Using Formulas to Help Solve Problems

How much concrete would you need for that pool?
How many outlets can be on one circuit?
How much insulation does a house need?
How big of a house will you be able to afford?

Objective: When done with this lesson, you will have demonstrated how to solve for a unknown amount using a formula.

Approximate completion time: 6 hours

 

 

Some questions and problems are so common that formulas have been developed to help us out. You have probably seen some famous formulas. Maybe you will discover a famous formula yourself one day.

Before we start using formulas to solve problems, let's look at the formulas other people have found.
(Did you know that when you find the 'perfect combination' for something, it's called a formula? It's no coincidence people talk about Formula1 Racing or Mama's Secret Formula...)

 

Visit these resources to help you create your own table of formulas: Math.about.com, Math.com,
Make your table like this and keep it handy for your "assignment 5.04.yourname.doc" .

Name of Shape
Picture
Area formula
rectangle
parallelogram
square
triangle
trapezoid
circle

 

Are you sure those formulas are true? Take some time to look at these investigations that help demonstrate why the formulas work.
Be patient as they take some time to load.

 

 

Ok, now it's time to solve some problems.

 

Example 1.

Problem: You have been hired to create a pond like that found at the Changdeokgung Palace. The shape of the new pond must be rectangular and the total area of the surface must be 1350 square feet to hold all the Koi the owner wants to keep in it. If the width of the pond is to be 45 feet, how long will the pond need to be?
Solution:

This is an area of a rectangle problem. For a rectangle,
Area = length·width . So substituting what we know,
1350 = length·45

To undo the multiply by 45, we will divide both sides by 45.
length = 1350÷45
length = 30

So the pond must be 30 feet long. Let's check to make sure that is correct.

Area = length·width
Area = (30 feet)(45 feet)
Area = 30 ·45·feet·feet
Area = 1350 square feet         Correct!

 

 

Example 2.

Problem: After your last successful pond building job, you've been hired again to make another koi pond. This time the owner is quite certain about what he wants, yet he cannot figure out how to do it. He knows the surface area of the water must be 192 square feet. He claims a 'perfect' koi pond must have a width that is only ¾ of the length. What dimensions should the pond be to make the owner happy?
Solution:

This is an area of a rectangle problem. For a rectangle,
Area = length·width . So substituting what we know,
192 = length·width. Yet we don't know the length or width.

Time for some algebra thinking!

Statements
Reasoning
Let L = the length of the pond We don't know either, yet the width depends on the length.
Then the width = ¾ L "a width that is only ¾ of the length"
Remember that "of" means multiply.
Area = length·width We know this is true.
Area = L·(¾ L) Substituting using our variables.
Area = ¾ L2 Simplifying
192 = ¾ L2 Substituting in for the area
4·(192) = 4·(¾ L2) Multiplying both sides by 4 to undo the divide by 4 in the fraction.
768 = 3 L2 Simplifying
768 ÷ 3 = L2 Dividing both sides by 3 to undo the multiply by 3.
256 = L2 Simplifying
Unsquaring the L2 requires square rooting
L = 16 Since 16*16 = 256

The length should be 16 feet and the width should be ¾ (16) = 12 feet.
That would make a pond 16 feet by 12 feet with an area of
(12ft)·(16ft) = 192 ft 2.

Did you notice how the ft units act like a variable when multiplying?
Just like x·x = x2, ft · ft = ft 2.

 

Example 3.

Problem: A landscape designer friend of yours has given you this drawing for a layout of a small walkway. The units are measured in feet. Find out how much the value of x must be and what the dimensions of the trapezoid must be.
Solution:

We know the formula for the area of a trapezoid is

Substituting in for what we know, we get the equation

Grabbing our algebra skills we can solve this equation.

Statements
Reasoning
The equation we have
Simplifying. (Remember PEMDAS)
Dividing both sides by 4 to undo the multiply by 4.
14 = 5x + 1 Multiplying both sides by 2 to undo the divide by 2.
13 = 5x Subtracting 1 from both sides to undo the divide by 1.
13/5 = x Dividing both sides by 5 to undo the multiply by 5.
x = 2.6 Simplifying

So the value of x is 2.6.
That makes the top base of the trapezoid 2.6 + 2 which is 4.6 feet.
The bottom base would be 4*(2.6) -1 = 10.4 - 1 = 9.4 feet.

 

 

Example 4.

Problem: Your good friend Pizza Pete wants to make a huge pizza with enough pizza to feed 100 people. He knows that it takes about 80 square inches per person to feed them. How big would the pizza need to be if it were circular and would feed everyone?
Solution:

We know the formula for the area of a circle is
Area = π·(radius)2     where π = pi = 3.1415926535 8979323846...

Since there are 100 people at 80 square inches each, the total area must be 100*80 = 8,000 square inches. Substituting this into the equation for area,

8,000 = π·(radius)2   

Now solving with algebra:

Statements
Reasoning
8,000 = π·(radius)2    The equation we have.
8,000 = π·r2    let r = the radius
Dividing both sides by π to undo the multiply by π.
2546.749... = r2 Simplifying.
Taking the square root of both sides to undo the squaring of r.
50.463... = r Simplifying.

So the value of r is about 50.463 inches.
Pete should make a pizza with a radius of about 50 ½ inches which would be about 2*(50 ½) = 101 inches in diameter. Wow! That's over 8 feet wide!

 

There are tons of formulas which require different levels of algebra skill to use and solve.
Please take some time to practice your algebra skills by visiting these sites:

Good practice with some helps
(about 1-2 hours)
Basic quick checks
(about 1-2 hours)

 

 

Now it's time to show how much you've learned.

Open up this assignment to practice your work. Be sure to do quality work!