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Let cos A = x

  !

Felicia has a ladder that is 11 ft long. She wants to lean the ladder against a vertical wall so that the top of the ladder is 9.5 ft above the ground. For safety reasons, she wants the angle the ladder makes with the ground to be no greater than 75°. Will the ladder be safe at this height? Show all your work and draw a diagram to support your answer.     

The wall is the opposite end of a right triangle.     

The ladder is the hypotenuse of that triangle.     

sine = opposite / hypotenuse     

sine of angle with ground  =  9.5 / 11     

                                          =  0.86364     

sine^–1 (0.86364)  =  59.7^o  call it 60^o   so, yes the ladder will besafe     

_.     

The ceiling of Xavier’s living room is a square that is 12 ft long on each side. To decorate for a party, he plans to hang crepe paper around the perimeter of the ceiling and then from each corner to the opposite corner. Xavier can buy rolls that each contain 10 ft of crepe paper. What is the minimum number of rolls he should buy? Show all your work.     

Needed for perimeter  =  4 • 12  =  48 ft     

Needed for diagonals  =  2 • 12 • sqrt(2)  =  33.94 ft     call it 34 ft     

Total crepe paper needed  =  48 + 34  =  82 ft     

Xavier needs to buy 9 rolls of crepe paper to have enough.     

_.     

Two trains started moving at the same time, train A from Boston to New York, and train B from New York to Boston.  The distance between the train stations in New York and Boston is 240 miles.  The average speed of train A is 60 mph, which is 3/4 of the average speed of train B.  How soon will the trains meet?     

Train A  =  60 mph     

Train B  =  80 mph     

Combined speed  =  140 mi / hr     

Distance = Rate • Time     ——>>     Time  =  Distance / Rate     

                                                           Time  =  (240 mi) / (140 mi / hr)     

                                                           Time  =  1.714 hr     

0.714 hr       ——>>     (60 min / hr) • ( 0.714 hr)  =  42.84 min     

Round 42.84 to 43       Trains will meet 1 hr & 43 min after they left     

_.     

Rhombus ABCD has perimeter 148, and one of its diagonals has length 28. what is the area of ABCD?     

call one diagonal p and the other diagonal q     

by Pythagoras,    (side_RHOMBUS)²  =  ( half p )² + ( half q )²     

You can do this because     

The diagonals of a rhombus intersect at a 90^o angle.     

The diagonals of a rhombus bisect each other.     

                                     37²  =  14²  + ( half q )²     

                                   1369  =  196  + ( half q )²     

                            ( half q )²  =  1173     

                            ( half q )    =  34.25     

                                          q  =  68.5     

             It was given that   p  =  28     

                              A_rhombus  =  (p • q) / 2     

                              A_rhombus  =  (28 • 68.5) / 2  =  1918 / 2     

                                          A  =  959     

_.      

We can use the combination formula: C(n, k) = n! / (k! * (n-k)!), where n is the total number of people (8) and k is the number to choose (2). So, we need C(8, 2) = 8! / (2! * 6!).

Calculate the factorials:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

6! = 6 * 5 * 4 * 3 * 2 * 1

2! = 2 * 1 = 2

Plugging in: C(8, 2) = (8 * 7 * 6!) / (2 * 1 * 6!). The 6! cancels out, leaving (8 * 7) / (2 * 1) = 56 / 2 = 28.

Alternatively, the formula simplifies to C(n, k) = [n * (n-1) * ... * (n-k+1)] / k!. For n=8, k=2: C(8, 2) = (8 * 7) / (2 * 1) = 56 / 2 = 28.

mercurym999

In a pet shop, there are 3 hamsters for every 2 guinea pigs, and there are 2 giant cloud rats for every 8 guinea pigs. If N is the total number of hamsters, guinea pigs, and giant cloud rats, and N > 0, then what is the smallest possible value of N?     

2 giant cloud rats for every 8 guinea pigs       

this is a proportion so it can be reduced to     

1 giant cloud rat  for every 4 guinea pigs     

1 rat per 4 pigs                                        1 rat     

                                                                4 pigs     

3 hamsters per 2 pigs     

6 hamsters per 4 pigs                             6 hamsters     

add them up                         least N  =  11 animals     

_.     

  Check the multiples of (6 x n.5) then see what works with 8.     

  when  6 x 1.5 =   9     then      9/8 —> remainder 1     

  when  6 x 2.5 = 15     then    15/8 —> remainder 7     

  when  6 x 3.5 = 21     then    21/8 —> remainder 5     

  when  6 x 4.5 = 27     then    27/8 —> remainder 3     

  Least children = 27     

_.     

x² + 3x – 4    =  (x – 1)(x + 4)     

x² + 9x – 10  =  (x – 1)(x + 10)     

When x=1, both (x² + 3x – 4) and (x² + 9x – 10) equal zero     

So, when x=1, (x² + 3x – 4) + (x² + 9x – 10) = 0 + 0  which     

puts a zero in the denominator, and you can't divide by zero.   

Therefore, when x = 1 the fraction is 1/0 which is undefined.       

_.     

A cube has side length . Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)     

"A cube has side length ."  Hard to solve a problem without that piece of information.     

Let's assign our own length to the side of the cube.  I think 1 is a pretty good number.     

You'll note that every edge of the tetrahedron is the diagonal of a side of the cube.     

If 1 is the length of the cube side, then sqrt(2) is the length of the tetrahedron edge.     

The volume of a regular tetrahedron in terms of the length of its edge is given as:     

     V  =  a³ / 6 • sqrt(2)             where a is the length of the edge of the tetrahedron     

     V  =  sqrt(2)³ / 6 • sqrt(2)     divide that sqrt(2) in the denominator into the numerator     

     V  =  sqrt(2)² / 6     

     V  =  2 / 6  =  1 / 3     

_.     

A kite is flying 82 ft. off the ground, and its string is pulled taut. The angle of elevation of the kite is 52°. Find the length of the string. Round your answer to the nearest tenth.     

     The string would be the hypotenuse of a right triangle.  sine = opposite / hypotenuse     

     sine(52)  =  82 / string       Use the trig tables or your calculator to find the sine of 52^o     

     0.7880  =  82 / string     

     string  =  82 / 0.7880     

     string  =  104.06  to nearest tenth is  104.1 ft     

_.     

     x = 26^o    

     You don't need all those calculations.  There's a theorem.     

     Theorem:  When a transversal line intersects two parallel lines,     

                       the alternate interior angles are equal in measure.     

_.     

Given that M(1,-6) is the midpoint of AB and A(-2,-1) is one endpoint, what is the sum of the coordinates of point B?     

Visualize or, better yet, sketch the following on an x-y coordinate grid.     

From endpoint A (–2,–1) to midpoint M (+1,–6) you went right 3 and down 5.     

To get from the midpoint to endpoint B, move the same directions and amounts.     

For x, from +1 go right 3 to +4.  For y, from –6 go down 5 to –11.  B is at (+4,–11).     

Adding the coordinates of B   ——>>   (+4) + (–11)  =  –7 is their sum.     

_.     

For three consecutive tiles with numbers \(a_i, a_{i+1}, a_{i+2}\), we require \(a_i + a_{i+1} + a_{i+2} = 3\).

Since each number is in {-1, 0, 1}, the maximum sum is \(1 + 1 + 1 = 3\), so \(a_i = a_{i+1} = a_{i+2} = 1\).

So, all 7 tiles must have number 1 to satisfy for all consecutive triples.

Each of the 7 tiles, all numbered 1, can be red or blue, giving \(2^7 = 128\) color assignments.

So there are 128 possible arrangements.

mercurym999

THX, mercurym999 !!!

Another method using the Law of Cosines  (and assuming a radius of 1 )

d^2  =  1^2  + 1^2  - 2 ( 1 * 1)  cos (127° -37°)

d^2  = 2  - 2  cos (90°)

d^2  = 2 - 2 (0)

d^2  = 2

d =  sqrt 2

For points P₁ = (cos 37°, sin 37°) and P₂ = (cos 127°, sin 127°), the distance is:

√((cos 127° - cos 37°)² + (sin 127° - sin 37°)²).

The angle difference: 127° - 37° = 90°. Simplify:

(cos 127° - cos 37°)² + (sin 127° - sin 37°)².

Expand:

= cos² 127° - 2 cos 127° cos 37° + cos² 37° + sin² 127° - 2 sin 127° sin 37° + sin² 37°.

Since cos² θ + sin² θ = 1:

= (1 + 1) - 2 (cos 127° cos 37° + sin 127° sin 37°).

Use the cosine angle sum identity:

cos 127° cos 37° + sin 127° sin 37° = cos (127° - 37°) = cos 90° = 0.

Thus:

= 2 - 2 · 0 = 2.

Distance:

√2.

a)  increases on   (0, inf)

     decreases on  (-inf, 0)

b)  derivative =  2x

c)  sign on f'(x) from (-inf, 0)  =  -

     sign on f'(x) from (0, inf) = +

\( \sin 4A = \sin A + \sin 2A \)

sin ( 2(2A))  = sin A + sin 2A

2sin2Acos2A  = sinA + sin2A

2sin2A ( 1 - 2sin^2A)  = sinA + sin2A

2sin2A  - 4sin2A(sin^2A) = sinA + sin2A

sin(2A) - 4sin(2A)* (sin^2( A))  = sin(A)

2sinAcosA - 8sinAcosA (sin^2A) = sinA

sinA [ 2cosA - 8cosA (1 -cos^2A) ] - 1 =  0

sinA  = 0     A =  0”

2cosA - 8cosA (1 -cos^2 A) = 1

2cosA - 8cosA + 8cos^3A - 1 = 0

8cos^3 A - 6cosA - 1 =  0

Let  A  = x

8x^3 - 6x - 1   =  0

Using WolframAlpha to solve this  cubic   gives  x ≈  .93969

cos A  =.93969

arccos (.93969)  = A  =  20°

Thx, Bosco  !!!

Another way

first term  = a

second term  = a + d

third term  =  a + 2d

a + ( a + d) + (a + 2d)  = 9

3a + 3d  = 9

a + d =  3

a =  3 - d

a    d           series

0    3            0  3  6

1    2            1  3  5

2    1            2  3  4

3    0            3  3  3   ( if  "d"  allowed to  = 0 )

4   -1            4  3  2

5   -2            5  3  1

6   -3            6  3  0

Same answer as Bosco !!!!

hmmm im getting $3 \pi,$ anyway i used wolframalpha for \[\left| z + \frac{1}{z} \right| \le 2.\] and I got this