CPhill

A pic for #2  might be helpful......

Here's the first one

Call the height of the water in the tank, H

And ......using similar triangles, we can express the radius, R, at this point thusly :

16 / 96  = R / H

1/6 = R/H

H/6  = R

Now  when the tank is 25% full, it will hold

(1/4) (1/3) pi 16^2 * 96   =   2048 pi  ft^3

So....to find the height, we have

2048 pi   =  (1/3) pi  (H/6)^2 * H     divide out pi, multiply both sides by 3

6144  =  H^3 / 36       multiply both sides by 36

221184  =  H^3

Take the cube root of both sides

48³√2     =   H

So

a +  b   =   48 +  2   = 50

Nice, Melody   !!!!

Look at the graph here :

https://www.desmos.com/calculator/jj4kxyynwi

The first statement is true

The second is true, as well  ⇒ year 5   = 8.8 ft   ,  year 7 = 15.9 ft

The third statement is false.....the initial  height is 1 ft

The fourth statement is also true.....when x  = 15, the tree has almost reached its max height  of about 30 ft

A circle has a sector with an area of 33\pi  and central angle of (11)/(6)\pi  radians. what is the area of the circle?

Note that (11/6) pi  rads  = 330°  =  330/360 =  11/12  of the whole area of the circle

So....to find the whole area we have

33pi  (12/11)  =

(33/11) * 12 * pi  =

 3 * 12  pi  =

36 pi

f(x)  = cx^2 + 2x    x < 2

f(x) = x^3 - cx        x ≥ 2

Let's just substitute 2  into each function for x

c(2)^2 + 2(2)  =   4c + 4

(2)^3 - c(2)  =  8 - 2c

Set these results equal  and solve for c

4c + 4  = 8 - 2c      add 2c to both sides, subtract 4 from both sides

6c  = 4      divide both sides by  6

c  =2/3

See the graph here that confirms the result :

https://www.desmos.com/calculator/ahns4csjpv

These  numbers  are very familiar

The  three known text sizes  [ from least to greatest ]  are

Phi^-1

Phi⁰

Phi²

The unknown text size  is Phi¹  = Phi  =  1.618

Note that the point  (1,0)  is the first point on the graph

And the point  (5,2)  is also on the graph

So.....the top  right graph is correct

Note that the bottom right graph couldn't be correct becuse no negative values for x  are allowed......these would make the quantity under the radical negative  which would not be a real number

Domain

x  cannot be any value < -5   because we would be taking the root of a negative number which isn't real

So....the domain  is   [-5, infinity )

The range is trickier

As small as the quantity under the radical can be is 0  [  when x  = -5 ]

So  1/2  * 0   =   0

And as x grows larger than -5, the function keeps increasing to infinity

So....the range is  [ 0, infinity )

The first answer is correct

f(x)  =  l x - 3 l  + 2

f(x + 2)  =   l (x + 2) -3 l + 2  =  l x - 1 l + 2

This shifts the graph 2 units to the left

Look at both graphs here :

https://www.desmos.com/calculator/8m6gjryhfu