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In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color, and at lest three of the squares are red.  How many ways are there to color the five squares?     

In order not to have two reds next to each other,     

the three reds must occupy squares 1, 3, and 5.     

That leaves squares 2 and 4 to fill, and two colors     

to do it with. There are only four ways.     

Blue Blue     Yellow Yellow     Blue Yellow     Yellow Blue     

So, the answer is four ways.     

_.     

Total permutations of 6 distinct children in 6 distinct seats: 6!=720

Fix sibling pair i to sit in the same column.

The pair has two siblings s1 and s2→ 2 ways to order them vertically (top-bottom or bottom-top).

There are 3 columns → 3 possible columns for them to occupy.

The remaining 4 children can be permuted in the remaining 4 seats arbitrarily → 4!=244! = 244!=24 ways.

Therefore,

3×2×4!=3×2×24=144

2 ways to order siblings vertically → 2×2=4

3×4×2=24

6×8=48

6!−408=720−408=312

312 ways 

A cylinder has a height of 16 feet and a radius of 19 feet. What is its volume? Use ​𝜋 ≈ 3.14 and round your answer to the nearest hundredth.     

Vol_cylinder  =  π r² h     where r is the radius of the base and h is the height of the cylinder     

                                    V  =  (3.14)(19²)(16)     

                                    V  =  (3.14)(361)(16)     

                                    V  =  18,136.64     

_.     

A man has a combination of $5 and $10 notes.     

If the total number of notes is 20, and the total value     

of money $120, what is the quantity of $10 notes?     

Call F the number of fivers     

Call T the number of ten-spots     

F + T  =  20          ——>>  F  =  20 – T      (equation 1)     

5F + 10T  =  120                                        (equation 2)     

Substitute (20 – T) in place of F in (equation 2).     

(5)(20 – T) + 10T  =  120     

100 – 5T + 10T  =  120     

                    5T  =  20     

                      T  =  4       This is the number of ten-dollar notes, which also     

                                        means there are F = 20 – 4 = 16 five-dollar notes.     

                                        We'll use the number of five-dollar notes later.          

check answer     

(eq 1)   16 + 4  =?  20     

                   20  =  20     

(eq 2)   (5 • 16) + (10 • 4)  =?  120     

                   80 +    40        =?  120     

                                  120  =  120     

_.     

What is the ratio of the volume of the cylinder to the volume of the cone, when x=2 and y=2?     

The ratio is 3 to 1     

Volume of a Cylinder .......  V  =  π r² h    where r is the radius of the base and h is the height.     

Volume of a Cone      .......  V  =  (1/3) π r² h     where r is the radius of the circular base and h is the height.     

_.     

\(Let\ \omega\ be\ a\ nonreal\ root\ of\ x^3 = 1.\\ Compute\ (1 - \omega + \omega^2)^6 + (1 + \omega - \omega^2)^6.\)

\(x^3=1\\ x=\sqrt[3]{1}\\ \color{blue}\sqrt[3]{1}\ is\ not\ a\ transcendental\ number.\ \sqrt[3]{1}=1 \\ \omega=1\\ (1 - 1 + 1^2)^6 + (1 + 1 - 1^2)^6=2 \)

  !

Two points on a circle of radius 1 are chosen at random. Find the probability that the distance between the two points is at most 1.5.      

The first point can be anywhere.     

The second point has to be within 1.5 of the first.     

The second point can be on either side of the first, so the span is 3.0.    

The circumference is 2 π r = 2 π     

So the probability is (3) / (2)(3.1416)     

Probability  =  (1.5) / (3.1416)  =  0.4775  =  47.75%     

_.     

Emily and Tammy are planting vegetables in their garden. Emily is planting 81 tomato plants and Tamara is planting 54 corn plants. If each row of their garden has the same number of plants in them, what is the GREATEST number of plants in a row?     

tomatos  =  3 rows with 27 plants in each row     

corn        =  2 rows with 27 plants in each row     

_.     

Let z_1 and z_2 be complex numbers such that |z_1| = 5 and (z_1/z_2) + (z_2/z_1) = 1. We are asked to find |z_1 - z_2|^2.

Step 1.
Let z_1 = 5e^{i\theta_1}, z_2 = 5e^{i\theta_2} (since their magnitudes are both 5).

Then
(z_1/z_2) + (z_2/z_1) = e^{i(\theta_1-\theta_2)} + e^{-i(\theta_1-\theta_2)} = 2 cos(\theta_1-\theta_2).

Given 2 cos(\theta_1-\theta_2) = 1, so cos(\theta_1-\theta_2) = 1/2. Thus (\theta_1-\theta_2) = 60^\circ or 300^\circ.

Step 2.
|z_1 - z_2|^2 = |z_1|^2 + |z_2|^2 - 2|z_1||z_2| cos(\theta_1-\theta_2)
= 25 + 25 - 255*(1/2)
= 50 - 25
= 25

Answer: 25

mercurym999

Felicia opens a chequing account that charges a monthly maintenance fee of $10.75. She starts her account by depositing $500 at the start of January. At the end of March, how much money will she have in her account? Assume that she does not make any deposits or withdrawals over this time period.     

End of Month     Maint Fee     Balance     

       Jan               10.75            489.25     

       Feb               10.75            478.50     

       Mar               10.75            467.75     

_.     

My sock drawer contains two red socks, two white socks, and two blue socks.  For three days in a row, I pull out two socks out of my sock drawer at random.  What is the probability that I get matching socks on all three days?      

The first sock can be anything                                prob  1 / 1     

Probability second sock matches                            prob  1 / 5     

The third sock can be anything                                prob  1 / 1     

Probability fourth sock matches                                prob  1 / 3     

If the above happens, all that's left is the last pair     prob  1 / 1     

To find the probability of every condition listed above     

happening, multiply individual probabilities together.     1 • (1/5) • 1 • (1/3) • 1  =  1 / 15     

_.     

  -2+(-7/2)=     

                                                             – 2 +(– 7/2)  =    

  Need a common denominator            – (14 / 2) – (7 / 2)  =  – (21 / 2)     

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