Chapter 4, Section 4

Chapter 4, Section 4

Setting up equations

Given an equation, most people can evaluate it for a given “x” and get a second coordinate or answer. Getting the equation in the first place is the hard part and that’s what we’re going to concentrate on today.

In general, follow the procedure below to come up with an equation that will lead to the answer:

1.read the problem – more than once –

and sketch a picture of the information that is given,

label the picture parts carefully – be sure not to reuse any letter names…there can only be one “x” in your problem.

In general, letters in the last half of the alphabet are variables (s, t, x, y) and letters from the first half are unknown constants (

2. rewrite the problem in your own words and

write out any facts that are in general knowledge that have to do with the problem…general equations are especially nice to have

3.rewrite the problem’s facts into the general information that you have…

substitute everything that you can into the organizing formulas

4.doodle around with the facts, get creative, try some special cases

5.check your answer – especially invoke a little reality testing

The text examples are really well done. I encourage you to study them carefully.

One other convention – leading to a skill that will come in handy in Calculus – when we ask for a quantity “as a function of x”…we mean you to get the quantity expressed in exactly one variable “x” and to replace the other variables in the expression with equivalent expressions in x. Note that the area formula for a triangle is a function of two variables (b, h)…

Here’s a problem:

The sum of two number is 5. The difference of the squares is, too. What is the product of the numbers?

What’s the first number?And the second?

What are the facts of the problem?

Now we’re at the doodling part:

So, did we answer the question?

Another problem:

Given . Select any point, x, with x > 0, you can sketch a right triangle from the origin to a point on the curve, then straight down to the x axis

Being able to recast y in it’s functional form is a survival skill in this section!

What is the area of the triangle in terms of the variable x?

What is the perimeter of the triangle in terms of the variable x?

Here’s another:

The perimeter of a rectangle is 54 feet. Express its area as a function of its width, w.

Perimeter is a function in TWO variables, you are to reduce it to one variable for this problem…this work is quite often problem specific and not general in nature…

The area of a rectangle is 54 ft2. Express its perimeter as a function of its width.

Suppose you have a circle centered at the origin. If you sketch a rectangle in the upper half plane with its base on the x axis and the two upper corners on the circle, what are the area and perimeter in terms of x? Hint the formula for the circle is with r a positive constant.

There’s a whole lot to this problem. Let’s sketch it first:

Note that we’ll have to use some geometric facts!

What are the formulas for area and perimeter of a rectangle?

Let’s fill them in with the facts from the problem:

Now are there some restrictions on which x’s I can use? Then let’s say those, too.

So here’s the whole answer, all in terms of x

A(x) =

P(x) =

Here’s another:

A wire of length x is bent into the shape of a circle.

Express the circumference in terms of x.

Express the area in terms of x

And another:

The hypotenuse of a right triangle is 8 feet. Express the area of the triangle as a function of x, the length of one of the legs. Hint: put a coordinate system on the triangle!

Let’s look at another number problem:

The base of a rectangle lies on the x axis, while the upper vertices line on the parabola . Express the area of the rectangle as a function of x.

Sketch the picture, label and fill in as much as possible

What are some other facts pertinent to the problem?

Doodle some:

The answer is:

Another one:

Let P be a point on the graph .

Express the distance from P to the origin as a function of x.

Express the distance from P to the point ( 0,  3) as a function of x.

Answer:

Another one:

Let A denote the area of a right triangle in the first quadrant formed by the y axis and the lines y = m and y = mx, for m >0. Express the area of the triangle as a function of m.

And another:

Suppose that P is a point on the line y = 3x  1. And Q is the point ( 1, 3).

Express the distance from P to Q as a function of x.

And a last one:

If the sum of two numbers is 8, find the largest possible value of their product.

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