Alan

"A bacteria culture starts with 600 bacteria. After 1 hour there are 1000 bacteria. Assuming the size of the culture grows exponentially, find the time required for the population size to double. (You may leave ln,log, and (or) e in your answer)."

Square both sides of the first equation:

(x²+y²)² = 1²    or   x⁴+ 2(xy)² + y⁴ = 1

So:  17/18 + 2(xy)² = 1 

       2(xy)² = 1/18

         (xy)² = 1/36

         xy = 1/6   or   xy = -1/6

.

Could also do it like this:

Let a=log₃x.     b=log₉x.      c=log₈₁x

Then x=3^a    x=9^b=3^2b   x=81^c=3^4c

So a=2b and a=4c

But a+b+c=7 so a + a/2 + a/4 = 7 or 7a/4 = 7 or a = 4

Hence x = 3⁴ = 81

Looks to me as though none of the specified possible solutions apply!!

No zeros in sight!

"In base 8, the four-digit numeral BBCC is the square of the two-digit numeral AA.  Find the ordered tirple of digits (A,B,C)."

From inspection it's clear that x = 2 satisfies the equation.

"The three-digit base x numeral 7yc is twice the three-digit base x numeral 3y7.  Find x and y."

Here's a little more detail:

As follows:

The longest crease will occur when A is folded onto B (or C), which gives a crease of length \(M=4\sqrt3\)

From the symmetry of the triangle the shortest crease will occur when A is centred between B and C, giving \(m=4\)