MAT 142 College Mathematics Module XPDimensional Analysis and ExponentialModelsTerri Miller revised December 13, 20101. Dimensional AnalysisThe purpose of this section is to convert between various types of units; feet to miles,kilometers per hour to meters per second, square feet to square inches, cubic meters to cubiccentimeters; etc. Dimensional analysis is the process of using a standard conversion tocreate a fraction, including units in that fraction and canceling units in the same way thatvariables are cancelled.We do need to know some relationships between units:1 mile = 5280 feet, 3 feet = 1 yard,1000 meters = 1 kilometer, 100 centimeters = 1 meterIf you write out the full process and make sure that the units cancel leaving only what youwant, you should be successful in converting units. The remainder of this section is examples.Example 1. Convert 6 feet to yards.Solution:6 feet =6 feet1·1 yard3 feet=6 yard3= 2 yardsExample 2. Convert 765 inches per second to miles per hour:Solution:765 in/sec =765 inchessecond·foot12 inches·1 mile5280 feet=765 miles12 · 5280 second=17 miles1408 second·60 second1 minute·60 minute1 hour=17 · 60 · 60 miles1408 hour=7650176mi/hr =392588mi/hr ≈ 43.4659 mi/hrExample 3. Convert 40 square feet to square inches.Solution:40 ft2=40 ft · ft1·12 in1 ft·12 in1 ft= 5760 in2Example 4. Convert 2 cubic yards to cubic feet.Solution:2 yd3=2 yd · yd · yd1·3 ftyd·3 ftyd·3 ftyd= 2 · 3 · 3 · 3 ft3= 54 ft3Example 5. Convert 75 centimeters per second to meters per hour:Solution:75 cm/sec =76 cmsecond·1 meter100 cm·60 second1 minute·60 minute1 hour=76 · 60 · 60 miles100 hour= 2736 m/hrFor more practice and to ensure that the process is clear, we will also do some with unfamiliar(made up) units.Example 6. Suppose we are given that 13 horks is equivalent to one plop, 7 plops areequivalent to one wooze, 5 hons is equal to one slop and 11 slops are equal to one murk.Convert:(1) 80 horks to woozes(2) 9 square plops to square horks(3) 2 cubic woozes to cubic plops(4) 12 horks per hon to plops per murk(5) 18 cubic horks per murk to cubic plops per slopSolution:(1)80 horks =80 hork1·plop13 hork·wooze7 plop=8013 · 7woozes =8091woozes ≈ 0.8791 woozes(2)9 plops2=9 plop1·13 horkplop·13 horkplop= 1521 horks2(3)2 woozes3=2woozes1·7 plopwooze·7 plopwooze·7 plopwooze= 686 plops3(4)12 horks/hon =12 horkshon·plop13 hork·5 honslop·11 slopmurk=66013plops/murk ≈ 50.7692 plops/murk(5)18 horks3/murk =18 horks3hon·plop13 hork·plop13 hork·plop13 hork·5 honslop·11 slopmurk=66013plops/mork ≈ 50.7692 plops/murk2. Exponential Functions2.1. Function Review. For our purposes, we will only consider functions which can begiven as a formula and whose input is some subset of the real numbers. In this case, recallthat a function is a rule that assigns a value (output) according to the formula for eachnumber you put in (input). The function has some name, the input variable is usually x butneed not be. Some examples of functions are:sam(x) = 2x + 1, tax(s) = 3s3+ 2s, g(t) =√t.2c 2010 ASU School of Mathematical & Statistical Sciences and Terri L. MillerFor the function sam, the input variable is x, for tax it is s, and for g it is t. This variableis also called the independent variable. Think of it as the one that you get to choose.Once you have followed the formula to produce a value, this value is called the functionvalue or the dependent variable. The process of following the formula for a particularinput is called evaluating the function. Let us look at our function sam. Suppose we want toevaluate sam at 3, then we would write sam(3). Once you have followed the formula, 2x + 1where x is 2, you get 5. Then we would write sam(3) = 5. Remember that the independentvariable in the formula is a place holder for the actual value that you will put in.Example 7. Evaluate:(1) tax(−1),(2) tax12,(3) tax(3),(4) tax(♥)Solution:(1) tax(−1) = 3(−1)2− 2(−1) = 3(1) + 2 = 3 + 2 = 5(2) tax12= 3122− 212= 3(14− 1 =34− 1 = −14(3) tax(3) = 3(3)2− 2(3) = 3(9) − 6 = 27 − 6 = 21(4) tax(♥) = 3♥2− 2♥2.2. The exponential function. The functions that we will be concerned with will havethe property that they grow or decay at a constant rate. Consider the following data collectedin a laboratory:time number of bacteria0 201 hr 352 hr 533 hr 794 hr 118A little calculation shows us that this bacteria grows at a rate of 50% per hour. This typeof function is an exponential function and they all have the formula f (x) = abkxwhereb > 0. b is the base of the exponential and k is a constant determined by the growth rate.You often see this written as y = abkx.There are several possible graphs for these functions depending on the values of a and k.Some examples of the graphs are given below. (Reminder, when you graph a function, theinput is on the horizontal and the output on the vertical.)3c 2010 ASU School of Mathematical & Statistical Sciences and Terri L. MillerNotice that four of these keep going up as you move from left to right. So the functionvalues, y, get bigger as the input, x, gets gigger. Such functions are said to be strictlyincreasing. The other two functions have the property that y gets smaller as x gets bigger,these functions are said to be strictly dedcreasing. Recall, the y-intercept is the pointwhere the graph crosses the x-axis. It is the point whose y value is the function evaluatedat x = 0; hence the coordinates are (0, f(0)) (if the function is called f).Let us examine the relationships between a and k that determine some of the traits of thegraph. We will be grouping them according to attributes a and k.The first set that we will look at all have a > 0.f(x) = 3x, g(x) = 2 ∗ 3x, h(x) =12∗ 3x, m(x) = 3−xWe will look at the first three together on one graph so that we can compare the graphsand determine how the size of a effects the graph; see Figure 1. In Figure 1, f is graphed ingreen, g is graphed in red, and h is graphed in blue. In Figure 2, we have graphed f and m,with m being red, so that we can look at the effects of a negative k.In Figure 1 see that all three of these functions will grow at the same rate, the difference isthe y-intercetp. The initial value is the value of the function when x = 0 or f(0). So, thea in the function abkxis the initial value. In Figure 2, we see that the difference is whetherthe function is growing (increasing) or decaying (decreasing). Hence we conclude
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