CPhill

Getting to them after I eat....hold on!!!!

3.

B and D are supplemental

4x - 20  + x  = 180      simplify

5x - 20  = 180     add 20 to both sides

5x = 200         divide both sides by 5

x = 40

So...B  =  4(20) - 20   =   60°

1.

A  and C are supplemental

A + 43   = 180    subtract 43 from both sides

A  = 137°

2.

A and C are supplemental...so...

(4x+ 5) + ( x + 15 )  = 180   simplify

5x + 20  = 180     subtract 20 from both sides

5x = 160     divide both sides by 5

x = 32

So  A  = 4(32) + 5    =  133°

 3600 * 24 * 365 * (75 - 14 )   =  $ 1,923,696,000

10)The two bases of a right conical frustum have radii 12 and 9. The two bases are 4 units apart. Let the volume of the frustum be V cubic units and the total surface area of the frustum be A square units. Find V + A.

The height  =  4

The volume, V,  is given by

(1/3) * pi * height  ( [radius base 1]^2 +  product of the base radii +[ radius base 2]^2 )  =

pi / 3  *  4  * ( 12^2 +  12*9 + 9^2)

pi / 3 * 4 * ( 144 + 108 + 81 )

pi / 3 * 4  * ( 333)

pi * 4 * 111

444 pi

The surface area A, is given  by :

pi ( sum of radii) * √ [ (difference of radii)^2 + height^2 ] + pi (radius 1)^2 + pi (radius 2)^2

pi ( 21) √ [ 3^2 + 4^2 ] + pi  [ 144 + 81 ]

pi [  21 * 5 + 225 ]

pi [ 105 + 225 ] 

pi [ 330]

330 pi

So

V + A  =

[444 +330 ] pi  =

774 pi

9)A 288 degree circular sector with radius 15 is rolled to form a cone. Find the height of the cone.

The perimeter of the base of the cone will be :

2 * pi * 15   (288/360)  =  

pi * 30/ 360 * 288

pi *  288/12  = 

24 pi

So....the radius of the cone is given by

24pi  = 2pi * r         divide both sides by 2 pi

12 =  r

The slant height will be the original radius of the sector  = 15

And this forms the hypotenuse  of  a   9 - 12 - 15     right triangle

And the radius forms one of the legs  = 12

So....the height forms the other leg  =  9

2)The radius of a cylinder is 5/6 its height. Find the total surface area of the cylinder if its volume is 150pi.

We have 

150 pi  =  pi [ 5/6 h ] ^2 * h         divide out pi

150  =  (25/36) h^3            multiply both sides by 36/25

150 (36/25)  = h^3

216  = h^3

6  =  h

So...r  = (5/6)h  = (5/6)(6)  = 5

So....the surface area  is

2pi * r  [ r + h ]  =

2pi * (5) [ 5 + 6 ] =

2pi * 55

110 pi  units^2

Thanks for the primer, guest.....I was not familiar with this technique...!!!!

Sorry, Julius....  I meant to say that  x  = 1   is the only critical number  !!!

Correct, Julius  !!!!

For all x > 5,   x - 5 will be positive, thus increasing.....

Also  [ don't worry about how I arrived at this ] ....if  x - 5  is the derivative, then  (1/2)x^2 - 5x + some constant   is the parent function....as shown here, no matter the added constant, this function always increases when x > 5

https://www.desmos.com/calculator/mssfiffnb0