power27

You want to find the least common multiple of 12 and 8 that is closest to 100.

12=3*2*2

8=2*2*2

so LCM(12,8)=24

List multiples of 24:

24

48

72

96

120
 

96 is the closest to 100, so the students want to buy 96/12=8 packages of hotdogs and 96/8=12 packages of buns.

If you draw a line from G to H, the quadrilateral becomes two issosolies triangles.

Use the pythagorean theorem to solve for GH using right triangle GHW. GH=sqrt(2).

Because GHW is a right issosolies triangle, angles WGH and WHG are 45 degrees. 105-45=60 so GHB is an equlateral triangle and GH=HB=GB=sqrt(2).

The perimiter is 1+1+sqrt(2)+sqrt(2)=2+2sqrt(2) or ~4.8

In an islsceles triangle two angles are the same and one is different. 40 could be the different angle or the angle there is two of.

The angles in a triangle sum to 180.

If 40 was the double angle, the third would be 180-40*2=100.

If 40 was the different angle the other two would be (180-40)/2=60.

100 is the largest so D is the answer.

Graph the line x=-1 and count how many times the red graph crosses it.

x=-1 crosses it 4 times, around 3, 1.5, -1.5, and -4.

The exterior angles of any polygon sum to 360 degrees. Since it is a 21-gon, 360/21=17 1/7 or ~17.14 degrees for each angle.

This can be written as a system of equations to solve.

Let b be the number of belts and w be the number of wallets.

b+w=42

7b+4w=213

Multiply the first equation by 4

4b+4w=168

Subtract this equation from the second.

(7b-4b)+(4w-4w)=213-168

The ws cancel out!

Now simplify.

3b=45

b=$15

This can be substituted back into the first equation

15+w=42

Subtract 15

w=$27

Double check to make sure these numbers work:

15+27=42, this one works

7(15)+4(27)=213 This one also works so this is the correct answer.

There are 26 letter possibilities and 10 numbers for 36 total options. There are 36*35*34*33=1,413,720 possible passwords.

To have a password with one number, there must be 3 letters and 1 number. There are 26C3 possible letter options and 10C1 possible numbers. 26C3=2600

10C1=10

2600*10=26000 possible combinations of characters with exactly one number. Multiply this by 4! to get 624000 passwords with one number.

624000/1413720~= 44.14% chance there is one number.

For this set of linear equations to have no solitution, the slopes must be the same so they are parallel lines.

First, rearange the equations so they are in y= form.

15x-18y=-19

subtract 15x

-18y=-19-15x

divide by -18

y=19/18 + 5/6x

The slope of this line is 5/6.

bx-12y=12

subtract bx

-12y=12-bx

divide by -12

y=(b/12)x-1

The slope is b/12.

For no solitution the slopes are the same so

5/6=b/12

multiply by 12

10=b

First square the equation and rearrange

3sqrt(x)=x+k

9x=x²+2kx+k²

0=x²+2kx-9x+k²

Factor to make it a quadratic

0=x²+(2k-9)x+k²​

For a quadratic equation to have one solitution, the discrmimnant must be 0.

b²-4ac=0

Substitute in the coefficents:

(2k-9)²-4(1)(k²)=0

Expand

4k²-36k+81-4k²=0

Simplify

-36k+81=0

Solve

-4k+9=0

-4k=-9

k=2.25

You're really not supposed to be discussing this yet... people could still be taking it until the 19th. I know all the schools in my area will have to take it at a date later than the 12th, which might be Monday or Tuesday.