CPhill

2)The face diagonal of a cube has length 1. Find the surface area of the cube.

The side  is     1/√2

So....the surface area  =  6*side^2   =  6 * (1/√2) ^2   =  6* (1/2)  =  3 units^2

9.   The area  is   (3x + 2)  (2x)  =   [ 6x^2 + 4x ]  units^2

10. 3x³ - 4x² + 2x + 1 - (3x³ + 4x² - 2x + 1)  =

3x^3  - 4x^2  + 2x  + 1  - 3x^3 - 4x^2 + 2x  - 1       combine like terms

-8x^2 + 4x

3.

Let the side of the hexagon  = S

Then....the area  of the hexagon =  6(1/2)S^2*(√3/ 2)  = 3√3/2 S^2  units^2  

Now ...to determine the length of Fido's leash, L, we have

(1/2)S             L

_____   =      ______

sin (30)         sin (60)

(√3/2)S  =  L

So....the area  that Fido can cover is     pi  (√3/2S)^2  =  (3/4)pi*S^2 units^2

So  the fraction of the yard that Fido can cover is

(3/4)pi*S^2               (3/4)  pi                    6 pi               pi          √3

___________  =    ___________  =    ______  =    _____  =   ___  pi

3√3/2 S^2                  3√3/2                  12√3               2√3          6

So   a  = 3   and b  =  6     

And the product ab  is 6*3   =  18

EDIT TO CORRECT AN ERROR....!!!

2.

Since ABE is equilateral....then angle ABE  = 60°

And since ABCD  is a square, then angle EBC =  90 - 60  = 30°

And since AC is a diagonal of the square, then  angle ACB  = 45°

Then, in triangle BPC,   angle BPC  must be   180  - 30 - 45  =  180  - 75  = 105°

1.  Let the largest angle   = a₅

Let the smallest angle   =  a₁

So

a₅ = a1  + 4d

a₅ - a₁  =  4d

But  a₅ - a₁  = 44

So  d  = 11

So the sum of the angles = 540

a₁ +( a₁+ d)  + (a₁ + 2d) + (a₁ + 3d) + (a₁ + 4d)   =  540

5a₁ + 10d  =  540

5a₁ + 10(11) = 540

5a₁  + 110   = 540

5a₁  = 430           divide both sides by 5

a₁  = 86°  =   the smallest angle

1/7  of the deer caught  had tags

Thus....we might assume that the 90 tagged ones represent 1/7  of  the total poplation

So....the total population  is    7 * 90   =     630

sin (xy)  = x + y

Use the Chain Rule on the left

cos(xy) *[ xy' +  y ]  =  1  + y'

cos(xy) * xy' + cos(xy)*y  =  1 + y'      ....collect the y' terms on the left.....

xy' cos(xy) - y'  =   1  -  y* cos(xy)

y '   [ x*cos (xy)  - 1 ]  =  1 - y* cos(xy)

y ' =    [  1  - y* cos(xy) ]

          _____________

              x* cos(xy) - 1

Thanks  ....!!!!

y  =   (-5/8)x^2 - 3x  + 4

y  - 4   =  (-5/8)x^2  - 3x       complete the square on the right

y  - 4  =  (-5/8)  [ x^2  +  24/5x + 576/100  -  576/100]

y  - 4  =  (-5/8) [ x + 24/10]^2  + 18/5

y - 4  -  18/5  =  (-5/8)[x + 24/10]^2

(y  - 38/5)  = (-5/8)[ x + 24/10]^2     multiply both sides by -8/5

(-8/5) (y - 38/5)  = ( x + 24/10)^2

The vertex is (   -24/10, 38/5)  =    ( -12/5, 38/5)

Using the form

4p (y - k)  =  (x - h)^2

4p   =  -8/5

p  = -2/5

This parabola  turns downward so the equation of the directrix  is

y  = 

38/5 - p    

38/5 -  (-2/5)   =   

8

So    y  =  8 

r  =      1  / [3 + 3 sin θ]

3r  =  1 / [ 1 + sinθ]

3r  =  1 / [ 1  +  y /r]

3r  =  r /  [ r + y]

r + y  =  1/3

r  =  (1/3)  - y

√[x^2 + y^2]  =  (1/3) - y       square both sides

x^2  + y^2   =  1/9  - (2/3)y + y^2

x^2  =  1/9 - (2/3)y

9x^2 =  1  + 6y

6y  =   -9x^2 + 1

y  = (-3/2)x^2  + 1/6

This is a parabola  ...  see  here  :  https://www.desmos.com/calculator/m0v1m8zsrd