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17) Candy brought 30 cookies to a party. List all the ways she can arrange them on a plate in equal rows.

Phantasy [73]

Answer:Candy can arrange in 1, 2, 3, 5, 6, 10, 15 ways

2, 3, 4, 6, 8, 12 number of people can sit at each table

Solution:

Candy brought 30 cookies to a party. List all the ways she can arrange them on a plate in equal rows.

Given that Candy brought 30 cookies to party

She has to arrange the ways she can arrange them on plate in equal rows

We have to list the factors of 30

factors of 30 = 1, 2, 3, 5, 6, 10, 15

So she can arrange in 8 ways

Possible arrangement are 1, 2, 3, 5, 6, 10, 15 rows

David is setting up tables for 24 people at the party. The same number of people will sit at each table, and no one will sit alone. How many people can sit at each table? List all possibilities.

Fcators of 24 = 1, 2, 3, 4, 6, 8, 12

Here we can eliminate 1 since given that no one will sit alone

Thus 2, 3, 4, 6, 8, 12 number of people can sit at each table

ANSWER NOT BY ME

Step-by-step explanation:

7 0

3 years ago

What is the volume of a cylinder with a diameter of 20 meters and a height of 4 meters?

Aleks [24]

Answer:

1256.64 cubic meters

Step-by-step explanation:

4 0

3 years ago

Read 2 more answers

Is the power produced directly proportional to the wind speed, give reasons for your answer?

yuradex [85]

Answer:

Is the power produced directly proportional to the wind speed.

No.

A proportional function can be described as y = a * x

So (0, 0) is on the graph, not (3, 0).

0 / 3 = 0

3800/6 = 633.33

7600/9 = 844.44

13600/12 = 1133.33

The quotient y/x should be always the same.

4 0

4 years ago

One number is 3 more then twice another. The sum of the numbers is 12

aev [14]

It’s 9. 9+3=12
3,6,9,12
The twice of 3 is six more than twice is nine.

3 0

3 years ago

Math pls help!! answers?

statuscvo [17]

Answer: Choice B) Infinitely many solutions

one solution: x = 8, y = -7/2, z = 0
another solution: x = -12, y = 13/2, z = 10

=======================================================

Explanation:

Here's the starting original augmented matrix.

$\left[\begin{array}{ccc|c} 1 & 0 & 2 & 8\\5 & 1 & 9 & 73/2\\-4 & 0 & -8 & -32\\\end{array}\right]$

We'll multiply everything in row 3 (abbreviated R3) by the value -1/4 or -0.25, which will make that -4 in the first column turn into a 1.

We use this notation to indicate what's going on: $(-1/4)*R3 \to R3$

That notation says "multiply everything in R3 by -1/4, then replace the old R3 with the new corresponding values".

So we have this next step:

$\left[\begin{array}{ccc|c} 1 & 0 & 2 & 8\\5 & 1 & 9 & 73/2\\1 & 0 & 2 & 8\\\end{array}\right]\begin{array}{l} \ \\\ \\(-1/4)*R3 \to R3\\\end{array}$

Notice that the new R3 is perfectly identical to R1.

So we can subtract rows R1 and R3, and replace R3 with the result of nothing but 0's

$\left[\begin{array}{ccc|c} 1 & 0 & 2 & 8\\5 & 1 & 9 & 73/2\\0 & 0 & 0 & 0\\\end{array}\right]\begin{array}{l} \ \\\ \\R3-R1 \to R3\\\end{array}$

Whenever you get an entire row of 0's, it always means there are infinitely many solutions.

-------------------

Now let's handle the second row. That 5 needs to turn into a 0. We can multiply R1 by 5, and subtract that from R2.

So we need to compute 5*R1-R2 and have that replace R2.

$\left[\begin{array}{ccc|c} 1 & 0 & 2 & 8\\0 & 1 & -1 & -7/2\\0 & 0 & 0 & 0\\\end{array}\right]\begin{array}{l} \ \\5*R1-R2 \to R2\ \\\ \\\end{array}$

Notice that in the third column of R2, we have 9-5*2 = 9-10 = -1. So we have -1 replace the 9. In the fourth column of R2, we have 73/2 - 5*8 = -7/2. So the -7/2 replaces the 73/2.

--------------------

At this point, the augmented matrix is in RREF form. RREF stands for Reduced Row Echelon Form. It seems a bit odd that the "F" of "RREF" stands for "form" even though we say "form" right after "RREF", but I digress.

Because the matrix is in RREF form, this means R1 and R2 lead to these equations:

$R1 : 1x+0y+2z = 8\\ R2: 0z+1y-1z = -7/2$

which simplify to

$R1: x+2z = 8\\R2: y-z = -7/2$

Let's get the z terms to each side like so:

$x+2z = 8\\x = -2z+8\\\text{ and }\\y-z = -7/2\\y = z-7/2\\$

Therefore, all of the solutions are of the form $(x,y,z) = (-2z+8, z-7/2, z)$ where z is any real number.

If z is allowed to be any real number, then we can simply pick any number we want to replace it. We consider z to be the "free variable", in that it's free to be whatever it wants. The values of x and y will depend on what we pick for z.

So the concept of "infinitely many solutions" doesn't exactly mean we can pick just any triple for x,y,z (admittedly it would be nice to randomly pick any 3 numbers off the top of my head and be done right away). Instead, we can pick anything we want for z, and whatever we picked, will directly determine x and y. The x and y are locked into place so to speak.

Let's say we picked z = 0.

That would lead to...

$x = -2z+8\\x = -2(0)+8\\x = 8\\\text{ and }\\y = z-7/2\\y = 0-7/2\\y = -7/2\\$

So z = 0 would lead to x = 8 and y = -7/2

Rearranging the items in alphabetical order gets us:

x = 8, y = -7/2, z = 0

We have one solution of (x,y,z) = (8, -7/2, 0)

Now let's say we picked z = 10

$x = -2z+8\\x = -2(10)+8\\x = -12\\\text{ and }\\y = z-7/2\\y = 10-7/2\\y = 13/2\\$

So we have x = -12, y = -13/2, z = 10

Another solution is (x,y,z) = (-12, 13/2, 10)

There's nothing special about z = 0 or z = 10. You can pick any two real numbers you want for z. Just be sure to recalculate the x and y values of course.

To verify each solution, you'll need to plug them back into the original equations formed by the original augmented matrix. After simplifying, you should get the same thing on both sides.

8 0

3 years ago