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If xy  =  3/2, then  y  =  (3/2)x^(-1)

If z  =  10x + (3/5)y,       then  z  =  10x + (3/5)(3/2)x^(-1)

If a minimum exists, it will exist when z' = 0:  z'  =  10 -(9/10)x^(-2)

Set z' = 0:    0  =  10 -(9/10)x^(-2)

                  -10  =   -(9/10)x^(-2)

           100x^2  =  9

                  x^2  =  9/100

                       x  = 3/10

We know the opposite side of a triangle (24), an angle (50 degrees), and we are trying to find the hypoteneuse (the length of the wire).....the sine function is what we need...so we have

sin(50) = 24/H    multiply both sides by H and divide both sides by sin(50).....this gives us

H = 24/sin(50) = about 31.33 feet in length

So we want to minimize  2x^4 + 8x^(-4)

Take the derivative and set this to 0

8x^3 - 32x^(-5) = 0       factor

8x^(-5) [x^8 - 4] = 0    ignore the frst factor, it can never be 0

x^8 - 4 = 0

x^8  = 4        take the ±roots....  x = ±(4)^(1/8)  = about ±1.189

Now, take the second derivative and see which critical points (if any) produce a positive result

24x^2 + 160x^(-6)   and both points will produce positive results, so both are minimums

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)

Nice one, Melody....!!!

Let y = (1/3)x^2  = g(x)

We are looking for the point where y = x + c  is tangent to g(x)

Beause the slope of y = x + c  = 1   we are looking for the point where the derivative of g(x) has a slope = 1.

So

g ' (x)  =  2/3x   ....set this = 1    and x = 3/2

And to find the y coordinate of this point we have  y= (1/3)(3/2)^2  = (1/3)(9/4) = 9/12 = 3/4

So, since the point (3/2, 3/4) is also on y = x + c    we have

3/4 = 3/2 + c      subtract 3/2 from both sides

c = -3/4

Thanks so much!

     (2.5 x 10^10) x (3.5 x 10^-7)     <--- When multiplying, numbers can be arranged in any order.

=  (2.5 x 3.5) x (10^10 x 10^-7)     <--- When you multiply numbers with the same base, add their exponents.

=  8.75 x 10^3

Let x = numer of crows.

Let y = number of cows.

Equation for number of animals:  x = y + 5

Equation for number of legs:      2x + 4y  =  52

Substituting the first equation into the second:  2(y + 5) + 4y  =  52

Distribute:                                                             2y + 10 + 4y  = 52

Simplifying:                                                                    6y + 10  =  52

Subtracting 10 from both sides:                                            6y  =  42

Dividing both sides by 6:                                                           y  = 7   <---   number of cows

Since  x  =  y + 5   --->   x  =  7 + 5   --->                               x  =  12     <---   number of crows

Which expression?

x^2 + 2y^2 = 16   solve for y..... subtract x^2 from both sides

2y^2 = 16 - x^2     divide both sides by 2

y^2 = 8 -x^2/2      take the positive root of both sides

y = (8 - x^2/2)^(1/2)

Now  we wish to maximize xy....so we have

x(8 - x^2/2)^(1/2)    take the derivative and set this to 0

(8 - x^2/2)^(1/2) -(1/2)(x^2)(8 - x^2/2)^(-1/2) = 0   factor

(8- x^2)^(-1/2)[(8 - x^2/2) -(1/2)(x^2)] = 0

So we have

[(8 - x^2/2) -(1/2)(x^2)] = 0

8 - x^2 = 0

x^2 = 8

x =√8  = 2√2

So y = (8 - 8/2)^(1/2) = (8 - 4)^(1/2) = (4)^(1/2) = 2

And the max product =   (2√2)(2) = 4√2 = about 5.65685

$$\\y=2(x+4)(x-2p) \qquad p\in R\\
$This is a concave up parabola$\\
$Roots are $ x=-4 \;\;and\;\;x=2p\\
$Axis of symmetry is $ x=(2p+-4)/2 = p-2\\
Min:\\
y=2(p-2+4)(p-2-2p)\\
y=2(p+2)(-2-p)\\
y=-2(p+2)^2\\
$You want this value to be $-18\\
-18=-2(p+2)^2\\
9=(p+2)^2\\
p+2=\pm3\\
p=\pm3-2\\
p=1 \quad or \quad p=-5$$

The Baking store is actually the name. My sister put that.

The domain holds all teh numbers that you enter for x.

When x = 0:  y  =  (1/2)(0) + 1  =  1

When x = 1:  y  =  (1/2)(1) + 1  =  1.5

Continue this, for x = 2, 3, 4, and 5.

Your answer will be the set of all these answers.

ln(2x + 3) = 7

In exponential form:  2x + 3  =  e^7

So:                                   2x  =  e^7 - 3

and:                                   x  =  (e^7 - 3) / 2

                                          x  =  546.817

Density is found by dividing the mass by the volume.

Sinc you know a point and a slope, use the point-slope form:  y - y1  =  m(x - x1)

x1 = -1     y1 = -3     m = -3                                  --->       y - -3  =  -3(x - -1)

                                                                                        y +31  =  -3(x + 1)

                                                                                        y + 3  =  -3x - 3

                                                                                              y  =  -3x - 6

Use the distributive property on both sides:

-3(5 - 15n)  =  (-3)(5) - (-3)(15n)  =  -15 + 45n

15(3n - 2)  =  (15)(3n) - (15)(2)  =  45n - 30

-3(5 - 15n)  =  15(3n - 2)

-15 + 45n  =  45n - 30

Subtract 45n from both sides:

-15  =  -30

When the variable disappears, there are two cases:

if the left side equals the right side, then all numbers work;

if the left side doesn't equal the right side, then no numbers work.

Since the left side doesn't equal the right side, no numbers work; your answer is the empty set.

Its simple mental math. Lets say you put 1 in place of x.

That would mean 2.1+3=7. Thats not possible.

Okay, now substitute 2 for x. That would mean 2.2+3 =7.

That should make much more sense.

You have the slope a little incorrect ...we have   (-13 - -1) /(-6-6) = (-13 +1) /(-12)  = -12/-12  = 1....whew!!! that makes things easier !!!  so we have.......

y - (-13) = (1) ( x - (-6))    simplify

y + 13 = x + 6        subtract 13 from both sides

y = x + 6 - 13      simplify

y = x - 7     that's slope-intercept form   .........add 7 to both sides and subtract y from both sides

7 = x - y  →    x - y = 7     and that's standard form

Hope that helps.....BTW.....you can "plug" in the points to make sure the equation is correct....I think you will find it is....

Whatever wood it would chuck

1+1=2.  Do you REALLY need to know this?

Guys. Come on. It's 1.

Multipy the total by 0.08, and then subtracts that answer from the total.

Remeber that a percent is a number out of 1. If you tipped, 15%, then you would multiply by 0.15.