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11 ÷ 3  =  3 2/3 in.  =  3.67 in. (approx)

What do want done with this function?

x = price of the book; what is the value of x at the vertex?

What would be the price?

The profit is determined by the number of books sold times the price per book.

We want to graph points where the x-value represents the cost per book, n, and where the y-value represents the amount of money taken in.

When n = 0, the amount of money taken in will be 0; again, when n = 72, the amount of money will again be 0.

This graph will be a parabola.

As n increases from 0, to 1, to 2, etc., the amount of money taken in will increase until it hits a maximum at 36; after 36, the amount of money taken in will decrease.

Let y = amount of money taken in:  x = price per book:  y = x(720-10x)

y  = 720 - 10x²

Either graph this and find the maximum point , or find the vertex (the maximum point) by completing the square.

y = -10x² + 720x

y  =  -10(x² - 72x)

y - 12960  =  -(x² - 72x + 1296)        <--- complete the square.

y - 12960  =  -(x - 3)²

Vertex:  (36, 12960)

The graph of this is a parabola that opens downward.  There are at least three ways to find the maximum value.

1)  You can graph it to find its vertex.

2)  If you are in calculus, find the first derivative and set that equal to zero.

     f'(t)  =  -2t + 8     --->     -2t + 8  =  0     --->     -2t  =  -8     --->     t  =  4     --->  

     f(4)  =  -(4)² + 8(4) - 4  =  -16 + 32 - 4  =  12

3)  If you want to do this by algebra: find the vertex by completing the square:

     f(t)  =  -t² + 8t - 4

     f(t) + 4  =  -t² + 8t                     Add 4 to both sides

     f(t) + 4  =  -(t² - 8t)                   Factor out the -1

     f(t) + 4 -16  =  -(t² - 8t + 16)     Complete the square by dividing -8 by 2 and squaring the answer

    f(t) - 12  =  -(t - 4)²

    f(t)  =  -(t - 4)² + 12      --->  vertex at (4, 12)   --->  max value = 12

Use the calculator at this site.

Find common denominators:

(1/7)x - (1/6)x  = (6/42)x - (7/42)x  =  (-1/42)x

(2/3) - (2/9)  =  (6/9) - (2/9)  =  4/9

Answer:  (-1/42)x + (4/9)

If you're having trouble finding common denominators, ask about them.

If the variable is in an exponent, find the log (or ln) of both sides:

log(3125)  =  log(5^e)              With logs, exponents come out as multipliers:

log(3125)  = e x log(5)

log(3125) / log(5)  =  e     --->     e  = 5

If you haven't had logs, think of it this way:

Is 3125 some power of 5?     Not 1, 2, 3, or 4;  but 5^5  =  3125.

Since 3125 = 5^5,  e must be 5.

a)  Relative mins and relative max occur where f' = 0.

     f(x)  =  x³ + ax² + bx + 9   --->   f'(x)  = 3x² +2ac + b

     f'(-1)  =  3(-1)² + 2a(-1) + b  =  0                       f'(3)  =  3(3)² + 2a(3) + b  =  0

                               3 - 2a +  b  =  0                                           27 + 6a + b  =  0

                                    -2a + b  =  -3                                                 6a + b  =  -27

                                                 Combining them:   --->               - (   -2a + b  =  -3     )

                                                                                                               8a  =  -24

                                                                                                                  a  =  -3, b = -9

b)   f'(4)  =  3(4)² +2a(4) + b  =  0    --->   48 + 8a + b  =  0     --->   8a + b  =  -48

     At inflexion point, f''(x)  =  0

                     f''(x)  =  6x + 2a   --->   f''(1)  =  6(1) + 2a  =  0     --->     2a  =  -6

                                                                                                            a  =  -3

                                                                                                            b  =  -24

30% of 100, or

100*0.3

is equal to 30.$$\mathrm{\ }$$

Can you write this in a form that I can write as well?

Not in the complicated way/

CPhill's answer is correct and much shorter than mine.

However, I'm assuming that you have the property of |x/y|  =  |x|/|y| for real numbers and now you are to prove the similar case for complex numbers; that is, when z1 = a + bi and z2 = c + di,

you need to prove that       |(a + bi)/(c + di)|  =  |a + bi| / |c + di|.

By definition:  |x + yi|  =  √(x² + y²)

So (right side):  |a + bi| / |c + di|  =  √[ (a² + b²) / √(c² + d²) ]

   and, since those are all real numbers, it can be rewritten as √[ (a² + b²) / (c² + d²) ].

Now, the left side:

| (a + bi) / (c + di) |

First, we'll need to write this as one complex number in the form x + yi; so let's multiply the numerator and denominator by the conjugate of the denominator, c - di:

| (a + bi)(c - di) / ( (c + di)(c - di) ) |

=  | (ac + bd + bci - bdi) / (c² + d²) | 

Splitting these into the real part and the imaginary part:

=  | (ac + bd)/(c² + d²) + (bc-ad)i/(c² + d²) |

This is now in the form x + yi where x = (ac + bd)/(c² + d²)  and  y = (bc - ad)/(c² + d²).

Thus, the above equals:  √ [ ( (ac + bd)/(c² + d²) )² + ( (bc- ad)/(c² + d²) )² ]

And (ac + bd)²  =  a²c² + 2abcd + b²d²  and  (bc - ad)²  =  b²c² - 2abcd + a²d²

So, the numerator becomes  a²c² + b²d² + b²c² + a²d²  =  a²c² + b²c² + b²d² + a²d²

     = (factoring)   (a² + b²)c² + (a² + b²)d²  =  (a² + b²)(c² + d²)

And the denominator is:  (c² + d²)²

to give the fraction, still under a radical sign:  

      √ [ (a² + b²)(c² + d²) / (c² + d²)² ]  =  √ [ (a² + b²) / (c² + d²) ]

which is what we had for the right side.

Enter:  1.01^40

The two angles are complementary; that is, when their sizes are added together, the sum is 90°.

From a calculator: 

sin(34°)  =  0.5592          sin(56°)  =  0.8290

cos(34°)  =  0.8290         cos(56°)  =  0.5592

tan(34°)  =  0.6745         tan(56°)  =  1.4826

Notice that the sin(34°)  =  cos(56°) and vice versa.

But the tan(34°) = 1/tan(56°) and vice versa.

sec(34°) = 1/cos(34°)     and  sec(56°)  =  1/cos(56°)

csc(34°)  =  1/sin(34°)    and  csc(56°)  =  1/sin(56°)

cot(34°)  =  1/tan(34°)   and  cot(56°)  =  1/tan(56°)

So, with the help of your calculator, you can find those decimal values; this also explains why most calculators do not contain those keys.

Since the sin can never be greater than 1 (but can be 1), the csc can never be less than 1 (but can be 1).

As the sin gets smaller and smaller, the csc gets larger and larger. At its smallest, the sin is zero; at its largest, the csc approaches an infinite size.

What is true for sin, is also true for cos.

It takes three dimensions to have a volume; a circle has only two dimensions.

A circle has an area, but no volume; by default, the volume of a circle is 0.

6(x)(y) + (y)

=  6(3)(4) + (4)                            Do multiplications before addition:

=  72 + 4

=  76

Find the z-score:  z  =  (score - average) / standard deviation

--->  z  =  (68 - 64) / 2.4   --->  z  =  1.6667

Finding normalcdf(1.6667, 100)  =  0.04775  --->   0.048

The general term in this expansion can be written as  nCr(10,10-n)(x²)ⁿ(1/ax)^10-n 

When the power of x is 11 we must have 2n - (10 - n) = 11, which means that n = 7

nCr(10, 10-7) = 120  so given that nCr(10,10-n)/a^10-n = 15 we have  120/a³ = 15 or a³ = 120/15 = 8

So a = 2

.

Ok thank you