CPhill

There are one billion ( 10⁹ ) nanometers in a meter

And there are  1609.344 meters in a mile.

And the diameter of each gold atom is 2 x .135 = .270nm

So we have.....

(1609.344 x 10⁹ ) nm /  (.270) nm = about 5,960,533,333,333   gold atoms...... (about 6 trillion)....!!!!

Nice BIG explanation, ND !!!   REALLY BIG !!!

I want to point out someting...."FOIL" is a useful method for expanding multiplied binomials. We could also do this....

(m - 50) (m + 18)

"Distribute" the "m" in the first set of parenthesis over the two terms in the other set of parenthesis...so we have......

m * m  + m *18   =  m² + 18m

Them, do the same thing with the "-50"    ...and we have......

-50m - 900

And combining like terms, we have

m² + 18m - 50 m - 900  =

m² - 32m - 900   ......just as NinjaDevo found

The only reason I mention this is that "FOIL"  can't be effectively used if we are multiplying polynomials that involve more than two terms. In that case, another "ditribution" method is often necessary......!!!!

LOL!!!!....thanks for that one........

Just add the numbers and divide by 2 as we normally would.

For instance, the average of 10 and -2  = (10 + -2) / 2  = 8/2 = 4. And this is true because 4 is an equal distance between 10 and -2  ....!!!

The angle of depression is .7 degrees. This means the observer is looking at  .7 degrees below the horizontal.

The angle between horizontal  and vertical = 90 degrees. So the remaining part of the angle is given by (90 -.7) = 89.3 degrees.

The distance the ship is from the lighthouse is the opposite side of this angle and the adjacent side is the height of the lighthouse.

Does that help??

So we know the angle of depression (42 minutes)  = .7°. And we can use the following to find the distance the ship is from the lighthouse (call this x)

tan (90 - .7) = x / ht of lighthouse

tan( 89.3) = x / 80 ft

Multiply both sides by 80 ft

(80 ft.) tan (89.3) = x = about 6547.8 ft = about 1.24 miles

Well, DS...I'm probably not that smart......I just know a few limited things  !!!!

Solving for x, we have

2x  > -10     and    9x < 18

x > -5           and      x < 2  

So, our solution would be

-5 < x < 2

Here's the number line graph:

I assume you want to solve this:

2cos^2(θ)−1 = 0

Add 1 to both sides

2cos^2(θ) =1

Divide by 2 on both sides

cos^2(θ) = 1/2

Take the square root of both sides

cos(θ)= ± 1/√2

So θ  = pi/4, 3pi/4, 5pi/4 and 7pi/4  in the interval  [0, 2pi]

The length and width of the frame is given by:

8 + 2(1+ 1/4) = 8 + 2 (5/4) = 8 + 5/2 = 8 + 2 + 1/2 = 10+1/2 inches

So the area is given by:

(10 + 1/2) ^2  = (10.5)^2 = 110.25 sq. in. = the area of the frame