CPhill

your team is winning 27-18. The announcer says they have scored five times on field goals and touchdowns and haven't missed a point-after-touchdown yet. Can you tell how many touchdowns and field goals your team has made?

Let x be the number of TDs they have scored and let y be the number of field goals they have scored.

Then x + y = 5

And the number of TDs (x) times the point value for each (plus the pt after), i.e., (7) = 7x

And the number of FGs times their point value each = 3y

So the number of total points they have scored is given by

7x + 3y = 27

So we have two equations here

7x + 3y = 27

x + y = 5

Let's solve this system...Using the second equation and subtracting x from both sides, we have y = 5 - x

Now, substituting for y in the first equation, we have

7x + 3(5 - x) = 27    Simplifying, we have

7x + 15 - 3x = 27

4x + 15 = 27  Subtract 15 from both sides

4x = 12   Divide both sides by 3

x = 3.......This is the number of TDs

and the number of field goals (y) is given by 5 - x   =   5 - 3  = 2

That makes sense......... 3 TDs and 2 FGs = 21 + 6 = 27

A machinist drilled a cone-shaped hole into a solid cube of metal as shown. If the cube has sides of length 7 centimeters, what is the volume of the metal after the hole is drilled? Use π ≈ 3.14 and round your answer to the nearest tenth

I don't see any illustration, but I think I can imagine what you might have. I think the radius of the cone is just 3.5 cm and its height = 7

Then, the volume of the cone = (pi) * (3.5)^2 * (7) / 3

= (3.14) * (3.5)^2 * 7 / 3 ≈ 89.75 cm^3

The total volume of the cube = (7)^3 = 343 cm^3

Therefore, the volune of metal that remains after the hole is drilled is just (343 - 89.75)cm^3 ≈ 253.25cm^3

I hope I've guessed correctly for you as far as the illustration goes...the math is correct, however !!

how do you convert 1 - i to polar form

First, let's find r. This is given by r = lzl = SQRT(a^2 + b^2) = SQRT(1^2 + (-1)^2) = SQRT (2)

Now, let us find θ. We have tan θ = b/a = -1 / 1.  Note that this is just a 45 degree angle in the 4th quadrant, i.e., [ -(pi) / 4]

And the polar form is given by

z = r (cos θ + i sin θ) =

√(2) (cos [ -(pi) / 4] + i sin [ -(pi) / 4])

Note one thing here...For completeness sake, any value of [ -(pi) / 4] + 2* pi *n  where n is a positive or negative integer would work since all these angles would give the same answer !!!

The ratio of 8 less than a number to 28 more than that number is 2 to 5. What is the number?

Let's call the number "n"

Then, 8 less than n = n-8, and 28 more than n = n + 28

So we have

(n-8) /(n+28) = 2/5

Cross multiply and set both products  equal

5(n-8) = 2(n+28)   Simpifying, we have

5n - 40 = 2n +56    Add 40 to both sides

5n = 2n + 96   Subtract 2n from both sides

3n = 96 Dividing both sides by 3, we get

n = 32

Check this

(32-8) /(32 +28) = 24 / 60  = 2 /5

And we're OK...........

How can I find the height of a scalene triangle given its side angles, 6, 8, and 10?

You don't have a scalene triangle, you have a right triangle. If its hypoteneuse (the side with the measure of 10) isn't the base, then it's height is either 6 or 8, depending on the orientation of the triangle.

If the hypoteneuse is the base, it's a little more difficult. Let us orient the triangle so that the hypoteneuse is the base and the side with the length of 8 is on the "left" side of the triangle. Using the law of sines, we can find the sine of the angle that lies between the side with length 8 and the hypoteneuse. We can just label this angle as "A."

sin 90 / 10 = sin A / 6    which gives us

sin A = 6 /10= .6

Now, draw a perpendicular from the base to the apex ....this is the height (h)

And using the law of sines again (and realizing that the perpendicular forms a right angle), we have

h / sin A = 8 / sin 90

h / .6 = 8

h = 8(.6)  = 4.8

That one was a little tougher !!!

V = pi* r^2 * h

where r is the radius of the cylinder and h is the height

BTW.....If you click on the "Formulary" tab at the top of the page, you will find a "cylinder" app that will calculate the volume for you !!!

I assume you want the sum of the vertex angles of a regular pentagon.

The measure of one of the vertex angles is given by (n-2)*(180)/n where n is the number of sides of the pentagon.

So we have (5-2)*(180)/5 = 3(36) = 108º

And since we have 5 of these, then 5(108º) = 540º

And that's it !!!