CPhill

Good job !!!

Let us find the area for 830 bricks

14  ft 9 in   =14. 75 ft

So  14.75  *  6  =   88.5 sq ft

The required wall square footage  = 

36 ft 6 in  = 36.5 ft  * 6ft   =  219 sq ft

So...the number of bricks required for the new wall can be found by a proportion

N / 219  = 830  / 88.5   multiply both sides by 219 

N =  219 * 830 / 88.5   ≈  2054 bricks    

It will take

[2054/ 300 ] hrs to construct  the wall

And the hourly rate  is  4000/8  = $500

So...the total cost   ≈  [2054/ 300] * 500   = $ 3423.16

The third answer  seems the closest   [the difference is due to rounding by me ]

A. deductible

Just add the amounts on lines 4,5 and 6 together, NSS

That will be your answer

1. The tangent to the circle x^2 - 2x + y^2 = 15 at(1,4)

Implicitly differentiating the function, we can find the slope of a tangent line to the circle

2x - 2 + 2yy'  = 0     

y'  =  [2 - 2x ] / [2y]

y' = [1 - x] / y

At (1,4)....the slope is

[1 - 1 ] / 4  = 0

So...the equation of the tangent line at (1,4) is

y = 4

Here's the graph : https://www.desmos.com/calculator/uqvstutnla

On the left side....Let's simplify under the radical, first

b^2             c

_____  -    ___       multiplyy the second fraction by 4a  on top and bottom

4a^2          a 

b^2            4ac

____    -  ____          and this gives us

4a^2        4a^2

[ b^2  - 4ac ]  

__________       taking the square root    of top and bottom  we have

    4a^2

√ [ b^2 - 4ac ]

____________

 √ [ 4a^2 ]

√ [ b^2  - 4ac ] 

____________

       2a

So...combine tis with the first fraction on the left and we end up with

-b               √ [ b^2  - 4ac ] 

__    ±       ____________        which gives us

2a                   2a

-b   ±  √ [ b^2  - 4ac ] 

________________       =    [the right side ]

        2a

EDIT 

Like the guest says  8C3    =  56  !!!

Melody....I just read about the proof of this....roughly, it goes 

Let  n = p^a  where p is a prime factor

Then...the  divisors of  n  = p^a   are

 1, p, p^2, p^3, p^4.......p^a

So...the number of divisors   =  (a + 1)

So

p^a * q^b    where p, q are prime factors

has

(a + 1)  (b + 1)   divisors

They are all correct....!!!!

We'll do (a)  last

(b) 

y^2  = x + 1    (1)

and

y  = m (x + 3)   square both sides

y^2  = m^2 (x + 3)^2  =  m^2 (x^2 + 6x + 9)  =  m^2x^2 + 6m^2x + 9m^2     (2)

Set the y^2 ' s   equal so that  (1)  = (2)

x + 1  = m^2x^2  + 6m^2x  + 9m^2

0 =  m^2x^2 + 6m^2x + 9m^2  - x  - 1

0 =  m^2x^2 + (6m^2 - 1)x  + (9m^2 - 1)

(c)

It can be shown [ using implicit differentiation ] that the slope of any tangent line to the  parabola  is =  1 / [2y]   = m

So

y = (m) (x + 3)

y =(1/ [2y]) ( x + 3)

2y^2 = x + 3         (3)

But

y^2  = x + 1   so

2y^2 = 2(x + 1)  (4)

Set  (3)  equal to (4)  and solve for x to find the x coordinate of the intersections

x + 3 = 2(x + 1)

x + 3 = 2x + 2

1  = x     [this is the x coordinate for the intersection of the line and the  parabola]

So....using this....we can find the y coordinates for the intersections

y^2 = x + 1

y^2  = 1 + 1

y^2  = 2

y = sqrt(2)  or  -sqrt(2) 

[ these are the y coordinates for the intersection of the line and the parabola ]

So....m = 1/ [2y] =  1/ [2 *±sqrt (2) ]  = 1 / [±sqrt(8)]

So...m^2  =  [ 1 / ±sqrt (8) ] ^2   =   1/8

(d)  The equations for the tangent lines  are

y = [1 / sqrt(8) ] (x + 3)    and

y = -(1/sqrt(8)] (x + 3)

And the intersection points are    [ 1, sqrt(2) ]  and  [ 1, - sqrt(2)  ]

(a)  Here's the graph :

https://www.desmos.com/calculator/jaa8mpc6d7