Dear Friends:

Today, while frustrated followers of this blog have me duct-taped to my computer desk, I will post answers to some of the previous two sets of questions. I had better hurry...I smell gasoline and a match! When your followers fire you, they really

**fire you!**

Here we go:

1) If you receive a solicitation from a charity (you're the first one), and the email has a directive that you send the message out to two friends, and they are each to do the same, and so forth, how many generations of mailing will it take (yours is the first, your two friends are the second, and so on) for a single mailing to go out to more than 1 million recipients? Everyone seems to receive chain letters. They work by a principle called geometric expansion. In this case, each phase of expansion is a power of 2. Your level is one, and you've sent out two messages. Your friends' level is two, and they've sent out 4 messages. Your friends' friends' level is three, and they've sent out 8 messages... The problem, restated, is, "TWO TO WHAT POWER EQUALS OR EXCEEDS ONE MILLION?" This problem is not as difficult as it may, at first, seem. By the

**19th level**, the total amount of recipients getting the mailing should reach 1,048,576. This assumes that no person breaks the chain.

2) Applying the same circumstances and parameters as in question 1, above, what would your answer be if each recipient were required to send a copy of the email to three friends, instead of two. The problem, restated, is , "THREE TO WHAT POWER EQUALS OR EXCEEDS ONE MILLION?" It is indeed amazing how much more quickly the numbers multiply when we ask that each recipient send the letter to THREE others instead of to TWO. By the

**12th level**, the total amount of recipients getting the mailing should be 1,594,323.

3) If an executive consistently deposits 10% of her gross income into a special savings account, and her gross income grows at an average rate of 5% annually (her employer is a tightwad), how many years will it take before she accumulates (interest notwithstanding) one year's gross income (her first year's gross income) in the bank? There are two variables...salary is increasing at 5% per year, and our executive is saving 10% of that amount each year... The first year she saves (.10)(1.05). The second year she saves (.10)(1.05)(1.05). In each sucessive year, she will save the previous year's amount, multiplied by 1.05. She will have accumulated one year's worth of salary (in savings) in approximately

**7.8 years**. If she had not been receiving raises, it would have taken her a full 10 years to have accumulated that same amount. Because of the compounding at 5%, she attained her objective 2.2 years early than she otherwise might have. She is quite a responsible, goal-oriented person...it seems to me that they should have given her annual raises (compounded) of at least 10%. Let's write her employer a nasty note.

1. How many cubic feet are contained in a box which is 3 feet by 3 feet by 5 feet? Multiply 3x3x5, and you have

**45 cubic feet**.

2. In the above question, what is the answer in cubic yards? This is tricky, as there are 9 cubic feet in a cubic yard (

*not 3*). 45 cubic feet divided by 9 cubic feet/ cubic yard =

**5 cubic yards.**

3. In the above question, how many square feet of surface area are there on the container? The container is rectangular, with six sides (

*not 4*). Using a bit of visualization, let's assume that the base is 3x3, or 9 square feet; let's assume that the top is also 9 square feet; let's assume that each side (there will be 4 sides in addition to the bottom and top, which we've just handled), is 3x5, or 15 square feet. If we add up all six sides, we get 9+9+15+15+15+15 =

**78 square feet**.

4. Would the container in the above question fit into a container which is 2 feet by 4 feet by 6 feet? Heck...sometimes pure math isn't enough, and you have to employ some visualization and logic. This is one of those times. This new container is 2x4x6, or 48 cubic feet...our original container was only 45 cubic feet, leaving a three cubic foot difference in available space. Sadly, this could never work because if any one of the dimensions of the first container exceeds a dimension of the second container,

**it just won't fit**.

5. What would be the diameter of the largest ball (sphere) which could fit into the container described in 1), above? As in the preceding example, we must visualize the shape of the container, and its limiting dimensions.

**The diameter of the ball could not exceed 3 feet**.

Faithfully,

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