CPhill

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(2)^(1/3)  * (14)^(1/3)   =    (28)^1/3  

Factors of 90 =

 1 2 3 5  6 9 10 15 18 30  45  90

We  can write

(1 * 90) * (2* 45 )  *  (3 * 30)  * (5 * 18)   * (6 * 15 )  * ( 9 *10)  =

   90    *     90        *      90      *     90      *     90        *   90     =

90^6

k  =  6

Agree with, EP.....I didn't see the  "each answer " part

We can use the Euclidian Algorithm to solve this

46899x + 13576y  = 1

46899  = (3)13576 + 6171

13576  = (2) 6171 + 1234

6171 = (5) 1234 + 1

1   =  6171 - (5) 1234

1 = 6171 -  5 [ 13576 - (2) 6171 ]

1 = (11) 6171 - (5) 13576

1 = (11) [ 46899 - (3)13576 ] - (5)13756

1 = 46899 (11) - 13576(38)

The first machne added

46899 (11)  = 515889  pieces of candy to the vat 

164x  +  37y  =  1

164    =   4(37)  + 16

 37   =  2(16) + 5

16 =  3(5) + 1

1 =  16 - 3(5)

1 =  16 - 3 [ 37 - 2(16) ]

1 = 7(16) - 3(37)

1 = 7 [ 164 - 4(37)]  - 3(37)

1 = 165 (7)  - 37(28) - 37(3)

1 = 164(7) + 37(-31)

So      (x, y)  =  ( 7, -31)

We are completing the square on x and y

Take 1/2 of 14 = 7, square it =49   add to both sides  

Take 1/2 of 10  =5, square it = 25    add to both sides

a. Fill in the blanks: x²+14x+ 49+y²+10y+ 25 = 74

b. Standard Form   =  ( x + 7)^2 + ( y + 5)^2 = 74

c. Center   ( - 7, - 5)

Radius  =   sqrt (74)

2.

The equation of a circle is (x −11) ²+(y+15) ²=144.

(a)   Johnny said that the center of the circle is located at (15, -11). Is Johnny correct? Explain your answer.

In the form   ( x - a)^2  + ( y + b)^2 = r^2....the center is  ( a, - b).....so....

The center is   (11, - 15)

(b)   Joyce said that the diameter of the circle is 144 units. Is Joyce correct? Explain your answer. 

144 =  radius^2    

12 =  radius

2 * radius = diameter

2 * 12 =

24   =  the diameter

Second question :

We have the following arithmetic sequence

15, 18, 21

The 7th row has     15 + 3(n - 1)   =  15 + 3(7 - 1)   =  15 + 3(6)  = 33 logs

So.....the total number of logs is

( 15 + 33) * 7 / 2 =

48 * 7  / 2  =

168 logs in total

The third answer is correct in each case

( i - d)(3)   is the rate at which the volume of liquid is changhing at 3 hrs....to find this we have 

[ 3(3)^2 + 2 ] - [ 4(3) - 1 ]  =

[ 27 + 2 ] - [11[  =

29  - 11  =  

18 gal per hour    (third answer )

Next one

We want to find when the debt = 0   .....so.....

-4x^2 + 10x + 60  = 0        divide through by - 2

2x ^2 - 5x - 30  =  0          

The graph here shows that the company will get out of debt after about 5.3 years

https://www.desmos.com/calculator/rtundu47as