Akhain1

We can solve this system of equations by adding and subtracting the equations strategically to eliminate one of the variables.

Adding the Equations:

Adding the two given equations eliminates the term ab^2:

(a^3 + 3ab^2) + (3a^3 - ab^2) = 679 + 615

Combine like terms:

4a^3 = 1294

Solving for a^3:

Divide both sides by 4:

a^3 = 1294 / 4 = 323.5

Subtracting the Equations:

Subtracting the second equation from the first equation eliminates the term 3a^3:

(a^3 + 3ab^2) - (3a^3 - ab^2) = 679 - 615

Combine like terms:

4ab^2 = 64

Solving for ab^2:

Divide both sides by 4:

ab^2 = 16

Relating a and b:

Since we know both a^3 and ab^2, we can try to express one variable in terms of the other.

From the equation for a^3, we can write a =∛(323.5).

Relating a - b:

We want to find a - b. Since we don't have a direct equation for b, we can try to manipulate the equation for ab^2.

Rewrite the equation for ab^2:

b^2 = a(ab^2) = a * 16

Substitute a = ∛(323.5):

b^2 = ∛(323.5) * 16

Now, we can express b in terms of a:

b = ± 4 * ∛(323.5)

Finding a - b:

Since a and b are real numbers, we can consider both positive and negative values of b. However, we only care about the difference a - b.

There are two cases:

Case 1: b = 4 * ∛(323.5)

a - b = ∛(323.5) - (4 * ∛(323.5))

Factor out ∛(323.5):

a - b = ∛(323.5) (1 - 4)

a - b = -3 * ∛(323.5)

Case 2: b = -4 * ∛(323.5)

a - b = ∛(323.5) - (-4 * ∛(323.5))

Factor out ∛(323.5):

a - b = ∛(323.5) (1 + 4)

a - b = 5 * ∛(323.5)

Conclusion:

Since we don't know the signs of a and b beforehand, both cases are valid. Therefore, a - b can be either -3 * ∛(323.5) or 5 * ∛(323.5).

Both answers are negative and positive multiples of the same cube root, so they essentially represent the same value with opposite signs. The absolute value of a - b is:

|a - b| = |(-3) * ∛(323.5)| = |5 * ∛(323.5)| = 5 * ∛(323.5)

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.

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.

Midpoint BC  =  [ (-9 + 1)/2 , (6 - 4) / 2 ] = [ -4, 1]

Midpoint AB  = [ (5 -9)/2 , (4 + 6) / 2 ] = [ -2, 5]   

Midpoint AC = [ (5 + 1) / 2 , (4 -4) / 2 ] = [ 3 , 0 ]   

Slope  AD  =  [ 4 - 1 ] / [ 5 - -4] =  1/3

Slope CE  = [ -4-5] / [ 1 - -2] =  -3

Slope BF =  [ 6-0] / [ -9-3]  = -1/2

Equation of line  through AD

y = (1/3)(x + 4) + 1                   (1)

Equation of line  through CE

y =  -3 ( x + 2) + 5              (2)

Equation of line  through BF

y = (-1/2)(x -3)       (3)

Set (1), (2)  equal  to find their x intersection

(1/3)(x + 4) + 1 =  -3(x + 2) + 5

x + 4 +  3  =  -9(x + 2) + 15

x+ 7 = -9x -18 + 15

x + 7 = -9x -3

10x = -10

x = -1

y = -3(-1+2) + 5 =  2

Intersection pt  =  ( -1 , 2)

Test in (3)

2  = (-1/2) (-1 -3) 

2 = 2         true