Calculus questions Again

You Already helped me with those questions but my professor made a typo.

1.   Suppose we have a population that increases
according to the formula P(t) = P₀
e^kt.  Our goal
is to fill out the following table:

Point	t	P(t)
A	0
B	10	20
C	30	50
D	40
E		2000

(i) 
Use
points B and C to write two equations. 
Then use them to solve for P₀ and k.  Finally, write the
population equation.

(ii) 
Use
(i) to fill in the rest of the chart. 
You may round each value to the nearest tenth.

(iii) 
Now
use your formula to determine the doubling time of this population.

(iv) 
Explain
how, without using an equation, you can now calculate the t values for which P(t) equals
1000. And how about 250?  Do this by
writing a clear brief essay, with complete English sentences.

(v) 
Now
on graph paper, draw a clear graph of this exponential function. Be sure to
label your axes, and to label, with their coordinate, each of points A through
E.

2.  Consider the following
situation:  A circle of with center O(0,0), radius 10m, is inscribed in a
square. The ray of angle 30^O, in standard position, intersects the
circle at point B, and continues to
intersect the square at point C. Let
A denote (10,0). 

(i) 
Sketch
the figure indicated in the above description.

(ii) 
Find
the exact coordinates of A, B, and C, and label them on your sketch.

(iii) 
Now
suppose we have arbitrary acute angle Q  (in
radians, instead of the 30^O).  Again draw the sketch!

(iv) 
Again
figure out the exact coordinates of A, B,
and C and label them on your sketch.
NB: You will use trig functions here!

(v) 
Now
figure out the equation you would have to solve to find Q to make
the area of ABCA exactly equal to the
area of the sector.  HINT: This means
area of sector is half area of triangle. (You cannot solve such an equation
exactly – this is an example of a TRANSCENDENTAL equation, so the theorems of
algebra do not apply.)