CPhill

3x^2 - 16x + 17  =  0

Product of the roots =   17/3  = ab

(ab)^2 = a^2*b^2  = 289 /  9

2ab =  34/3

Sum of the roots = 16/3  = a + b         square both sides

a^2 + 2ab + b^2  = 256 / 9

a^2 + (34/3) + b^2  =  256 /  9

a^2 + b^2  =  256 / 9  - 34 /3  =  154 / 9

Quadratic  is

x^2  - (154/9)x + 289 / 9

8 * 45  =  18 * x

360  = 18 * x

360  / 18 =  x

20 (min)   = x

Call the side, s

s^2 = 10s + 38

s^2  - 10s  =  38

s^2 -10s + 25 =  38 + 25

(s - 5)^2  = 63       take  both roots

s - 5  = sqrt (63)                   s - 5 =  -sqrt (63)

s = 5 + sqrt (63)                  s =  -5 -sqrt (63)    reject

s ≈ 12.94

https://web2.0calc.com/questions/a-circle-lies-inside-a-quarter-circle-as-shown-below

Ceiling [ 15 * 60 *1 /  (.3048^2) / 400] =  25 gallons

f(1)  = -1

f(2)  = 0

...

f(1000) = 0

Sum  = -1

\(\frac{x^2}{x - 1 + x^3} for x > 1 \)

The graph here shows that there is  no min for any x

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

The graph will approach 0 as x →  infinity

https://web2.0calc.com/questions/help-with-geometry_22606

W               A    15      X

                                   x

                              60 C

                                  3x

Z             B      20       Y

Let ACX = Angle A

Let BCY = Angle B

A + B + 60  =  180

A + B =  120 

tan ACX =   15/x

tan BCY  =  20 / (3x)

tan (A + B)  =  tan (120)

[ tan A  + tan B] /  1 - tanAtanB  =  -sqrt 3

[ 15/x + 20/(3x) ] / [ 1 - (15/x)(20/(3x)) ]  = -sqrt 3

[ 15 (3x) + 20x ]  / [ 3x^2 ]  

_____________________    =  -sqrt (3)

1 - 300 / (3x^2 )

[ 45x + 20x ] / [3x^2]

________________  =   -sqrt 3

[3x^2 - 300] / [3x^2]

65x =  sqrt (3) [300 - 3x^2]

3sqrt (3)x^2 + 65x - 300sqrt (3)  =  0

Using the Quad Formula and taking the positive value of x ≈ 5.5403

3x = 16.6209

AC  = sqrt [  15^2  + (5.5403)^2 ]  ≈ sqrt [ 225.69 ] 

BC = sqrt [ 20^2  + (16.6209)^2 ]  ≈ sqrt [676.25]

Law of Cosines

AB^2  =  AC^2 + BC^2 - 2 (AC * BC) cos (60)

AB^2  = 225.69 + 676.25 - 2 sqrt (225.69*676.25 ] * (1/2)

AB^2 = 225.69 + 676.25  - sqrt [ 225.69* 676.25 ]

AB^2  = 511.270

AB  = sqrt (511.270)  ≈  22.6

https://web2.0calc.com/questions/help-with-algebra_96