class 051031

If I take a 10% chance of injury every day, then what are the chances I won't get injured today?

Obviously, 90%.

If I take a 10% chance of injury every day, then what are the chances I'll be injured by the day after tomorrow?

There's a 90% chance I'll be at it again tommorrow, and a 90% chance of that that I'll be at it again the next day, and a 90% chance of that that I'll get through that third day.  So the probability is 90% * 90% * 90% = .9 * .9 *.9 = .729.

Here the thing that's changing is my probability of being uninjured.  That probability for each day is 90% or .9.  We multiply by .9 with every successive day, so .9 is the growth factor.

Since growth factor is 1 + growth rate, and since 1 - .1 = .9, we conclude that the growth rate here is -.1.

If I take a one-tenth-of-one-percent chance with disaster every day, then what are my chances of getting through one day without disaster?

One-tenth of a percent is .1 % or .001.

This is the chance of disaster.

What is the chance of no disaster today?

The chance of no disaster today is 1 - .001 = .999.

The chance of no disaster over the next three days is .999 * .999 * .999.

The chance of no disaster over the next t days is .999^t.  This can be expressed as a function of t:

p(t) = .999^t.

On what day will there be less than a 50% chance that I haven't encountered a disaster?

We could keep multiplying by .999 until we get down to .5.  You could do this but it would take awhile and you probably wouldn't count it accurately.

We could use trial and error, which would be a little quicker.  Starting with t = 500 and t = 1000 we get two results, one greater than .50 and one less.  Then we refine our estimates.  In about 10 trials we can get it down to between 692 and 693 days.

If we let p(t) = .50 and solve for t, we'll find out.

What are the growth rate and growth factor for this function?

Growth rate is .999, growth factor is -.001.

1.  Interest on an initial principle of $7000 accumulates annually at the rate of 3.7 percent per year. 

How much money will there be in 1, 2, 3, and 10 years after the initial investment?

What is the growth rate for this situation?

What is the growth factor?

What is the function that tells you the principal P(t) as a function of clock time t?

2.  Give the basic points of the basic exponential function y = 2^t, and use these points to sketch a graph of this function.

• What then would be the basic points of the function y = .5 * 2^t + 3?  Use these points to sketch a graph of this function.

What are the basic points of the exponential function y = 3^t?

• Could this function be a vertical shift of the basic exponential function y = 2^t?
• Could it be a vertical stretch of the basic exponential function y = 2^t?
• It is possible to get the graph of the y = 3^t function by shifting and stretching the graph of y = 2^t?  If so what kinds of stretches and shifts would be required?

If a 50ft. whale weighs 35 tons, how much would a 60 ft. whale weigh?

• Could I not just say 35/50 = x/60 and then solve for x?
• or do I have to use the y = k x^p?

Why we have to use proportionality of the cube

The weights of 3-dimensional object are distributed through their volumes.

Volume can be thought of as being occupied by tiny cubes.

Two geometrically similar 3-dimensional objects can be thought of as being occupied by the same number tiny cubes. The cubes making up the larger object are just bigger.

If one object is longer than the other, then its cubes will be longer also. Of course if the cubes just got longer and
didn't get wider and higher, they wouldn't remain cubes, just like a whale that's longer but not wider and thicker wouldn't
be geometrically similar to the original whale.

So when the cubes get longer, they also get wider and higher.

In this case the larger whale is 60/50 = 1.20 times as long as the first, so its little cubes would have to be 1.2 times
longer than those of the first. That alone would give them 1.2 times the volume of the first, but we're not done because the
cubes would then have to get 1.2 times wider, and which would give us 1.2 times 1.2 times the volume of the first. Of course
we're still not done because the cubes have to get 1.2 times higher, giving us 1.2 times 1.2 times 1.2 times the volume of
the first.

This is how we end up with 1.2^3 times the volume of the first, not just 1.2 times.

Solving by Ratios

The ratio form of the solution would be of the form (W2 / W1) = ( L2 / L1)^3. You have to use the weight ratio on one side and the length ratio on the other, and should always do this when using proportionality. If you use x instead of W2 for the unknown weight of the second whale your equation would read

(x / 35) = (60 / 50)^3, or

x / 35 = 1.2^3 or

x / 35 = 1.73 so that

x = 35 * 1.73 = 61, meaning that the second whale weighs 61 tons.

Solving by Functional Proportionalies

All this is equivalent to saying that the volumes of geometrically similar objects are proportional to the cube of their lengths, or using V for volume and L for length, V = k * L^3.

The weight of a whale is distributed through its volume, so we also know that W = k * L^3, where W is the weight.

The 50-ft whale weights 35 tons so any geometrically similar whale will obey the same proportionality as this one. The proportionality is

W = k * L^3, which gives us
35 tons = k * (50 ft)^3 so that
k = 35 tons / (50 ft)^3 = 35 tons / (12500 ft^3) = .00028 tons / ft^3, approximately.

This tells us that the proportionality is

W = .00028 tons / ft^3 * L^3.

So the 60-foot whale, with L = 60 ft, would have weight

W = .00028 tons / ft^3 * (60 ft)^3 = .00028 tons / ft^3 * 216,000 ft^3 = 61 tons, approximately.

So the ratio solution gives us the same thing as the proportionality solution. </h3>