CPhill

40x - 80 = 160

Add 80 to both sides

40x = 240    divide both sides by 40

x = 6

Yep.....it's just a 3-4-5 Pythagorean "Triple" triangle "scaled" up by a factor of 100

It's correct....!!!

We can use the Pythagorean Theorem for this.....the hypotenuse = 500 and one leg = 400

So.....the height - h - is given by

h = √[500^2 - 400^2 ] = 300 ft

If  "C" is the cost per funnel.....then.....the number we make doesn't matter.....each one costs "C" dollars to make......!!!

The side must be 240/5 = 48km

We have a "formula" to find the area given the side length....I won't derive it for you, but it is given by:

A = 1.7204 s^2   where s is the side length....so we have

A = 1.7204 (48)^2 = 3963.8 km ^2

I'll try this one, too

Ada could choose any of 9 gifts and the other 9 people have 9! ways to choose theirs. Note that some of these include the possibiiity that Billy does draw his own gift.

After she draws hers, we want to look at the total scenarios that have Billy "stuck" with his own gift.

In this scenario, Ada draws 8 gifts (since she doesn't draw her own, nor Billy's)....and the other 8 people have 8! ways of choosing amongst the remaining 8 gifts. Notice that some of these include the possibility that they actually draw their own gifts, but that's not a stated consideration.

So  ... P[ Billy draws his own gift l Ada doesn't draw her gift ] = 

[8 * 8! ] /  [9 * 9!] = (8/9)(1/9) = 8/91  ..... the same as found by Melody....

z/3 = z - 2    multiply both sides by 3

z= 3(z - 2)   simplify

z = 3z - 6     add 6 to both sides and subtract z from both sides

6 = 2z          divide both sides by 2

3 = z

We need to be careful here... we have a SSA situation.....using the Law of Sines we have

sinA / 10  = sin C / 7

7*sin 80 / 10 = sin C

sin^-1( 7* sin 80 / 10)  = about 43.6°

And angle B = (180 - 80 - 43.8) = 56.4°

We need to check to see if B could be  the supplemental to 56.4°

But. this is impossible since, if A is 80°, the angle supplemental to 56.4° would be 123.6° and these two angles added together would be > 180°

So....we only have one triangle.....

3^n = 513 ....    let's just take the log of both sides....so we have

log 3^n  = log 513    and by a log property we can write

n * log 3   =  log 513           divide both sides by log 3

n = log 513  / log 3      

And you can evaluate the right side for yourself with a calculator....

{Hint......the answer should be between 5 and 6 }