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V _{square pyramid}  =

base area  * height  / 3  =

5^2  * 6 / 3  =

25 * 2  =

50  cm^2

V  =   pi  * (diameter / 2)^2  * height  /  3 =

[ pi  * ( 5/2)^2  * 6  ] / 3  =

pi  *  (2.5)^2 * 2  =

3.14 * 12.5 =

[ 12.5 pi ]   in ^3  =   39.25 in ^3

First one

New area  =  Old Area  * (Scale Factor)^2   =  30  * (3)^2  =   30 * 9   =  270  cm^2

Second one 

The ratio of their volumes  is  (5/3)^3   =   125 / 27  =   125 : 27

The graph shifts 7 units left and 2 units up.

For the first one....the park will  have an area  =

Old Park Area  *  (Scale Factor)^2  =

37625  * 4^2  =    602,000 yd^2

Second one

The volume od the new prism  will have a volume  (Scale Factor)^3  as great  = (1.5)^3  =  3.375 times as great

Third  one 

The volume of the new cone will be 

Old volume * (Scale Factor)^3  =

2880 * (1/4)^3   =  2880/ 64  =   45 m^3

N (divisible  by 7 or 10)   = N(divisible by 7)  + N (divisible by 10)   -  N (divisible by 7 and 10)  =

(14)   +  (10)  -   1   =

23

Using the Law of Sines

20 / sin C  = BC   /  sin A

20  / ( 1 /sqrt (2) )   =  BC  / (1/2)

sqrt (2) * 20 * (1/2)  =  BC   =   10 sqrt (2)  units

Call the height of the helicopter, h

And call the distance that Amanda is from a point directly below the helicopter , x

So....the distance that Jeremiah is from this point is  4000 - x

So  we have that

tan (60)  =  h / x   ⇒   x tan (60)  = h       (1)

and

tan (30)  =  h / ( 4000 - x)  ⇒    tan(30) (4000 - x)  =  h      (2)

Equating  (1)  and (2), we have that

x tan (60)  =  (4000 - x) tan (30)   simplify

xsqrt (3)  =  (4000- x)  (1/ sqrt (3) )          multiply both sides by  sqrt 930

3x  =  4000 - x      add x to both sides

4x  = 4000   divide both sides by 4

x  = 1000

So...Amanda's distance from the helicopter is      cos (60)  =   1000/ Amanda's distance to helicopter  =

1000/ cos (60)  = Amanda' s distance to the helicopter = 1000/(1/2)  =  2000 ft 

Jeremiah's distance to the helicopter is

cos (30)  =  ( 4000 - x )  / Jeremiah's distance to the helicopter

cos (30)  =  3000 /  Jeremiah's distance to the helicopter

Jeremiah's distance to the helicopter   =   1000/ cos (30)  ≈  3464 ft 

3.  The height of the helicopter  is  1000 tan (60)  ≈  1732 ft

4. Jeremiah  is  (4000 - 1000)   = 3000 ft from the rope

5. Amanda is 1000 ft from the rope

Thank you!  I think I missed something obvious in the lesson.  Do you think you could explain this next question just in case?

Not too bad, ACG  !!!

Note that...if AP  = BP....then triangle APB is an  isosceles right triangle

So.....AB  =  sqrt (4^2 + 4^2)  =  sqrt (32)   =  sqrt (16 * 2) =  4sqrt (2)

If  P  is the midpoint of BD....then   BP  = 4  and BD  = 8

Note that BDC is a right angle....and since angle B is bisected, then  CBD  = 45°  .....so  since BDC  = 90°......then , by default  angle BCD  = 45°

So...triangle BDC  is also isosceles with BD  = DC  = 8

So......BC  = sqrt  (8^2 + 8^2)  =  sqrt (128)  = sqrt (64* 2)  =  8 sqrt (2)

So....the perimeter is

BA + AP + PD + DC + BC =

4sqrt (2) + 4 + 4 + 8   + 8sqrt (2)  =

16  + 12sqrt (2) units  ≈   32.97  units

Thank you!

Could you explain the last 3 steps in the second part of #2 solution? Im not sure how you got the two.

All of the angles  = 45°....and sin (45°)  = cos(45°)

So.....every statement is true.....!!!

The first is known as the "Intersecting Chords Theorem"

Specifically.....

AP * PC  =  DP * PB   ...so....

12 * 6  =   5 * PB

72  = 5 * PB     

72/5  = PB  =  BP

Second one

Arc length  =

(2 pi) * r  *  (45 / 360)  =

(2 pi) * 6  *  1/8  =

12 /8 pi  =

(3/2) pi   inches

Triangle ABD is isosceles.....and angle BAD   = 45

So angle DAC = 45....so....angle CAB is bisected....which sets up the following relationship :

CA / AB  = CE / BE

But triangle ABC is a right triangle   ...so   BC   =  sqrt (10^2 + 6^2)   =  sqrt (136)  = 2 sqrt (34)

So

10  / 6  = CE / BE

So....CB  is divided into 16 equal parts....and  CE  is 10 of these....so  CE  = (10 / 16) * 2 sqrt (34)  =

(5/4) sqrt (34)  = 1.25 sqrt (34)

The measure of angle  ACB  =   arcsin (6/(2sqrt (34)) 

So....the area of triangle ACE  = (1/2)(10)(1.25)sqrt (34)sin (arcsin (6/(2sqrt (34)) )  =

(1/4) * 10 * 1.25 * 6   =     

18.75  units^2

Thank you so much @CPhill and i have two more qustion I'm about to post

ROQ  =  measure of minor arc RQ  =   360  - (280)   =   80°

Second one.......since this is a cyclic quadrilateral......Angles B + D   = 180

So we have

(2x + 3) + (4x + 3)  = 180    simplify

6x + 6  =180  subtract 6 from both sides

6x  =  174     divide both sides by 6

x  =  29

So   ...C  = 2(29) + 1  =    59°