"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=3a x=9b=32b x=81c=34c
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 \(m=4\)