CPhill

\(x\cdot\lfloor x\rfloor=70 \)

Note that √70  ≈ 8.366

So... if we divide 70 by 8  we get  70 / 8  =  35 / 4 = 8 3/4  = 8.75

So.....8.75 * 8  = 70

And ....  the floor of 8.75  =  8

So

8.75 * floor [ 8.75 ] =

8.75  * 8    =

70

Then  x  = 8.75

ln ( x + 3)  - ln x  = 4

ln [ (x + 3) / x ]  = 4

e^ [ ln [ (x + 3 ) / x ] ]  = e^4        { remember that e^(ln a)  = a  }

[ x + 3 ] / x  = e^4      cross-multiply

x + 3  = xe^4   rearrange as

x - xe^4  =  - 3       factor out  x

x [ 1 - e^4]  = -3 

x  = - 3 / [ 1 - e^4]   

√[25 - x^2 ] + √[ - ( x - 2) ]  =

√[25 - x^2 ]  + √ [2 - x ]

The domain for the first function  is  [ -5, 5 ]

The domain for the second function is  ( - infinity, 2]

Because  we must take the most restrictive interval, this is   [ -5, 2 ]

So....the width of the interval  is  7 units   [ 8 integers ]

So the graph, here : https://www.desmos.com/calculator/ioiruyyh4v

Because CE  is a chord meeting a tangent, the measure of the angle supplemental  to angle PCQ =

(1/2) ( m arc CDE)  =   (1/2) (m arc CD + m arc DE ) = (1/2) ( 70 + 30)  =  50°

So....angle  PCQ  =  180 - 50  =  130°

And angle  CQP is vertical to angle DQE  =  40°  

So....in triangle PCQ, angle PCQ  = 130  and angle CQP = 40

So  angle CPQ  =  180 - 130 - 40  =  10°

And angle  QPE  equals this  = 10°

And angle PQE is supplemental to  angle DQE  = 140°

So...in triangle PQE,  angle QEP  = 180 - 140 - 10  = 30°

So m arc CA  is twice this  = 60°

So   m arc AE  =  360 - m arc DE - m arc CD  - m arc CA  =

360  - 30 - 70 - 60  =

360  - 160  =

200°

This doesn't  "look " to be correct......[ I don't think I made any mistakes, but maybe I did !!!!]

I'm thinking that the volume is supposed to be 3w^3 - 4w^2 - 15

Let the width = w

Then the height is 3w + 5

And the volume  is   Width * Heiht * Length....so we have

3w^3 - 4w^2 - 15w  = w * ( 3w + 5) * L          

w (3w^2 - 4w - 15) = w (3w + 5) * L     divide both sides by w

3w^2 - 4w - 15  = (3w + 5) * L          factor the left side

(3w+ 5) (w - 3) = (3w + 5) * L      divide both sides by (3w + 5)

(w - 3)  =  L  =  Length

We have ways to award gold....15 ways to award silver and 14 ways to award bronze

So

16 * 15 * 14   =

3360  ways

71.  Arc length  =  2 * pi (8) (120/360)  = (16/3)pi mm ≈ 16.76 mm

72. 120 * pi  / 180  =   (120/ 180) *pi  =  (2/3) pi  rads

73.  Sector area  = (1/2) *r^2* (theta in rads)  =  (1/2) * (8)^2 * (2/3 pi)  =

(64/ 3) pi  mm^2  ≈  67.02 mm^2 

74.   (x - 3)^2  + (y + 4)^2  = 4 

75. x^2 - 6x + y^2 + 4y - 3  = 0

x^2 - 6x + y^2 + 4y  =  3            complete the square  on x and y

x^2 - 6x + 9  + y^2 + 4y + 4  =  3 + 9 + 4

(x - 3)^2 + (y + 2)^2  = 16

center ( 3,  -2)    radius  = 4 

We have 5 digits which are odd...

So....the number of "odd looking" 4 digit numbers is

(5)(5)(5)(5)  =  5^4   =  625

tan  =  sin  / cos

tan  =  ( 7/25) / (24/25)

tan  =  (7/25) * (25/24)  =  ( 7/24) * (25/25)   =  7/24 * 1   =  7/24

First one 

The measure  of   angle AED  =  (1/2)(arc AD + arc BC)

angle AED  =  (1/2)(38 + 162)

angle AED  (1/2) (200)  =   100°

Second one ....this is the intersecting chord theorem

It says  that  

DE * EB  =  AE  * EC

 3 * 16   =   4 * EC

48  = 4 * EC       divide both sides by 4

12  = EC