Section V: Quadratic Equations and Functions. Module 4: Applications Involving Quadratic Functions. distance rate time

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1 Habeman / Kling MTH 95 Section V: Quadatic Equations and Functions Module 4: Applications Involving Quadatic Functions EXAMPLE: Pete s Plymouth tavels 00 miles aveaging a cetain speed. If the ca had gone 10 mph faste, the tip would have taken 1 hou less. Find Pete s aveage speed. SOLUTION: Let epesent Pete s aveage speed in mph and let t epesent the amount of time (in hous) the 00-mile tip took. Since the infomation given in the poblem involves distance, ate, and time, we will use the fomula d = t (distance = ate time). Fist, let s oganize what we know in a table: distance ate time 00 t t 1 (This ow epesents infomation about the actual two hunded mile tip.) (This ow epesents infomation about the tip if the ca had gone 10 mph faste.) Using the fomula d = t we obtain the following equations: 00 = t and 00 = ( + 10)( t 1). Since we ae tying to find Pete s aveage speed, we need to eliminate the time vaiable. If we solve 00 = t fo t, we obtain the equation so we can substitute the expession 00 Now let s solve this equation fo : t = 00, fo t in the equation 00 = ( + 10)( t 1) : 00 ( ) 00 = ( + 10) ( ) 00 = ( + 10) 1 00 = =

2 = = + 10 = + ( 000 ) = = 0 ( + 50)( 40) = = 50 o = 40 Since a negative value does not make sense fo an aveage speed, we can say that Pete s aveage speed was only 40 mph. (Thee must have been some ough teain o Pete is just a slow dive!) EXAMPLE: Find the equation of the quadatic function that passes though the points (, 1), (1, 17), and (, 31). SOLUTION: Using the standad fom of a quadatic function, f ( x) = ax + bx + c, we can substitute the values fo x and f (x) fom each point to come up with thee equations. Using the point (, 1) we obtain 1 = a( ) + b( ) + c 1= 4a b + c 1. The point (1, 17) yields 17 = a + b + c and the point (, 31) yields 31 = 4a + b + c 3.

3 We ve numbeed these equations so that we can efe to them late: 3 1= 4a b + c 17 = a + b+ c 31 = 4a + b + c 1 3 Now, if we subtact equation fom equation 1, we will eliminate the vaiable c. Likewise, if we subtact equation 3 fom equation, we will eliminate the vaiable c. 1 1= 4a b + c 17 = a + b + c 17 = a b c 3 31 = 4a b c 4 18 = 3a 3b 5 14 = 3a b Now if we add ou newly obtained equations 4 and 5, we can eliminate the vaiable a = 3a 3b = 3a b 3 = 4b So, b = 8. Now, substituting b = 8 into equation 5, we obtain 14 = 3a 8 6 = 3a. = a Finally substituting a = and b = 8 into yields 17 = c 17 = 10 + c. c = 7 So, f( x) = x + 8x + 7 is the equation of the quadatic function that passes though the points (, 1), (1, 17), and (, 31).

4 EXAMPLE: Assume that the numbe of lites of wate emaining in the bathtub vaies quadatically with the numbe of minutes which have elapsed since you pulled the plug. a. If the tub has 38.4, 1.6, and 9.6 lites emaining at 1,, and 3 minutes, espectively, since you pulled the plug, find a function V (t) expessing the volume of wate t minutes afte you pulled the plug. b. How much wate was in the tub when you pulled the plug? c. When will the tub be empty? d. In the eal wold, the numbe of lites of wate in the tub can neve be negative. What does the model pedict is the least amount of wate in the tub? Is this numbe easonable? e. Daw a gaph of the function in the appopiate domain. Use a dotted cuve fo any potion of the gaph that is outside the easonable domain. f. What is the easonable domain and ange fo this model? g. Why is a quadatic function moe easonable fo this poblem than a linea function would be? 4 SOLUTION: a. Let V (t) epesent the volume (in lites) of wate emaining in the bathtub t minutes afte you pulled the plug. Then we can wite odeed pais of the fom ( t, V( t )). Based on the given infomation, the odeed pais we have ae (1, 38.4), (, 1.6), and (3, 9.6). We need to find an equation of the fom V () t = at + bt + c. Using the thee odeed pais, we obtain the following thee equations (which we numbe again fo easy efeence): = a + b + c 1.6 = 4a + b + c 9.6 = 9a + 3b + c Now, if we subtact equation 1 fom equation, we will eliminate the vaiable c. Likewise, if we subtact equation 1 fom equation 3 we will eliminate the vaiable c. 1.6 = 4a + b + c = 9a + 3b + c = a b c = a b c = 3a + b = 8a + b

5 5 Now we can multiply equation 4 by and add it to equation 5 to eliminate the vaiable b = 6a b = 8a + b 4. 8 = a So, a =.4. Substituting a =.4 into equation 4 yields 16.8 = 3(.4) + b 16.8 = 7. + b. So, b = 4. Finally, substituting a =.4 and b = 4 into equation 1 yields 38.4 = c 38.4 = c. So, c = 60. Theefoe, Vt () =.4t 4t+ 60 is the function expessing the volume of wate t minutes afte you pulled the plug. b. To find the amount of wate in the tub when you pulled the plug, we need to find the volume in the tub when t = 0 so we just need to compute V (0). Since V (0) = 60, we know that thee wee 60 lites of wate in the tub when you pulled the plug. c. The tub will be empty when V() t = 0, so we need to solve.4t 4t + 60 = 0. We can use the quadatic fomula to solve this equation. t ± = (.4) 4 ± 0 = 4.8 = 5 ( 4) ( 4) 4(.4)(60) Thus, the tub will be empty in 5 minutes. d. Since the gaph of y = V() t is a paabola that opens upwad, we know that the least amount of wate that this model pedicts will occu at the vetex. t b 4 = = = = + = a and V(5).4(5) 4(5) 60 0 So, the model pedicts that the least amount of wate in the tub is 0 lites, which is easonable. [Note: If V (t) would have been negative at the vetex, it would not have been easonable and we would not include this t-value in the domain of ou model.]

6 6 e. The model is only valid on the inteval [0, 5], so we will highlight this potion of the cuve. Volume (in lites) Time (in minutes) since you pulled the plug Figue 1: The gaph of y = V() t. f. As mentioned in pat e, the easonable domain fo this model is [0, 5]. The easonable ange fo this model is [0, 60] which can be seen in Figue 1. We also could have deduced this without the gaph because we found in pat b that thee wee 60 lites of wate in the tub when you pulled the plug. g. A quadatic function is moe easonable than a linea function since a linea function would imply that the wate was daining at a constant ate. Since thee is moe pessue when the tub is fulle, the wate should dain moe quickly when you fist pull the plug