A culture of bacteria obeys the law of unlimited growth. If 500 bacteria are present initially and there are 800 after 1 hour, how many will be present in the cultrue after 5 hours? How long until there are 20,000 bacteria?

Hmmm, what is "the law of unlimited growth"? In any case, this looks like exponential growth to me.
The most basic formula for exponential growth is
A = P*er*t where r is the rate of growth, t is the time, P is the initial amount and A is observed growth.
Here 800 = 500*er*1hr so
ln(1.6units) = r*1hr or ln(1.6units)/1hr = r or
A = 500*(1.6)t
Substitute 5 for t to answer the first question, i.e.
A = 500*(1.6)5
To answer the second part set A=20,000
20,000=500*(1.6)t or
40 = (1.6)t
It looks to be slightly less than 8 hrs. You can verify this.

A ship, offshore from a vertical cliff known to be 100 feet in height, takes a sighting of the top of the cliff. If the angle of elevation if found to be 25◦, how far offshore is the ship?

To solve the first question, we use the formula A = P * e^(rt), where A is the observed growth, P is the initial amount, r is the rate of growth, t is the time. In this case, the initial amount is 500 and the observed growth is 800 after 1 hour. Let's solve for r.

Using the formula, we have 800 = 500 * e^(r * 1). Dividing both sides by 500 gives us 1.6 = e^(r * 1). To solve for r, we take the natural logarithm (ln) of both sides, giving us ln(1.6) = r * 1. Therefore, r = ln(1.6).

To find the number of bacteria after 5 hours, we can plug in the values into the formula A = P * e^(rt). In this case, P = 500, r = ln(1.6), and t = 5. So we have A = 500 * e^(ln(1.6) * 5).

For the second question, we are given that there are 20,000 bacteria. We can use the same formula A = P * e^(rt), with P = 500, r = ln(1.6), and A = 20,000. So we have 20,000 = 500 * e^(ln(1.6) * t).

To find t, we can solve this equation for t by dividing both sides by 500, giving us 40 = e^(ln(1.6) * t). To isolate t, we take the natural logarithm of both sides, which gives us ln(40) = ln(1.6) * t. Therefore, t = ln(40) / ln(1.6).

Simplifying, we find t to be approximately 7.92 hours.