Word Problems

[snowseekr]

Maybe I'm overthinking this problem. Any help is appreciated!

The domestic postage rate for first class letters weighing 13 ounces or less was 39 cents for a letter weighing one ounce or less plus 24 cents for each additional ounce (or part ounce). Write postage P (cents) as a function of weight w (ounces).

I was thought it was P = 39 + 24w but that doesn't make since to me.

 

[Loren]

P = 39 + 24w

You haven't specified what the w stands for. Always start your problems with something like...

Let w be the weight of the letter in ounces.

Now, then, have you analyzed what is wrong with your formula? Do you see that if w represent the weight of the letter, then the first ounce is getting counted twice? Bear in mind that it is only the number of ounces after the first ounce that will cost 24 cents each.

 

[snowseekr]

So would the function be P= 39 + (24(w-1))
? I'm still somewhat confused. Thanks for helping figure out what I was stuck on.. I knew I was counting the first ounce twice, but I'm not sure how I go about changing that.

 

[mmm4444bot]

So would the function be P = 39 + ( 24(w - 1) )

This looks like an improvement.

What is the postage if I give you a letter that weighs 1.65 ounces?

That's supposed to be 63 cents, right?

Here's what happens when w = 1.65 in your function:

P(1.65) = 54.6

The problem is that we cannot add fractional parts of 24 cents; we may only add multiples of 24 cents.

Have you heard of the Least Integer Function (aka "ceiling" function)? It's one of the step functions, and I'll denote it as follows. (Actually, the strange-looking double-square brackets that I see in texts might belong to a different step function, but I need to type something.)

[[x]] = the smallest integer greater than or equal to x.

What it does is round non-whole numbers (x) up to the next integer. If x is already an integer, then [[x]] does not change x.

[[0.50]] = 1

[[1.00]] = 1

[[1.50]] = 2

[[1.65]] = 2

[[1.84]] = 2

[[2.00]] = 2

[[2.01]] = 3

[[2.40]] = 3

This function might come in handy for your exercise (hint, hint).

If you have not learned about step functions, then I think your only other choice is to write a piecewise function definition.

P(w) = 39, for 0 < w <= 1

P(w) = 63, for 1 < w <= 2

P(w) = 78, for 2 < w <= 3

and so on, all the way up to the last "piece" of the domain of function P: 12 < w <= 13.

 

[snowseekr]

Thank you so much!
ok so can I do P= 39 + 24[w] ?? would that be the correct way of using the ceiling function?

 

[mmm4444bot]

can I do P= 39 + 24[w] ?? would that be the correct way …

Do you ever try experiments to answer your own questions? (This question is rhetorical.)

Try your new function out. Test it out by trying some evaluations.

P(w) = 39 + 24 * [[w]]

Tell me, what is P(1.65)?

 

[Loren]

Possibly it would be acceptable to take care of the fractional ounces in the naming of the variable. For instance, could you name as follows:?

Let n represent the number of ounces rounded up to the nearest integer if the weight is not an integer.

If that is acceptable, your formula needs one little adjustment and you have it made.

 

[mmm4444bot]

Let n represent the number of ounces rounded up to the nearest integer if the weight is not an integer.

Hmmm. I'm so lazy that I wonder why I didn't think of that.