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) = P_{0} 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 P_{0} 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.)
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