8.7 Triangles are the Answer! Solving Applied Problems
Bob Brown; Morgan Chase; and Marilyn Nielson
Applications
Here’s an age-old forestry problem: How tall is this tree?
The Hendricks Creek survivor elm stands on lands managed by the Wisconsin Board of Commissioners of Public Lands in northern Wisconsin. USDA Forest Service photo by Linda Haugen.
What do you need to know to find the height of this tree? How could you collect that data? What tools would you need? What about measuring the height of the tree if you are standing on a slope? We’ll apply right triangles to this situation and many others!
Instruments for Measuring Angles
Outside of a surveying environment, we’ll generally use a clinometer or relaskop to measure vertical angles (up and down) and a compass to measure horizontal angles (back and forth). A clinometer will have two scales – one in percent and another in degrees or topographic. We’ll use the percent or topographic scale when measuring slopes and heights because the calculations can be done in your head.
Slopes
Soon enough, you’ll learn how to measure the slope angle using a clinometer or relaskop in the field!
Slopes on maps
What if you’re presented with a topographic map and asked to find the slope? The distance you measure on the map represents the horizontal distance of the slope triangle (after you have converted it from the map distance to a ground distance). If you were to walk that same path on the ground, you’d be walking the slope distance.
You’ll find the slope angle of a map segment using [latex]Slope Angle in Percent = \frac{VD}{HD} * 100[/latex], where the the vertical distance (VD) is the number of contour lines crossed by that segment multiplied by the contour interval – and the horizontal distance is the ground distance along the segment.
Examples: Slope on a Topographic Map
Type your examples here.
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Slopes for Ladders and Ramps
We’ll see several situations where calculating the slope angle is needed given two of the sides of the slope triangle. In these cases, you’ll likely have a slope angle that you need to stay below or above for safety or accessibility. For example, a ladder is set up for optimal safety if its vertical distance is four times the horizontal distance it lies from the base of the object being climbed (400% slope or about 76°).
To meet accessibility requirements under the Americans with Disabilities Acts (ADA), ramps need to have a slope ratio of 1:12 (8.3%) or less and are limited to a maximum horizontal distance of 30 feet.
Examples: Ladders and Ramps
Type your examples here.
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Slopes for Trails
On a trail, the distance you walk is the slope distance (SD), and it’s the hypotenuse of the slope triangle. The horizontal distance (HD) is the distance measured on a map and the vertical distance (VD) is the difference between your starting and finishing elevations.
The Forest Service has developed guidelines for ADA accessible trails that include the following parameters:
- 1:20 (5 percent) for any distance
- 1:12 (8.33 percent) for up to 200 feet (61 meters)
- 1:10 (10 percent) for up to 30 feet (9 meters)
- 1:8 (12.5 percent) for up to 10 feet (3 meters)
- 1:7 (14 percent) into and out of drain dips for up to 5 feet (1.5 meters) where the cross slope does not exceed 1:10
- Cross slopes should not exceed 1:20 (5 percent)
Trails built with a grade less than half the side slope (slope of the hillside the trail traverses) are more resistant to erosion. In general, trails with an average slope greater than 40% are considered extremely challenging.
Given these parameters, we can design trails over a variety of terrain!
Examples: Trails
Type your examples here.
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Heights
Let’s leave the world of slope for a moment to
Heights on a Slope
Gaps
Gaps on a Slope
Plots on a Slope
You’re on one side of a river (or a canyon, or a swampy muskeg, or a busy highway) and need to know the distance to the other side. Right triangles can help and it’s like surveying magic!
You’ll need to know one of the legs of the triangle and one of the angles. Which ones can you readily measure?
Use a tape to measure the distance you walk from the object you sighted on the other side to the point you’ll sight back to the object (side b). Use a compass to measure angle α. Viola! You have an angle and a side of a right triangle. Use these to find side a.
(In this particular diagram, you’d need to subtract the distance you stood from the river’s edge from a to get the width of the river.)
[latex]\text{tan }\alpha= \frac{a}{b}[/latex]
so
[latex]b \cdot \text{tan }\alpha= a[/latex]
In general, we call side b the baseline and angle α is our “angle to point”.
Examples
Tree Heights
Check out the sweet triangle we can make with a tree! Which parameters can we measure easily in the field?
On flat ground, if you’ve got the angle to the top of the tree, the angle to the bottom of the tree, and you know your distance from the tree – you can build a couple of straightforward triangles.
The tangent of angle alpha will be height A divided by length D1. So, we can find height A by multiplying the tangent alpha by the distance from the tree, and we can find height B by multiplying the tangent beta by the distance from the tree. So, the total height of the tree (A+B) is equal to:
[latex]D\small 1 \cdot \tan\alpha+D\small 1 \cdot \tan\\beta = Height[/latex]
Simplified
[latex]D\small 1 (\tan\alpha+\tan\beta) = Height[/latex]
We’ve got a general formula that will work with trees on a slope too, but let’s take some time to look at slope first!
Examples
A) You are standing 66 feet from a tree on level ground. Your eye is at 5’6″. The angle to the top of the tree is 12°. How tall is this tree?
B) You are standing 100 meters from a tree on level ground. Your eye is 171 cm tall. The angle to the top of the tree is 8°. How tall is this tree?
Slope
The slope of a surface is a measure of its steepness. In some cases, such as a walkway or ramp or street, a shallow slope is safer than a steep slope. In other cases, such as a roof, a steep slope may be preferred because it allows rainwater or accumulated snow to move off the roof surface more easily than a shallow slope.[1]
We will first look at slope in terms of vertical and horizontal distances, and we will then look at slope in terms of angles. Come on, let’s hit the slopes!
Slope as a Ratio
Slope is defined as the ratio of the vertical rise to the horizontal run.
- A line that is increasing in height has a positive slope.
- A line that is decreasing in height has a negative slope.[2]
- The slope of a horizontal line is 0; it is not increasing or decreasing.
A slope may be expressed as a ratio, a decimal, or a percent grade. For example, consider this loading ramp with a vertical rise of 23.5 inches and a horizontal run of 132 inches.
As a ratio, the slope is 23.5 : 132. Dividing the rise by the run gives a decimal value of approximately 0.18. Moving the decimal point two places to the right gives a grade of approximately 18%.
Check out Forest Measurements by Joan DeYoung. It has an excellent discussion of slope and its measurement and applications in the field of forestry.
Practice Exercises
Express each slope in three ways: as a ratio, a decimal, and a percent grade.
Slope as an Angle
The steepness of a line may also be described by its angle of elevation above the horizontal (or its angle of depression below the horizontal).
Notice that the vertical rise is the side opposite the angle, and the horizontal run is the side adjacent to the angle. Therefore, trigonometry tells us that [latex]\text{tangent}=\frac{\text{opposite}}{\text{adjacent}}=\frac{\text{rise}}{\text{run}}[/latex]. The tangent of the angle is equal to the slope.
Or, thinking about it in reverse, the inverse tangent of the slope is the angle.
[latex]\text{tan}^{-1}\left(\frac{\text{rise}}{\text{run}}\right)=\theta[/latex]
When we are looking at slope as a right triangle, the following terms will be synonymous:
Run = Horizontal Distance (HD) (also the distance if calculated from a map)
Rise = Vertical Distance (VD) (you can get this by counting contour lines on a topo map!)
Distance along the Hypotenuse = Slope Distance (SD)
Examples
A) SD = 124′ Slope Angle = 5° Find HD and VD.
B) Slope Angle = 15° HD = 125′ Find SD and VD.
C) HD = 120′ VD = -12′ Find SD and Slope Angle.
D) You’re on a slope of 12° and measure along the ground a distance of 100 feet (SD). How far is this as a horizontal distance?
E) A slope rises 10 feet for every 150 horizontal feet. What is the slope angle in degrees?
F) On a map, you place a proposed 400-yard fenceline. The fence runs directly up a slope that is 8°. What will the actual length of the fenceline be on the ground?
G) Two points on a map are 4 inches apart. The map scale is 1:24000. The starting elevation is 1800’and the endpoint elevation is 2200′.
- What is the distance between the two points on the ground (not accounting for slope).
- What is the slope?
- What will the actual distance be if measured on the ground?
Practice Exercises
For exercises 5 through 8, determine the angle of elevation for each slope. Round to the nearest hundredth of a degree.
For proper drainage, the ground around a building should slope downwards, away from the building.
9. The minimum downward grade of the ground is 1%. Assuming this grade, if a point on the ground is 50 feet horizontally from the base of the house, how much lower is the ground at that point?
10. The preferred minimum downward grade of the ground is 2%. Assuming this grade, if a point on the ground is 50 feet horizontally from the base of the house, how much lower is the ground at that point?
11. The maximum acceptable downward grade of the ground is 10%. Assuming this grade, if a point on the ground is 50 feet horizontally from the base of the house, how much lower is the ground at that point?
12. A motion sensor needs to be installed on the outside of a warehouse door 12 feet above the ground. The sensor should go off if anyone approaching the warehouse gets within 20 feet of the door. What angle from vertical should the sensor point away from the building to detect someone at the appropriate distance from the warehouse? Round your answer to the nearest degree.
13. A ramp is being constructed to the entrance to a public building that is 2.5 feet higher than the level courtyard in front of the entrance. Assuming that the ramp will be continuous with no switchbacks or level platforms,[5] what is the minimTum horizontal distance required for the ramp so that it will comply with ADA regulations?
Tree Height + Slope
When we measure trees in the field, we’ll usually be standing on slope (at least in this part of the world!). The distance that we measure by pulling a tape or pacing chains will be a slope distance (SD). We can get the slope angle with a clinometer. We can also get the angle to the top of a tree and the angle to the bottom of a tree with a clinometer.
How can we put these measurements together to arrive at the tree’s height?
Our first step is the get our horizontal distance from the tree. We can derive this from the slope distance (that we measured by pacing or pulling a tape/chain) and the slope angle (that we measured using a clinometer or similar tool).
[latex]\text{Horizontal Distance} = \text{Slope Distance}\cdot \text{Cos}(\text{slope angle})[/latex]
Then, we can calculate the total height of the tree by using the angles we measured to its top and bottom.
[latex]\text{Tree Height}= \text{Horizontal Distance}\cdot (\text{Tan angle to top - Tan angle to bottom})[/latex]
In this class, our work with tree height is limited to applications of trigonometric functions. We can also use percent slope (% slope) for the same calculations. Forest Measurements by Joan DeYoung has a great discussion and explanation of the arithmetic for working with percent slope rather than slope angle.
Problem Set 8.4
Express each slope in three ways: as a ratio, a decimal, and a percent grade.
1. HD = 100 ft VD = 15 ft
2. HD = 66 ft VD = 3 ft
3. HD = 100 ft VD = 8 inches
4. HD = 50 m VD = 28 cm
5. To be safe, for every four feet of height it gains, a ladder should be placed one foot from the structure it is accessing. In this scenario, what is the slope of the ladder in percent grade?
6. The downward grade of the ground from my house out toward the shed is 2%. If the shed is 100 feet horizontally from the base of the house, how much lower is the ground at that point?
| Calculate the missing values. | ||||
| Slope Distance (ft) | Slope Angle | Horizontal Distance (ft) | Vertical Distance (ft) | |
| 7 | 125.6 | 5° | ||
| 8 | 113.9 | -12° 15’ | ||
| 9 | 17° | 145.8 | ||
| 10 | -20° 20’ | 98.1 | ||
| 11 | 195.8 | -35.2 | ||
| 12 | 145.3 | 15.9 |
Find the horizontal distance (HD) in the following scenarios.
13. Slope angle = 14º SD = 100 ft
14. Angle of depression = 8º SD = 66 ft
15. Slope angle = 2º SD = 50 ft
16. Angle of depression = 1º SD = 100 ft
17. Slope angle = 8º SD = 25 ft
18. Slope angle = 4º SD = 33 ft
19. A slope rises 15 feet for every 125 horizontal feet. Find the slope angle in degrees to the nearest hundredth of a degree.
20. SD = 122 ft’ Slope Angle = 5° Find HD and VD.
21. Slope Angle = 12° HD = 100 ft Find SD and VD.
22. HD = 120 ft VD = -8 ft Find SD and Slope Angle.
23. You measure a distance of 197.3 feet on a slope of -27° 15’13”. Find the horizontal and vertical distances in feet to the nearest tenth of a foot.
24. A plot radius of 27.4 feet is laid out on a map. The slope is 16°. Find the length of the radius on the ground.
25. Two points on a map are 5.4 inches apart. The scale is 1:24000. The starting elevation is 1,200 feet and the ending elevation is 1,450 feet. (a) Calculate the distance between the two points on the map in feet to the nearest foot. (b) Find the angle of the slope. (c) Calculate the distance if measured on the ground.
26. Two points on a map are 4.2 cm apart. The scale is 1:24000. The starting elevation is 800 feet and the ending elevation is 1,200 feet. (a) Calculate the distance between the two points on the map in feet to the nearest meter. (b) Find the angle of the slope. (c) Calculate the distance if measured on the ground (in meters).
23. You are standing 66 feet from a tree on level ground. The angle to the top of the tree is 12° and the angle to the bottom is 2°. What is the height of the tree?
24. You are standing 100 meters from a tree on level ground. The angle to the top of the tree is 8° and the angle to the bottom is 5°. How tall is this tree?
25. You are on one side of an impassable canyon and want to measure the distance across the canyon. You stand directly across from a large pine and mark your spot. Then, you turn 90º and pace one chain along the canyon rim (this is your baseline). You use your compass to sight back to the large pine. This angle is 10º. What is the distance across the canyon?
26. You’re on a mountainside next to a waterfall and you want to know how wide the river is at the falls. There’s a big pine on the other side of the waterfall. You stand across from the pine and then turn 90º and walk 200 feet down the slope. You sight back to the pine at an angle of 5º from your baseline. Then, you remember: The baseline you walked down the slope won’t be a true horizontal distance because the mountainside is so steep. If you want your measurement of the waterfall to be accurate, you need to take the distance you walked (200 ft) and correct it from a slope distance to a horizontal distance. You use your clinometer and measure the slope angle as 4º. Now – use the true horizontal distance for the baseline and calculate the width of the waterfall.
27. Find the widths of these gaps. The baseline is the distance you measure on one side of the gap. The angle to point is the angle you measure back to a point on the other side of the gap. The slope angle is the slope of the ground along your baseline. (a) Baseline HD = 130 ft angle to point = 48° 17’10” (b) Baseline SD = 110 feet, slope angle = – 10°, angle to point = 28° (c) Baseline SD = 150 feet, slope angle = +7° 15’ 23”, angle to point = 62° 47’ 32”
28. You’re on a slope of 6° and standing 66 feet downhill from a large Ponderosa pine. You sight 18° to the top of the tree and 2° to the bottom. What is the height of the tree?
29. You’re on a slope of 9° and standing 100 feet uphill from a large western larch. You sight 5° to the top of the tree and -12° to the bottom. What is the height of the tree?
Practice finding the heights of the following trees, given the information in the table.
| Slope Dist out | Slope ° | Horizonal Dist | >° top | >° bottom | Height (ft) | |
| 30. | 90 ft | 0 | 53° | -9° | ||
| 31. | 15.75 m | -7° | 32° | -4° | ||
| 32. | 2.50 chains | 11° | 47° | -15° | ||
| 33. | 75 ft | -20 | 60° | +12° | ||
| 34. | 60 ft | +6° | 51° | +7° | ||
| 35. | 25 m | -15° | 49° | -15° |
- Source: https://www.nachi.org/roof-slope-pitch.htm ↵
- We won't worry about negative slopes in this textbook, because we can always express a negative slope using a word like "decrease", "decline", or "depression" in combination with a positive number. ↵
- Source: https://www.ada-compliance.com/ada-compliance/403-walking-surfaces ↵
- Source: https://www.ada-compliance.com/ada-compliance/405-ramps ↵
- If you were curious, this is the maximum rise allowed for this situation; a rise of more than 2.5 feet would require a switchback or a level platform. ↵
An instrument used for measuring angles of slope, elevation, or depression of an object.
https://www.forestry-suppliers.com/Images/Original/1316_43895_p1.jpg
a handheld optical instrument primarily used in forestry for measuring tree diameters, heights, and basal area
an instrument containing a magnetized pointer which shows the direction of magnetic north and bearings from it
A scale designed to measure heights directly in feet when using a horizontal baseline of 66' (one chain).
Horizontal leg of a right triangle when measuring slope or height.
Hypotenuse of the right triangle used in slop or height calculations.
Applications
You're a wildlife technician working on a behavioral study of Oregon spotted frogs (Rana pretiosa). You are looking at the time required for larvae to metamorphose into adults and the reaction time of adult frogs to prey items under a variety of conditions in the lab. What units of time will you use to measure the development of larvae into adult frogs? What units of time will you use to measure an adult frog's reaction time? Are these units of measurement different? Why?
You're measuring stream habitat using a meter tape. Do you record measurements to the nearest meter? The nearest centimeter? The nearest millimeter? If you record your measurements to centimeters, you might measure a riffle section as 5.38 meters long and 1.46 meters wide. Should you report the area of this riffle as 7.8548 square meters?
Objectives
When you've finished this section, you will be able to perform the following:
- Choose a level of precision for applied math calculations.
- Round to that level of precision.
In the first few modules, we rarely concerned ourselves with rounding; we assumed that every number we were told was exact and we didn't have to worry about any measurement error. However, every measurement contains some error. A standard sheet of paper is [latex]8.5[/latex] inches wide and [latex]11[/latex] inches high, but it's possible that the actual measurements could be closer to [latex]8.4999[/latex] and [latex]11.0001[/latex] inches. Even if we measure something very carefully, with very sensitive instruments, we should assume that there could be some small measurement error.[1]
Exact Values and Approximations
A number is an exact value if it is the result of counting or a definition.
A number is an approximation if it is the result of a measurement or of rounding.
Exercises
Identify each number as an exact value or an approximation.
- An inch is [latex]\frac{1}{12}[/latex] of a foot.
- This board is [latex]78[/latex] inches long.
- There are [latex]14[/latex] students in class.
- A car’s tachometer reads [latex]3,000[/latex] rpm.
- A right angle measures [latex]90^\circ[/latex].
- The angle of elevation of a ramp is [latex]4^\circ[/latex].
Accuracy and Significant Figures
Because measurements are inexact, we need to consider how accurate they are. This requires us to think about significant figures—often abbreviated "sig figs" in conversation—which are the digits in the measurement that we trust to be correct. The accuracy of a number is equal to the number of significant figures. (By the way, the terms "significant digits" and "significant figures" are used interchangeably.) The following rules aren't particularly difficult to understand but they can take time to absorb and internalize, so we'll include lots of examples and exercises.
Significant Figures
- All nonzero digits are significant.
Ex: [latex]12,345[/latex] has five sig figs, and [latex]123.45[/latex] has five sig figs. - All zeros between other nonzero digits are significant.
Ex: [latex]10,045[/latex] has five sig figs, and [latex]100.45[/latex] has five sig figs. - Any zeros to the right of a decimal number are significant.
Ex: [latex]123[/latex] has three sig figs, but [latex]123.00[/latex] has five sig figs. - Zeros on the left of a decimal number are NOT significant.
Ex: [latex]0.123[/latex] has three sig figs, and [latex]0.00123[/latex] has three sig figs. - Zeros on the right of a whole number are NOT significant unless they are marked with an overbar.
Ex: [latex]12,300[/latex] has three sig figs, but [latex]12,30\overline{0}[/latex] has five sig figs.
Another way to think about #4 and #5 above is that zeros that are merely showing the place value—where the decimal point belongs—are NOT significant.
Exercises
Determine the accracy (i.e., the number of significant figures) of each number.
- [latex]63,400[/latex]
- [latex]63,040[/latex]
- [latex]63,004[/latex]
- [latex]0.085[/latex]
- [latex]0.0805[/latex]
- [latex]0.08050[/latex]
In 1856, the first official measurement of the height of Mount Everest—called Sagarmatha in Nepal and Chomolungma in Tibet—was announced. The height was determined to be exactly [latex]29,000[/latex] feet, but there was concern that people would think this was only a rough estimate rounded to the nearest thousand feet. Therefore, the height was announced as [latex]29,002[/latex] feet, so that everyone seeing that number would believe that the measurement was correct to the nearest foot.[2] Yes, to demonstrate the correctness of the measurement, an incorrect measurement was announced.
Instead of fudging a number like [latex]29,000[/latex] to show that it is correct to the nearest foot, we can write it with an an overbar to indicate that the zeros are significant. Putting [latex]29,00\overline{0}[/latex] in a newspaper headline in 1856 would probably have confused people, but you can handle it because you're in a math class. Writing [latex]29,00\overline{0}[/latex] is our way of saying "Really, to the nearest foot, it's exactly [latex]29,000[/latex] feet!"
Exercises
Determine the accuracy (i.e., the number of significant figures) of each number.
- [latex]29,000[/latex]
- [latex]29,\overline{0}00[/latex]
- [latex]29,0\overline{0}0[/latex]
- [latex]29,00\overline{0}[/latex]
Two things to remember: we don't put an overbar over a nonzero digit, and we don't need an overbar for any zeros on the right of a decimal number because those are already understood to be significant.
Accuracy-Based Rounding
As we saw in Chapter 0.4, it is often necessary to round a number. We often round to a certain place value, such as the nearest hundredth, but there is another way to round. Accuracy-based rounding considers the number of significant figures rather than the place value.
Accuracy-based rounding:
- Locate the rounding digit to which you are rounding by counting from the left until you have the correct number of significant figures.
- Look at the test digit directly to the right of the rounding digit.
- If the test digit is 5 or greater, increase the rounding digit by 1 and drop all digits to its right. If the test digit is less than 5, keep the rounding digit the same and drop all digits to its right.
Exercises
Round each number so that it has the indicated number of significant figures.
- [latex]51,837[/latex] (three sig figs)
- [latex]51,837[/latex] (four sig figs)
- [latex]4.2782[/latex] (two sig figs)
- [latex]4.2782[/latex] (three sig figs)
When the rounding digit of a whole number is a [latex]9[/latex] that gets rounded up to a [latex]0[/latex], we must write an overbar above that [latex]0[/latex].
Similarly, when the rounding digit of a decimal number is a [latex]9[/latex] that gets rounded up to a [latex]0[/latex], we must include the [latex]0[/latex] in that decimal place.
Exercises
Round each number so that it has the indicated number of significant figures. Be sure to include trailing zeros or an overbar if necessary.
- [latex]13,997[/latex] (two sig figs)
- [latex]13,997[/latex] (three sig figs)
- [latex]2.596[/latex] (two sig figs)
- [latex]2.596[/latex] (three sig figs)
The height of Mount Everest has changed over the years due to plate tectonics and earthquakes. In December 2020, it was jointly announced by Nepal and China that the summit of Mount Everest has an elevation of [latex]29,031.69[/latex] ft.[3]
- Round [latex]29,031.69[/latex] ft to two sig figs.
- Round [latex]29,031.69[/latex] ft to three sig figs.
- Round [latex]29,031.69[/latex] ft to four sig figs.
- Round [latex]29,031.69[/latex] ft to five sig figs.
- Round [latex]29,031.69[/latex] ft to six sig figs.
Accuracy when Multiplying and Dividing
Suppose you needed to square the number [latex]3\frac{1}{3}[/latex]. You could rewrite [latex]3\frac{1}{3}[/latex] as the improper fraction [latex]\frac{10}{3}[/latex] and then figure out that [latex](\frac{10}{3})^2 = \frac{100}{9}[/latex], which equals the repeating decimal [latex]11.111...[/latex]
Because most people prefer decimals to fractions, we might decide to round [latex]3\frac{1}{3}[/latex] to [latex]3.33[/latex] and find that [latex]3.33^2=11.0889[/latex]. The answer [latex]11.0889[/latex] looks very accurate, but it is a false accuracy because there is round-off error involved. Only when we round to three sig figs do we get an accurate result: [latex]11.0889[/latex] rounded to three sig figs is [latex]11.1[/latex], which is accurate because [latex]11.111...[/latex] rounded to three sig figs is also [latex]11.1[/latex]. It turns out that because [latex]3.33[/latex] has only three significant figures, our answer must be rounded to three significant figures.
Don't round off the original numbers; do the necessary calculations first, then round the answer as your last step.
Exercises
Use a calculator to multiply or divide as indicated. Then round to the appropriate level of accuracy.
- [latex]8.75\cdot12.25[/latex]
- [latex]355.12\cdot1.8[/latex]
- [latex]77.3\div5.375[/latex]
- [latex]53.2\div4.5[/latex]
- Suppose you are filling a 5-gallon can of gasoline. The gasoline costs [latex]\textdollar4.579[/latex] per gallon, and you estimate that you will buy [latex]5.0[/latex] gallons. How much should you expect to spend?
Bonus material: Here is a comic strip from xkcd.com showing that including a lot of decimal digits can give a false sense of accuracy.
