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f(n + 1)  = 3f(n)

Means that every successive term is just  3 times  the previous one....so....

f(0) = 4

f(1) =  3 *4  = 12

f(2)  = 3*12  =36

f(3) =  3 * 36  = 108

The triangle rule states that the sum of the 2 smallest sides must be greater than the bigger side. 

Thus, \(\color{brown}\boxed{E}\) will not work, because \(9 + 5 <15\)

I think we can  solve this with similar triangles

Call the side length of the  cube, s

The diagonal  across the cube's base =  s * sqrt (2)

1/2 of this     =  s / sqrt 2

We can construct a triangle with a base =  radius of cone -  s/sqrt 2   =   3  - s/sqrt 2

And the height of this triangle =  s

And the other triangle has a base of  the radius of the cone = 3  and a  height of 8

And these triangles wil be similar....so.....

3/ 8 =  (3 - s/sqrt 2) / s       cross-multiply

3s =  8 (3 - s /sqrt 2)  

3s + 8s / sqrt 2  =  24

s ( 3 + 8/sqrt 2)  = 24

s =  24  / (3 + 8/sqrt 2) ≈   2.77  

I hope this is correct.

SInce the height is 8, and the diameter is 6 (which is greater than 4), I think the cube has to end halfway up the height. This means that the answer is 4. I'm not very confident about my answer, so someone else can try to double-check it.

I hope this is correct.

It can be any of the answer choices provided.This is because they are all less than 9+15 = 24.

Volume of the cylinder =  pi * R^2 * h

So  the frustum's   volume =  (2/3)* pi * R^2 * h        (1)

Volume of a frustum =    (1/3) * pi * h *  [  R + r + Rr ]      (2)

Set  (1)   = (2)

(2/3)pi*R^2*h =   (1/3)pi * h * [ R + r + Rr ]         simplify

2R^2  =  R + r + Rr

2R^2 - R  =   r + Rr

R (2R - 1 ) =  r (1 + R)

R ( 2R - 1 ) / (1 + R)  = r

r / R =      R(2R - 1)  /   [ (1 + R) * R ] =

(2R - 1) / (R + 1)

(-2, 3)

We need  one more 3 and one more 5 to produce a perfect square

So.....

3 x 5 =   15

I think this website only displays when users are typing, not guests.

Sorry....I didn't see anyone typing.....!!!

Whatever x's  make the denominator = 0  are the asymptotes.....these are also the roots of the polynomial

So

x^2 + 5x + 2  =  0

By Vieta, the sum of the  roots =  -5

So

a + b =   -5

You sent your answer as I was typing mine lol.

I hope this is correct.

Surface Area: 6*6 = 36 for base. sqrt(391) is the distance from the midpoint of a side to the top of the pyramid, due to Pythagorean Theorem. 3sqrt(391) is the area of one of the sides. The answer after we multiply that by four and add 36 is 12sqrt(391) + 36.

Volume: The volume of a pyramid is the area of the base times the height, divided by 3. the base is 6*6 = 36. Since the distance from the midpoint of a side to the top of the pyramid is sqrt(391), and the distance from that to the center of the pyramid is 3, we use the Pythagorean Theorem. We find that the height of the pyramid is sqrt(382). Our answer is 12sqrt(382).

First, find the slant height   = sqrt [ 20^2 - 3^2 ] =    sqrt [ 391]

So   the area of  one triangle   = (1/2)(6)sqrt [391]  =  3sqrt [391]

So  the surface  area  = 4 * 3sqrt [391]  + surface area of the base  =

12sqrt [391] + 6^2  =     12 sqrt [ 391 ] + 36  =   12 [ sqrt (391) + 3 ] units^2

To find the volume  find the height  

The diagonal across the base =  6sqrt 2

So (1/2)  of this =  3sqrt (2) =  sqrt (18)

So the height  =   sqrt [ 20^2 - (sqrt 18)^2 ]  =  sqrt  (382)

So the volume =  (1/3) area of base * height  =  

(1/3) 6^2 * sqrt (382)  =

12 sqrt (382)  units^3

(8 choose 4) = 70 is the answer if they cannot repeat positions. (8^4) = 4096 is the answer if they can repeat positions.

geno3141's answer is correct, but it can be simplified to 1/8.

It's not but it will soon

Let the roots be m, n

The product of the roots =   5

Therefore     m * n =  5

But

1/m * 1/n  =  1/5

So

a*b  = 1/5

I posted this question before she uploaded the answer

Melody gave an answer. Are you having difficulties understanding her solution?

We factorize it first.

\(11! + 12! + 13! = 11! (1 + 12 + 12 \times 13) = 13^2 \times 11!\)

All prime factors of \(11!\) are smaller than or equal to 11 since \(11! = 11 \times 10 \times 9 \times \cdots \times 2\times 1\).

The largest prime factor is 13.

Since you didn't reply, I will assume that it denotes the symmetric difference \(A \oplus B = (A \cup B) - (A \cap B)\).

First, \(C - B\) is just the collection of elements in C that is not in B, i.e., \(C - B = \{2, 4\}\).

Now, we calculate \(A \oplus (C - B) = A \oplus \{2 ,4\}\).

Note that \(A \cap \{2, 4\} = \varnothing\) since the two sets do not have common elements. Then \(A \oplus \{2, 4\} = A \cup \{2, 4\} = \{1,2,4,5,6,8\}\).

Hence, we have \(A \oplus (C - B) = \{1,2,4,5,6,8\}\).

We can use the tangent

tan 57°  =   height  /   21

21*tan 57°   =   height  ≈   32.3 ft

I think you mean that 18 is the number of degrees of freedom!  Look in some chi-square distribution tables (e.g. https://en.wikibooks.org/wiki/Engineering_Tables/Chi-Squared_Distibution) along the row for DF = 18 until you get to a number close to 21.6, then look at the column heading for the probability.  

(NB: In the reference above the probabilities represent greater than rather than less than, so you would need to subtract the resulting probability from 1 for your case.)

Write a quadratic equation ...

Hello SBAHarper!

\(v_0t\cdot sin\ \alpha-\frac{g}{2}t^2=h\)

The launch height is zero.

There are two unknowns to be determined:

Launch angle \(\alpha\) and launch velocity \(v_0\).

So two equations:

\(I.\ {\color{blue}v_0\cdot sin\ \alpha}\cdot 1sec-\frac{g}{2}\cdot (1sec)^2-5feet\cdot\frac{0.3048m}{feet}=0\\ II.\ {\color{blue}v_0\cdot sin\ \alpha}\cdot 4sec-\frac{g}{2}\cdot (4sec)^2-5feet\cdot\frac{0.3048m}{feet}=0\\ g=9,81m/s^2\\ I.\ v_0\cdot sin\ \alpha=\dfrac{6.429m}{1sec}\\ \color{blue}\alpha=arcsin\ \dfrac{6.429m}{v_0\cdot 1sec} \)

\(II.\ v_0\cdot sin\ \alpha\cdot 4sec-4.695m=0\\ \color{blue}\alpha=arcsin\ \dfrac{4.695m}{v_0\cdot 4sec}\)

These two functions do not intersect.

Ergo: The goal of 5ft after 1 second and 5ft after 4 seconds cannot be realized.

  !