Multiples
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
3, 6, 9, 12, 15, 18, 21, 24
4, 8, 12, 16, 20, 24
5, 10, 15, 20, 25
6, 12, 18, 24
7, 14, 21
8, 16, 24
9, 18
10, 20
11, 22
12, 24
13 - 25

Perfect Cubes
8

Primes
2, 3, 5, 7, 11, 13, 17, 19, 23

If yes to the first, no to the second, and yes to the last, then the answer is 3

If no to the first, yes to the second, and no to the last, then the answer is 8.

You can’t have a prime that is also a perfect cube. and you can’t have a perfect cube that is a multiple of 3, because it would at least give you 27.

So you have a 50/50 of getting it right. I would go with 8 as my answer. although 3 as your answer would be only if you got yes for the first and last question.

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2.)
log(2sin(x)) + log(sqrt(3) + 2sin(x)) = log(6)
log(2sin(x)(sqrt(3) + 2sin(x)) = log(6)
2sin(x)(sqrt(3) + 2sin(x)) = 6
sin(x)(sqrt(3) + 2sin(x)) = 3
2sin(x)^2 + sqrt(3)sin(x) = 3
2sin(x)^2 + sqrt(3)sin(x) - 3 = 0

using the quadratic formula

sin(x) = (-b ± sqrt(b^2 - 4ac))/(2a)

sin(x) = (-sqrt(3) ± sqrt((sqrt(3))^2 - 4(2)(-3)))/(2(2))
sin(x) = (-sqrt(3) ± sqrt(3 + 24))/4
sin(x) = (-sqrt(3) ± sqrt(27))/4
sin(x) = (-sqrt(3) ± 3sqrt(3))/4
sin(x) = ((-1 ± 3)sqrt(3))/4
sin(x) = (-4sqrt(3)/4) or (2sqrt(3)/4)
sin(x) = -sqrt(3) or (sqrt(3)/2)

since you can’t have a value that is above 1 or below -1

sin(x) = sqrt(3)/2
x = 60

ANS : pi/3

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3. find sin(2arctan3)
no calculators

sin(2tan^-1(3)) = .6

Sorry i used a calculator. Its been awhile since i’ve done problems like that.

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4. for a given arithmetic sequence the sum of the first 50 terms is 200, the sum of the next fifty terms is 2700. What is the first term of the sequence

an = a1 + (n - 1)d
Sn = (n/2)(a1 + an)

a(50) = a1 + (50 - 1)
a(50) = a1 + 49

S(50) = (50/2)(a1 + a(50))
200 = 25(a1 + a(50))
8 = a1 + a1 + 49
2a1 = -41
a1 = -41/2
a1 = -20.5

a(100) = -20.5 + (100 - 1)
a(100) = -20.5 + 99
a(100) = 78.5

a(100) = (100/2)(-20.5 + 78.5)
200 + 2700 = 50(58)
4900 = 4900

ANS : -20.5

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5.) a garden hose is wound on a reel
the first 8 turns are each of length 21 inches, the next 8 turns are each of length 23 inches, the next 8 turns are each of length 25 inches and so on. Fin the total number of turns required to roll up a 200-ft hose

8 = 21
8 + 8 = 21 + 23
8 + 8 + 8 = 21 + 23 + 25

200 * 12 = 2400 inches

an = a1 + (n - 1)d
an = 21 + 2(n - 1)

Sn = (n/2)(a1 + an)
2400 = (n/2)(21 + (21 + 2(n - 1)))
2400 = (n/2)(42 + 2(n - 1))
2400 = (2n/2)(21 + n - 1)
2400 = n(n + 20)
2400 = n^2 + 20n
n^2 + 20n - 2400 = 0

n = (-b ± sqrt(b^2 - 4ac))/(2a)

n = (-20 ± sqrt(20^2 - 4(1)(-2400)))/(2(1))
n = (-20 ± sqrt(400 + 9600))/2
n = (-20 ± sqrt(10000))/2
n = (-20 ± 100)/2
n = (-120/2) or (80/2)
n = -60 or 40

since you can’t have a negative

n = 40

Thats just for how many sets of 8 turns it would take to get 200 ft

40 * 8 = 240

ANS : 240 turns

Work from the inside out.
θ = arctan3
tanθ = 3
sinθ = 3/√10
cosθ = 1/√10

sin(2θ) = 2(sinθ)(cosθ)
sin(2θ) = 2sin(arcsin 3/√10)cos(arccos 1/√10)
sin(2arctan3) = 2*(3/√10)(1/√10) = 2*3/10 = 0.6
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4. For a given arithmetic sequence the sum of the first 50 terms is 200, the sum of the next fifty terms is 2700. What is the first term of the sequence?

(for k = 0 to 49) ∑(a + bk) = 50a + 1225b = 200
(for k = 50 to 99) ∑(a + bk) = 50a + 3725b = 2700

Subtracting the first sum from the second we have

2500b = 2500
b = 1

Plug back into the first equation.

50a + 1225b = 200
50a + 1225*1 = 200
50a = -1025
a = -20.5

The first number in the sequence is -20.5.

If you try all the possible answers (yes-yes-yes, yes-yes-no, yes-no-yes,…) the only set of answers that ed could give that would narrow down erin’s choices to one number is yes, no, yes.

A prime multiple of three that is not a perfect cube.

The only prime multiple of 3 is 3!