i need help with precal questions!!!! please?

Question

if you could show me how to and answer any/all of these that would be great

1. ed is thinking od a number from 1to 25 inclusive. Erin asks him wheter the number is a multiple of thee whether itis a perfect cubeand whether it is prime. fter hearing the answer to all three questions, erin knows what the number is. What is the Number? show work and explain thinking

2.
0 is less than or equal to x which is less than or equal to 2pi
log(2sinx)+log(sqrtof 3 +2sinx) = log6
find x as a multiple of pi
(i kno you can take away the logs but after that?)


3. find sin(2arctan3)
no calculators

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

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 areeach of length 25 inches and so on. Fin the total number of turns required to roll up a 200-ft hose

Answer 1

1. ed is thinking of a number from 1 to 25 inclusive. Erin asks him whether the number is a multiple of three whether it is a perfect cube and whether it is prime. after hearing the answer to all three questions, Erin knows what the number is. What is the Number? show work and explain thinking

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

Answer 2

3. Find sin(2arctan3).

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.

Answer 3

1. Summarize the questions as M3, PC and Prime.
(a) M3= yes. Since 8 is the only perfect cube, "no" must be the answer to the second question. Since any multiple of 3 is not a prime, "no" must be the answer to the third question. Thus, 3 is the answer.
(b) M3= no. If PC is "yes", 8 is the answer.
If PC is "no", If Prime is "yes', several possibile answers exist, such as 1, 5,7,11,13,17,19 and 23. If prime is "no", than several possible answers exist such as 2,4,10,14,16,20,and 22.
Erin must know something that I don't.

Answer 4

The first one is 3!

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!