CPhill

We can find the remainder by substituting x = 4 into the function....so we have

(4)^3 - 7(4) - 6  =

64 - 28 - 6  = 

30

The area of the pool  = 24 * 16 = 384 sq ft

So the total area = twice this = 768 sq ft

Call the patio width, x

So.....the whole area can be represented by

[24 + 2x] * [ 16 + 2x]  = 768     simplify

384 + 80x + 4x^2  = 768        divide through by 4

96 + 20x + x^2  = 192       subtract 192 from both sides and rearrange

x^2 + 20x - 96   = 0        factor

[x + 24] * [ x - 4]   =  0       set each factor to 0 .....   and we have that

x = -24  or x  = 4       reject the negative answer

So.....the patio width = 4 ft

Let the smaller angle = x

And the larger = 11x

So.....

x + 11x   = 180    simplify

12x  = 180         divide both sides by 12

x = 15°    this is the smaller angle

And the larger angle  =  11 (15°)    = 165°

Since  x^2 + y^2  = 1     then

(1/4)^2 + y^2   = 1

1/16 + y^2  = 1

y^2   = 1 - 1/16

y^2  = 15/16       ..... take pos/neg square roots........

y = ± √(15/16  = ± √(15)/4

But....y is negative in this quadrant, so the answer is :  - √(15)/4

(65/11)x + 15 - (10/11)x - 19   =

(65/11)x - (10/11)x + 15  - 19   =

(55/11)x  - 4  =

5x   -  4

The chance that any die shows a prime number = 4/10......so we have

C(4,2) * (.4)^2 * (.6)^2   =   28/45  = about 62.22%

30= -6(k+4)    divide both sides by -6

-5 = k + 4     subtract 4 from both sides

-9  = k

x^2+8y=16
 3x+4y=12

Multiply the second equation by 2  .....this gives us     6x + 8y  = 24

Rearrange this to   ....    8y = 24 - 6x      and substitute this into the first equation....so we have

x^2 + [24 - 6x] = 16     simplify

x^2 - 6x + 8  = 0       factor

(x - 4) (x -2) = 0     and setting both factors to 0, we have that x = 2 and x = 4

And using the second equation to find y in both cases

3(2) + 4y = 12              3(4) + 4y  = 12

6 + 4y = 12                   12 + 4y   = 12

4y = 6                             4y  =  0

y = 3/2                             y = 0

So....the intersection points are (2, 3/2)  and ( 4, 0 )

Here's a graph  :  https://www.desmos.com/calculator/ac9ayppsom

The sum of these probabilities gives us the answer

P(5 heads)  = C(7,5) * (.5)^5 * (.5)^2  +

P(6 heads) = C(7,6) * (.5)^6 * (.5)^1 +

P(7 heads)  = (.5)^7

=  29 / 128     =  about 22.66%