Math for Grownups (18 page)

First, you should measure the pots. Because they are rectangular prisms, each has height, width, and length. Using a tape measure, you get these results:

Height
=
20 inches

Width
=
15 inches

Length
=
15 inches

The potting soil will go
inside
the pots, of course. So you need to know the volume—how much the container will contain. The volume of a rectangular prism is length times width times height, so you multiply:

20 in. • 15 in. • 15 in.

4,500 in
³

Wait! What’s that in
³
all about? Remember how you indicated square feet (ft
²
) when you calculated the area of an object? Well, volume is measured in cubed amounts. Therefore, you’ll need 4,500 cubic inches of potting soil. But there’s a problem: The gardening center sells bags of potting soil measured in cubic
feet
, so you’ll need to convert.

You have
cubic inches
and you want
cubic feet
. There are 12 inches in a foot, but how many cubic inches are there in a cubic foot?

Turns out the answer is pretty simple—and pretty intuitive.

1 ft
³
=
12
³
in
³

In other words, 1 cubic foot equals 12 • 12 • 12 in
³
, or 1,728 in
³
. So, to find out how many cubic feet there are in 4,500 cubic inches, just divide.

4,500 in
³
/
1,728 in
³
=
2.6 ft
³

Okay, so you need to buy 2.6 ft
³
of potting soil—for one pot. But you have four containers to fill, so you’ll need to multiply.

2.6 • 4
=
10.4 ft
³

At the garden center, you find a brand that is sold in 1.5-ft
³
bags. Arggh!!! How many bags will you need? Back to division!

10.4
/
1.5
=
6.9333 …

You can’t buy 6.93333 … bags of potting soil, so you should round up to 7 bags.

That wasn’t so bad, was it? Want to share your tomatoes?

What if you had containers with circular bases, instead of rectangular containers? The process is the same, but you’ll have to use a different formula for volume.

That’s because a container with a circular base is probably a cylinder.

The formula for the volume of a cylinder is

V
=
πr
²
h

π
= 3.14 …

r
is the radius of the base

h
is the height of the cylinder

Now
π
(pronounced “pi”) is a very important number. Mathematically speaking, it’s the ratio of the circumference of a circle to its diameter—but you don’t need to remember that. It’s enough to know that
π
is a very long number that rounds to 3.14.

Finding the height is pretty easy; just measure from the bottom of the container to the top. But how do you find the radius?

The radius is half the width of the circle (half of the diameter, in other words). Instead of trying to find the center of the base of the container, just measure the base from one side to the other—as close to the center of the circle as possible—and divide by 2.

Then you can plug everything into the formula to find the volume.

What if you had a cylinder-shaped container that was 3 feet tall and had a base 2 feet wide? How much potting soil would you need? The height is 3 feet, and you need to divide the width of the base by 2 to find the radius (2 / 2 = 1).

Now you can use the formula.

V
=
πr
²
h

V
= 3.14 • 1
²
• 3

Remembering your order of operations, you know you should handle the exponent first. (Which is super-easy in this case, because 1
²
is 1!)

V
= 3.14 • 1 • 3

Now multiply:

V
= 9.42 ft
³

You’ll need 9.42 ft
³
of potting soil for this container.

Units of measure are a big deal when you apply math to everyday situations. Here are some basics to keep in mind:

When you’re measuring volume, you’ll get a cubed unit. That’s because you’re measuring the space inside a three-dimensional figure. Get it? Three-dimensional→unit
³
.

When you’re measuring area, you’ll get a squared unit. That’s because you’re measuring the space inside a two-dimensionl figure (a plane). Two-dimensional→unit
²
.

And when you’re measuring length, you’ll get a plain old unit. That’s because you’re measuring a one-dimensional figure (a line). One-dimensional→unit
¹
, or unit.

But what about the surface area of a three-dimensional figure? Is that measured in cubic units? Nope. Because you’re measuring area, it’s still measured in square units. (Think of taking a box apart at its seams. You get a flat, two-dimensional object, right?)

And what about the perimeter of a two-dimensional figure, like the wedge-shaped flowerbed? Is that measured in square units? No again. Because perimeter is length, it’s measured in units. (Think of stretching out the curved garden hose. You get a line, right?)

Good news! Rick landed a great summer job doing landscaping for a local company. All day long, he hauls mulch, cuts lawns, and trims shrubbery.

Bad news! When getting out of the truck one morning, Rick slipped on a puddle of water and fell. His left leg is in a cast—and he can’t do the landscaping any longer.

Good news! Rick’s boss also runs a gardening supply business, and his office assistant is out on maternity leave. Rick can keep working at the company, taking orders for mulch, topsoil, compost, and gravel.

Bad news! Rick isn’t sure he can do the math required for the job. He’s fine with folks who know exactly how many cubic yards of mulch they need. But he’s worried about the customers who don’t know how to figure that out for themselves.

On day 1, he gets his first test.

Susan has recently lost her job. She used to have a landscaping service take care of her weekly lawn trimming and biannual mulch delivery. But this year, she needs to cut some costs, and the landscaping service is at the top of the list of luxuries that have to go.

Her hubby will take over the mowing and the task of spreading the mulch, but she’ll be responsible for ordering the mulch. She calls Rick, who has to help her figure out how much mulch she’ll need.

Rick knows one thing for sure: Even though mulch covers a two-dimensional space, it’s measured in cubic yards. That’s because, once it’s spread, the mulch itself has a thickness. (He reminds himself of this by thinking of the mulch layer as a really, really short rectangular box.) Typically, his boss recommends a layer of 4" of mulch for ordinary flower gardens.

“What are the dimensions of your flower beds?” Rick asks Susan. She tells him that they’re 24' by 10' and 28' by 11'. Rick suggests that it would be best for him to do the math and then call Susan back. She agrees.

On a piece of paper, Rick writes down the dimensions of Susan’s flower beds.

24'
×
10'

28'
×
11'

Because the mulch must be 4" deep, he needs to add another dimension.

24'
×
10'
×
4"

28'
×
11'
×
4"

It’s a good thing that Rick included the units in his notes. Otherwise, he might have been tempted to think of 4" as 4', and that would have made his answer way, way too large.

Rick needs to have all of his measurements in the same units. But should he use feet or inches? His final answer will have to be in cubic yards, so he decides that using feet is the best option, because it will be easier for him to convert cubic feet into cubic yards than to convert cubic inches into cubic yards.

He knows that 4 inches is the same as 0.33333 … feet. He decides that rounding to the nearest hundredth is probably just fine, so he substitutes 0.33 for 0.3333…. And he plugs in the new number:

24'
×
10'
×
0.33

28'
×
11'
×
0.33

Then he multiplies and gets 79.2 ft
³
and 101.64 ft
³
.

These two numbers are the amounts of mulch needed for the two flower beds, so Rick can add them together to find the total amount of mulch he needs.

79.2 ft
³
+ 101.64 ft
³
= 180.84 ft
³

But he’s not finished yet. Rick’s boss sells mulch in cubic yards, not in cubic feet. Rick needs to make one last conversion before calling Susan back.

Rick knows that there’s a conversion chart somewhere in the office, but darned if he can find it. So he has to think a little bit more.

One cubic foot is like a 1 foot by 1 foot by 1 foot cube. That could also be expressed as 1 • 1 • 1 = 1. And 1 cubic yard is like a 1 yard by 1 yard by 1 yard cube. How many of the first cubes will fit into the second?

The answer is 27. That’s because the cubic yard is 3 feet by 3 feet by 3 feet, or

3 • 3 • 3
=
27

(Another way to write this is 3
³
= 27.)

Because there are 27 ft
³
in 1 yd
³
, Rick needs to divide his original answer by 27 to find out how many cubic yards Susan will need to order.

180.84
/
27
=
6.69777 …

Rounding up, Rick sees that Susan needs 7 yd
³
of mulch, and he feels much better about his math abilities.

Many gardeners use a little fertilizer to help their baby lettuces and carrots grow. These come in a variety of applications—from granules to sprays to concentrated liquids that must be mixed with water before they are sprayed.

Perfect Petunia liquid plant fertilizer is one of the latter. It must be mixed with water before it can be sprayed on plants. The directions say to mix at a rate of 8 fluid ounces per 16 gallons.

What if your sprayer only holds 1 gallon? How much Perfect Petunia fertilizer should you add to the sprayer?

A ratio comes in handy here. Eight fluid ounces of fertilizer to 16 gallons of water, or 8:16, is how we could express this. If you remember, ratios can also be expressed as fractions: 8/16 is still another way to say this.

To find the correct ratio of fertilizer to water, you start by recording what you know. You know that the ratio is 8/16, with 16 being the number of gallons. You know your sprayer holds 1 gallon. So what you have is a ratio with an unknown, like this:
x
:1. In order to solve the problem, these two ratios must be equal. And, ta-da! That means you create a proportion. (A proportion is nothing more than two ratios set equal to one another.)

To find
x
, you have to isolate it. And that requires cross multiplication in this case—just multiply the numerator of one fraction by the denominator of the other.

We still haven’t isolated
x
, but we’re closer. Next, divide each side of the equation by 16.