CubeyThePenguin

Rewrite:

\(3^a = 3^{4(b+2)}\)

\(5^{3b} = 5^{a-3}\)

Get rid of the exponents:

a = 4(b + 2)

3b = a - 3

Then you can solve.

8 = 2^3

15 = 3 * 5

LCM(8, 15) = 2^3 * 3 * 5 = 120

Notice that \(25^x - 9^y = (5^x + 3^y)(5^x - 3^y)\). Hope this gets you started.

\(\frac{6t(15) - 2(5) + t(3)}{15} = \frac{12 + (2-t)}{3}\)

\(\frac{93t - 10}{15} = \frac{14 - t}{3}\)

\(3(93t - 10) = 15(14-t)\)

\(279t - 30 = 210-15t\)

\(294t = 240 \quad \rightarrow \quad \boxed{t = \frac{120}{147}}\)

Suppose the original point's coordinates were (x, y). After reflecting across y = x, the new coordinates are (y, x).

So,

S (-6, 1)   ==>  S' (1, -6)

T (-6, 8)   ==>  T' (8, -6)

and so on.

While you'll have to write the email yourself, here's some tips about email etiquette:

Here are the points 1/4 of the way:

(0, 0)

(2.25, 1.5)

(4.5, 3)  [midpoint]

(6.75, 4.5)

(9, 6)

We want the sum of the coordinates of the point in bold, which is 3.75.

The two planes and the airport form a right triangle with hypotenuse 97 and one leg 72.

\(\text{other leg = eastbound plane} = \sqrt{97^2 - 72^2} = \boxed{65} \) km

The product of a positive number and a negative number is negative.

There are a total of 5 choose 2 = 10 pairs. Out of these 10, 4 choose 2 = 6 pairs do not contain a negative number.

So, the probability that the product is negative is 1 - 6/10 = 2/5

Surface area = (area of square) + 4 * (area of triangle)

                     =  5^2 + 4(5 * 8/2)

                     = 105 in^2

Volume = 1/3(Bh) = 1/3(area of base) * (height)

V = 1/3 * 25 * 6 = 50 in^3