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Problem 1:

Since ABCD is a regular tetrahedron with volume 18, each of its four triangular faces has the same area. Let this area be K.

Step 1: Finding the Height of Tetrahedron ABCD

The volume of a tetrahedron is given by:

Volume = (1/3) * Base Area * Height

In this case, the base area is K (area of one face), and the volume is 18. We need to find the height (h) of the tetrahedron.

18 = (1/3) * K * h

Solving for h:

h = (18 * 3) / K = 54 / K

Step 2: Relating Tetrahedron ABCD to Pyramid EFGH

Each face of pyramid EFGH is half the size of a face of tetrahedron ABCD. This is because each face of EFGH is formed by connecting the midpoints of the edges of a face in ABCD.

Therefore, the base area of pyramid EFGH is K/2.

Step 3: Finding the Height of Pyramid EFGH

Since each face of pyramid EFGH is the centroid of the corresponding face in ABCD, the height of EFGH (h') is two-thirds of the height of ABCD (h).

h' = (2/3) * h = (2/3) * (54 / K) = 36 / K

Step 4: Finding the Volume of Pyramid EFGH

Using the formula for the volume of a pyramid:

Volume = (1/3) * Base Area * Height

The base area of EFGH is K/2, and the height is 36/K.

Volume of EFGH = (1/3) * (K/2) * (36 / K) = 6

Therefore, the volume of pyramid EFGH is 6.

a. You are choosing five people out of 12, so the number of ways is C(12,5) = 660.

To avoid ever having the same group of 5 members review a book, we need to ensure there are enough total members (z) such that any group of 5 can be chosen without repeating a combination.

Here's the key idea:

We care about the number of distinct groups of 5 members we can form, not just the total number of possible selections.

Combinations vs. Permutations:

Combinations: Order doesn't matter (e.g., John, Mary, Sarah is the same group as Sarah, John, Mary).

Permutations: Order matters (e.g., John reviewing first is different from Mary reviewing first).

In this case, since the order the members review the book doesn't matter, we're interested in the number of combinations of 5 members we can choose from a group of z people.

Using Combinations Formula:

The number of ways to choose 5 members out of z for a book review can be calculated using the combinations formula:

nCr = n! / (r! * (n-r)!)

where:

n = Total number of members (z)

r = Number of members chosen for review (5 in this case)

Finding the Smallest z:

We want to find the smallest positive integer z such that the number of combinations of choosing 5 members (nCr) is greater than or equal to the total number of days (400). In other words:

nCr >= 400

Trial and Error with Combinations:

We can try different values of z and calculate the corresponding nCr using the formula. The smallest z that satisfies the condition will be our answer.

For z = 5 (5 choose 5): 1 (There's only one group possible - all 5 members) - not enough.

For z = 6 (6 choose 5): 6 (We can choose 5 members out of 6 in 6 ways) - still not enough.

For z = 7 (7 choose 5): 21 (We can choose 5 members out of 7 in 21 ways) - finally enough!

Therefore, the smallest positive integer z is 7.

There are 6 students in the science club, and 3 students are a member of both clubs.

The number to students that are members of only the math club is 3-3 = 0

The number of students thare are members of only the science club is 6 - 3 = 3.

There are a total of 3 students who are in the math or science club but not both.

Problem: The real numbers $c, b, a$ form an arithmetic sequence with $a\ge b\ge c\ge 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root?

Using the discriminant, you can work out that the answer is -7 - 4*sqrt(3).

To calculate the total cost of the Chromebook with the payment plan, we can leverage the concept of compound interest. Here's how:

Monthly Interest Rate:

The annual interest rate is 6%. Since the payments are compounded monthly, we need to convert this to a monthly rate.

Monthly interest rate = Annual interest rate / Number of compounding periods per year

Monthly interest rate = 6% / 12 months = 0.5% per month (converted to a decimal)

Number of Payments:

The payment plan involves 18 equal monthly payments.

Cost of Chromebook:

The Chromebook's cost is $499.99.

Formula for Monthly Payment:

We can use the following formula to calculate the monthly payment amount:

Monthly payment = Loan amount * (Monthly interest rate / (1 - (1 + Monthly interest rate)^(-Number of payments)))

Calculation:

Plug in the values:

Loan amount = $499.99

Monthly interest rate = 0.005 (converted from 0.5%)

Number of payments = 18

Calculate the monthly payment:

Monthly payment = $499.99 * (0.005 / (1 - (1 + 0.005)^(-18)))

Monthly payment = $28.61 (rounded to two decimal places)

Total Cost:

Since there are 18 equal payments of $28.61 each, the total cost will be:

Total cost = Number of payments * Monthly payment

Total cost = 18 payments * $28.61/payment

Total cost = $514.98 (rounded to two decimal places)

Therefore, by taking the payment plan with monthly compounding interest, you will end up paying a total of $514.98 for the $499.99 Chromebook.

Since children must be next to two adults, let's consider the children as a single unit. There are now 5 "units" to arrange around the table (4 adults + 1 unit of 4 children).

Here's how to solve the problem:

Total Arrangements without Restriction:

Initially, ignoring the child requirement, we can arrange 5 units (adults and children) around the circular table.

For circular arrangements of n distinct objects, there are (n-1)! ways.

In this case, there are (5-1)! = 4! = 24 ways to arrange the units.

Overcounting due to Rotation:

However, in a circular arrangement, rotating the entire configuration doesn't create a new seating order.

So, we've overcounted the arrangements by the number of ways to rotate 5 objects in a circle.

There are 5 positions, and rotating one slot to the right fills all the positions eventually.

Therefore, we've overcounted by a factor of 5.

Correcting for Overcounting:

To get the actual number of unique arrangements, we need to divide the initial arrangements by the overcounting factor:

Unique Arrangements = Total Arrangements / Overcounting Factor

Unique Arrangements = 24 arrangements / 5 rotations

Unique Arrangements = 4.8 (Since the answer deals with arrangements, we can't have a fraction of a seating)

Accounting for Child Arrangement:

So far, we've treated the children as a single unit. But within that unit, the 4 children can be arranged in 4! ways.

Final Answer:

To get the total number of arrangements where each child sits next to two adults, we multiply the number of unique arrangements for the units (adults and children) by the number of ways to arrange the children within their unit:

Total Arrangements = Unique Unit Arrangements * Child Arrangements

Total Arrangements = 4 * 4! = 4 * 24 = 96

Therefore, there are 96 ways to seat the children and adults around the table such that each child sits next to two adults.

Let's say the distance between the café and the boba shop is d, and Carissa walks the same distance in reverse on her way back.

Let's break the trip down into three parts: walking uphill, walking across flat ground, and walking downhill.

When walking uphill, Carissa walks at a speed of 100 meters per hour, so it takes her d/100 hours to complete this part of the trip.

When walking across flat ground, Carissa walks at a speed of 120 meters per hour, so it takes her d/120 hours to complete this part of the trip.

When walking downhill, Carissa walks at a speed of 150 meters per hour, so it takes her d/150 hours to complete this part of the trip.

So, the total time for the round trip is:

d/100 + d/120 + d/150 + d/100 + d/120 + d/150 = (11/600)d

To find the average speed, we need to divide the total distance (2d) by the total time, which is (11/600)d:

average speed = 2d / ((11/600)d) = 1200 / 11 ≈ 109.1 meters per hour

Therefore, Carissa's average speed during the entire round trip is approximately 109.1 meters per hour.

The third equation has no y-term, so let's eliminate the y-term from the combination of the first two equations.

x + y + 3z  =  10     --->     x -2     --->     -2x - 2y - 6z  =  -20

4x + 2y + 5z  =  7       --->                      4x + 2y + 5z  =  7

Add down the columns:                             -6x - z  =  -13   

Since  kx + z  =  3        --->         z  =  3 - kx

Substituting these:        -6x - (3 - kx)  =  -13

                                    -6x - 3 + kx  =  -13

                                      6x + 3 - kx  =  13

                                          6x - kx  =  10

                                          x(6 - k)  =  10

If  6 - k  =  0,  there is no value for x that will result in a product of 10, so  k = 6  results in no possible solution.

for 3 it's the easiest find the side lengths ratio - because the pyramid has 3 45-45-90 triangles, and cube it, multiply by 4, and substract from original cube

for 2 there's a formula for volume of a tetrahedron knowing only  all six side lengths - it appears in murdereous maths, and just plug the sidelengths in - inefficient to do that tho

for 1. try find the ratio of the side lengths using the pythogorean theorem, and also properties of equilateral triangles

Let's track the amount of water in the beaker throughout the process.  Initially, there are 200 mL of water.

First drainage and addition:

Ruth drains 200 mL of solution, which includes some water. Since the solution is well mixed, the proportion of water drained will be the same as the proportion of water in the whole solution.

Proportion of water = 200 mL water / (800 mL acid + 200 mL water) = 1/5

Therefore, the amount of water drained is 1/5 * 200 mL = 40 mL.

After draining, there's 200 mL water - 40 mL drained water = 160 mL water remaining.

Then, Ruth adds 200 mL of pure water, bringing the total water content to 160 mL + 200 mL added water = 360 mL.

Second drainage and addition:

Following the same logic as the first drainage, 40 mL of water gets drained out of the 200 mL solution removed (1/5 proportion).

There's 360 mL water - 40 mL drained water = 320 mL water remaining.

Finally, adding 200 mL more water brings the final amount of water to 320 mL + 200 mL added water = 520 mL.

Therefore, there are now 520 mL of water in the beaker.

Workers and Days:

We know 5 workers can complete the job in T days (initially unknown).

Impact of Additional Worker:

Hiring one more worker reduces the completion time by 12 days, meaning it takes (T-12) days with 6 workers.

Work Relationship: There's a constant amount of work to be done. We can represent this with the following equation:

Work = Rate x Time

In this case, Work is constant (the job itself).

Rate represents the combined speed of the workers (more workers = faster rate).

Time is the number of days to complete the job.

Translating to Equations:

From the work relationship, we can write equations for both scenarios (5 and 6 workers):

5 Workers: Work = (Rate of 5 Workers) * T days

6 Workers: Work = (Rate of 6 Workers) * (T-12) days

Since the Work is the same, we can equate both expressions:

(Rate of 5 Workers) * T days = (Rate of 6 Workers) * (T-12) days

Relating Rates to Workers:

We can assume the rate of each worker is constant (say, w units of work per day).

Therefore, the Rate of n Workers = n * w. Substituting this into the equation from step 5:

5w * T = 6w * (T-12)

Solving for T:

Expand and solve for T:

5wT = 6wT - 72w

T = 72 days (This is the original completion time with 5 workers)

New Completion Time Requirement:

We need to reduce the completion time by 32 days. This means the new desired completion time is (T-32) days.

Workers Needed for Faster Completion:

We can again use the work relationship:

Work = (Rate of x Workers) * (T-32) days

We know the Work is constant and the original time (T). We need to find the number of workers (x) required to achieve the new completion time (T-32).

Solving for Additional Workers:

Since the Rate of each worker is w (assumed constant), we can rewrite the equation from step 9:

5w * T = x * w * (T-32)

We know T from step 7 (72 days). Substitute and solve for x (additional workers):

5 * 72 = x * (72 - 32)

x = 12

Answer: We already have 5 workers. To achieve the 32-day reduction, we need to hire 12 additional workers. Therefore, a total of 5 + 12 = 17 workers are needed.

1.

B              

       3        E

3            1

C    1   D                           A

For convenience, let BC = 3  and the radius of the semi-circle =  1

BC = BE

Let D be the center of the semi-circle   and let E  be the point where the semi-circle touches AB

Triangle EAD  similar to Triangle CAB

DE / EA = BC /  CA

1 / (BA - 3)  = 3 / CA

CA =  3 (BA - 3)

AC = 3BA - 9

BA^2  = 3^2 + (3BA - 9)^2

BA^2  = 9 + 9BA^2 - 54Ba + 81

BA^2  = 9BA^2 -54BA + 90

8BA^2  - 54BA + 90 =  0

Solving this produces .....BA =  15/4

AC = 3(15/4)  - 9  =  9/4

AC /  BC  =   (9/4) / 3  =  9 / 12  =   3 / 4

Note that  the sequence is

1000 , 1000(-1/3)  , 1000(-1/3)^2 , 1000(-1/3)^3   ........

We only need to  consider the terms that are positive  (these will be the odd terms)

So

1000 (1/3)^(n -1)  >  1         where n is  the nth term

(1/3)^(n -1)  >  1/1000    take the log of  both sides

log (1/3)^(n -1) >  log (1/1000)       and we can write

(n -1) log (1/3) > log (.001)      

Divide  both sides by log (1/3)....since this is  negative, we need to reverse the inequality sign

n - 1   <  log (.001) / log (1/3)

n <    1 +  log (.001) / log (1/3)   ≈  7.28

Note  that  the 7th term    =  1.37

Note that the 9th term = .152

So  only  terms   1 , 3 , 5  and 7   are >  1

x( x + 2)  + x ( x+ 1) =  kx (x + 1) (x + 2)

x^2 + 2x  + x^2 + x  =  kx ( x^2 + 3x + 2)

2x^2 + 3x  = kx^3 + 3kx^2 +  2kx

kx^3 + (3k-2)x^2 + (2k - 3)x  =  0

x [ kx^2  + (3k -2)x + (2k -3)  = 0

(3k - 2)^2  - 4 [k *(2k -3)]  < 0

9k^2 - 12k + 4 - 8k^2 + 12k  < 0

k^2 + 4  <  0

k^2  <   -4

k < 2i      and  k > -2i

k = (-2i, 2i)