Math Q&A

Grade 7 through university

grade 8

percents

Question

I scored 21 out of 37 in my last test. How do I work that out as a percent?

[question g8percent27]

The first thing to understand is that “21 out of 37” is a fraction:

The rule for converting fractions to percents is: multiply by 100%. In this case:

Note on rounding: the info given for this question, 21 (out of 37), is 2 s.d. (two significant digits). Use 2 s.d. for the answer as well. See also [question g9rounding03].

Answer 2

Percents become easier to understand when you realize that a percent is just a decimal in disguise. A few examples:

   watch the decimal
 places!

So, to calculate your test score as a percent, convert it into a decimal…

…and “visually” convert to a percent:

Again, use 2 s.d. for the answer.

grade 9

maximizing volume of a cylinder

Question

I don’t see how to max the volume of a cylinder. Can’t I just keep increasing r?

[question g9max-cylinder01]

Answer

This is a tricky concept. In some ways it’s the same as maximizing the area of a rectangle with fixed perimeter, but the details are more difficult. The basic situation is this:

Maximize	Given (value that’s fixed)	Best Shape
area of rectangle lw	perimeter 2l + 2w	square, l = w
vol. of cylinder πr²h	surface area 2πr² + 2πr²h	height = diameter, h = 2r

For a rectangle, it’s easy to choose trial values of l and w so the perimeter is fixed: just increase l by the same amount as you decrease w. For a cylinder, the trial-and-error for r and h takes a couple of extra steps:

1. Fix the value of the surface area. Any value will do, since we are looking for the best shape rather than a particular size; let’s say, 200π cm² (we’ll see why the factor of π in a moment).

2. The equation 200π = 2πr² + 2πrh can’t be solved (yet), but let’s rearrange it to isolate h:

This is a formula for h in terms of r. (See why 200π cm² was chosen?)

3. For each trial value of r, use the formula to find h, and then use r and h to determine the volume:

r h V = πr³h 5 cm 15 cm 1178 cm³ 6 cm 10.7 cm 1206 cm³ 7 cm 7.3 cm 1122 cm³ 6.1 cm 10.3 cm 1203 cm³ 5.9 cm 11.0 cm 1208 cm³ 5.8 cm 11.4 cm 1209.1 cm³ 5.7 cm 11.8 cm 1208.9 cm³

To the nearest tenth, we have r = 5.8 cm, h = 11.4 cm. Notice that for the last three values, h ≈ 2r. Finer calculations would reinforce h = 2r as the maximizing shape.

Another way to put it is that the volume of a cylinder is maximized when the height equals the diameter, as in the picture.

[To be absolutely sure that h = 2r takes math with more power than you can draw on in grade 9. If you wind up taking Calculus in three years’ time, you’ll learn some techniques for settling questions like this.]

grade 12

Calculus: product rule, chain rule

Question

I tend to get the product rule and the chain rule mixed up. I guess I don’t really see the difference. Can you explain it?

[question g12calc-rules01]

Answer 1

The product rule applies when you multiply two functions together; hence its name. In other words,

 z = u×v, where u and v are both functions of x

You use the chain rule when you apply two functions one after the other. That is,

 z is a function of y, and y is a function of x

Now let’s look at the derivatives. The chain rule is the easier one:

(not a rigorous proof of the chain rule, but it gets the idea across). For the product rule, think of a rectangle, like this:

The value of δz is the area of the upside-down L-shape:

Use this to calculate :

It’s easy to put these rules into function notation. Just let u = u(x), v = v(x), z = p(x) for the product rule:

For the chain rule, put y = f(x), z = g(y):

Answer 2

Maybe flow diagrams will help. Think of our two functions, f and g, as little computer programs with inputs and outputs. Here’s the product rule:

Notice the product rule has parallel “processing.” But in the chain rule, processing is in series:

So we’ve cleared up the difference between f(x)×g(x) and g(f(x)).

Now; what about those derivatives? We’ll use these ideas:
 (*) and (+)

(Why are (*) and (+) true? Hint: draw a graph for (*) and a rectangle for (+).)

To get the product rule, do (*) twice, in parallel, and then (+):

As usual, divide through by x and take the limit:

The chain rule is actually a little easier: just use (*) twice, in series.

That is…

… or you may prefer

Remember: product >> parallel, chain >> series.