1.6: The Great Pacific Garbage Patch

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INTRODUCTION

In this collaboration, you will set up and solve several proportions. Proportions are very forgiving on how you set them up. As long as you match “units,” your proportion will be set up correctly.

For example, imagine a scale on a map indicates that 2 inches represents 5 miles. If two items are 4.5 inches apart on the map, how far apart are they on the earth? Take a minute to think about this in your group before moving on.

\[ \textbf{Setup 1:}\;\;\; \dfrac{2\:inches}{5\:miles} = \dfrac{4.5\:inches}{?\:miles} \nonumber \]

For this setup, we have inches in the numerator on both fractions, and miles in the denominator, so our units match and can be used to give us 11.25 miles for the unknown value. The reason behind setting it up this way is that the ratio of the distance on a map to the actual distance is constant.

\[ \textbf{Setup 2:}\;\;\; \dfrac{2\:inches}{4.5\:inches} = \dfrac{5\:miles}{?\:miles} \nonumber \]

For this setup, we have both original given values in the numerators. We put the second value and the unknown second value as denominators. So again, our units match and this setup will give the correct answer, 11.25 miles. The reason behind setting it up this way is that the ratio of two distances on a map is the same as the ratio of the actual distances.

Both setups are examples of proportions. Proportions are equations in the form of $\dfrac{a}{b}$ = $\dfrac{c}{d}$. One approach to solving proportions is to write them as to $ad = bc$. This form is often easier to work with than the original setup.

For example: $\dfrac{2}{4.5} = \dfrac{5}{x}$

Is equivalent to: $2x = 4.5\times5$

$2x = 22.5$

$x = 11.25$

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

scaling linear distances by a factor will result in scaling areas by the square of that factor.

By the end of this collaboration, you should be able to

compute a scale factor between two quantities having the same dimension.
use a scale factor to convert units in a variety of contexts.

PROBLEM SITUATION: A LOT OF TRASH

The Great Pacific Garbage Patch has been receiving media attention since its discovery in 1997. The size of the Garbage Patch is not exact, because it has no specific border, and identifying a rough border depends upon the cutoff chosen for particle density (pollution level). The size of the Garbage Patch is usually measured as somewhere between 700,000 km² (about the size of Texas) and 15 million km² (about twice the size of the contiguous United States). To illustrate how large that is, the Great Pacific Garbage Patch is often described as “twice the size of Texas.”

(1) Is the description, “twice the size of Texas,” a reasonable estimate of the size of the Great Pacific Garbage Patch? Explain.

(2) Statistics books often use pie charts and bar graphs to portray quantities and sizes. In contrast, the image below, called an “infographic” (or “pictograph”), uses images rather than a chart or graph to represent the size of the objects in the Problem Situation.

$Image showing the Garbage Patch in the Pacific Ocean.$

(a) Think about why the shape of Texas is a good image to help represent the Garbage Patch. Would a simple square or circle be more helpful or less helpful? Why?

(b) Does the size of the Garbage Patch in the above infographic seem reasonable? Explain why or why not.

The infographic above claims to show the size of the Garbage Patch in the Pacific Ocean. To better understand how large the infographic is claiming the Garbage Patch to be, we will create a scale factor. A scale factor is a number that when multiplied by a given quantity results in a proportional second quantity. The scale factor for this infographic will tell us the proportion of real-world kilometers to centimeters on the map, or how many kilometers are represented by one centimeter on the map.

(3) Let’s say the distance (in centimeters) from the point shown in Australia to the point shown in Alaska on the infographic is 10.5 cm. If the actual distance from Australia to Alaska is 12,000 km, how many km does each cm on the infographic represent? Round to the nearest kilometer.

(4) Now you are going to estimate the total area of both Texas shapes from the given map. Let’s imagine someone quickly sketched a rectangle to roughly represent the “two-Texas area.”

$An image showing a drawing of a two-Texas area as approximate rectangles. The heigh is 3 cm, and the width is 5cm.$

(a) Using the measurements in the sketch above, and the scale factor (km/cm) from Question 3, what would be the actual height and width (in kilometers) of the sketched rectangle on the infographic map?

(b) What actual area (in km²) is represented by the rectangle of the “two-Texas area”? Write down your steps for doing this calculation.

As discussed in Question 4, using a 3 cm by 4 cm rectangle somewhat accurately represents the upper estimate of area of the Great Garbage Patch of 15,000,000 km².

Now, let’s visualize it in a slightly different way, using two squares:

$Image showing “two-texas” shapes in the Pacific Ocean. Over laying the two-texas shapes are two squares, the smaller representing a Garbage Patch of about 1,400,000 km2, and the larger one representing a Garbage Patch of about 15 million km2.$

One square that represents the lower estimate of 700,000 km²
Another square one that represents the upper estimate of 15,000,000 km²

(5) (a) What should be the dimensions of the smaller square that represents the area of 700,000 km²?
Round your answer to two decimal places.

(b) What should be the dimensions of the larger square that represents the area of 15,000,000 km²?
Round your answer to two decimal places.

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

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