 -3x + 1 ]

 [ (x + 1) (x + 3) / [2x -1 ]  *  [ 2x^2 - 3x + 1 ] / [x^2 + x]  =

 (x + 1) ( x+ 3) / ( 2x - 1)   *  [ (2x -1) (x -1)] / [ x ( x + 1) ]  =

(x + 1) ( x + 3) / [ x ( x + 1) ]  *  [ (2x - 1) (x - 1) ] / (2x -1)   =

(x + 3) / x   *   (x - 1)  =

(x + 3) (x -1)  /  x

500 = fixed cost

n 8 x 8 = labor cost

n x 5 x3.1 x 8 = money brought in from sales

500 + n 8x8  = n x 5 x 3.1 x 8

            500 + 64n = 124 n

             500 = 60n

               n = 8.33          ~~  9 employees