Exit icket Sample Solutions Bob can paint a fence in hours, and working with Jen, the two of them painted a fence in hours. How long would it have taken Jen to paint the fence alone? Let xx represent the time it would take Jen to paint the fence alone. hen, Bob can paint the entire fence in hours; therefore, in one hour he can paint of the fence. Similarly, Jen can paint of the fence in one hour. We know that it took them two hours to xx complete the job, and together they can paint of the fence in one hour. We then have to solve the equation: + xx = + = + = xx = 33. hus, it would have taken Jen 33 hours and minutes to paint the fence alone. Homework Problem Set Sample Solutions 1. If two inlet pipes can fill a pool in one hour and 3333 minutes, and one pipe can fill the pool in two hours and 3333 minutes on its own, how long would the other pipe take to fill the pool on its own?. + xx =. We find that xx = 33. 7777; therefore, it takes 33 hours and 4444 minutes for the second pipe to fill the pool by itself. 2. If one inlet pipe can fill the pool in hours with the outlet drain closed, and the same inlet pipe can fill the pool in. hours with the drain open, how long does it take the drain to empty the pool if there is no water entering the pool? xx =. We find that xx = ; therefore, it takes hours for the drain to empty the pool by itself. 267 his file derived from ALG II--E-1.3.0-07.2015
3. It takes 3333 minutes less time to travel miles by car at night than by day because the lack of traffic allows the average speed at night to be miles per hour faster than in the daytime. Find the average speed in the daytime. tt 3333 = + tt 66 We find that tt =. he time it takes to travel miles by car at night is minutes, which is 33 hours. Since = 4444, the average speed in the daytime is 4444 miles per hour. 33 4. he difference in the average speed of two trains is miles per hour. he slower train takes hours longer to travel miles than the faster train takes to travel miles. Find the speed of the slower train. tt tt + = We find that tt = 33, so it takes 33 hours for the faster train to travel miles, and it takes hours for the slower train to travel miles. he average speed of the slower train is 3333 miles per hour. 5. A school library spends $8888 a month on magazines. he average price for magazines bought in January was 7777 cents more than the average price in December. Because of the price increase, the school library was forced to subscribe to 77 fewer magazines. How many magazines did the school library subscribe to in December? 8888 xx + 00. 7777 = 8888 xx 77 he solution to this equation is., so the average price in December is $.. hus the school subscribed to 3333 magazines in December. 6. An investor bought a number of shares of stock for $, 666666. After the price dropped by $ per share, the investor sold all but 44 of her shares for $,. How many shares did she originally buy? xx = xx 44 + his equation has two solutions: 3333 and. hus, the investor bought either 3333 or shares of stock. 268 his file derived from ALG II--E-1.3.0-07.2015
7. Newton s law of universal gravitation, FF = GGmm mm, measures the force of gravity between two masses rr mm and mm, where rr is the distance between the centers of the masses, and GG is the universal gravitational constant. Solve this equation for GG. GG = FFrr mm mm 8. Suppose that = xx+yy xxxx. a. Show that when xx = aa and yy =, the value of tt does not depend on the value of aa. aa+ When simplified, we find that tt = ; therefore, the value of tt does not depend on the value of aa. b. For which values of aa do these relationships have no meaning? If aa is 00, then xx has no meaning. If aa =, then yy has no meaning. 9. Consider the rational equation RR = xx + yy. a. Find the value of RR when xx = and yy = 33 44. So RR = 66. RR = + 33 44 269 his file derived from ALG II--E-1.3.0-07.2015
b. Solve this equation for RR, and write RR as a single rational expression in lowest terms. here are two approaches to solve this equation for RR. he first way is to perform the addition on the right: RR = xx + yy = yy xxxx + xx xxxx xx + yy = xxxx. he second way is to take reciprocals of both sides and then simplify: RR = xx + yy = yy xxxx + xx xxxx =. (xx + yy) xxxx In either case, we find that RR = xxxx xx+yy. 10. Back in Lesson 15 you looked at the problem: Consider an ecosystem of rabbits in a park that starts with rabbits and can sustain up to 6666 rabbits. An equation that roughly models this scenario is 6666 PP = +, tt + where PP represents the rabbit population in year tt of the study. You found that the rabbit population in year was 41 rabbits. A. Solve this equation for tt. Describe what this equation represents in the context of this problem. 6666 6666 tt = PP 6666 his equation represents the relationship between the number of rabbits, PP, and the year, tt. If we know how many rabbits we have, < PP < 6666, we will know how long it took for the rabbit population to grow that much. If the population is, then this equation says we are in year 00 of the study, which fits with the given scenario. 270 his file derived from ALG II--E-1.3.0-07.2015
B. At what time does the population reach rabbits? If PP =, then tt = 6666 333333 = = ; therefore, the rabbit population is in year 6666 of the study. Extension:. Suppose that Huck Finn can paint a fence in hours. If om Sawyer helps him paint the fence, they can do it in 33 hours. How long would it take for om to paint the fence by himself? Huck paints the fence in hours, so his rate of fence painting is fence per hour. Let denote the percentage of the fence om can paint in an hour. hen fence = + fence per hour (33 hours). 33 = = + + 33( + ) = = = + = So, om can paint of the fence in an hour. hus, om would take the fence by himself. = 77. hours to paint 271 his file derived from ALG II--E-1.3.0-07.2015
12. Huck Finn can paint a fence in hours. After some practice, om Sawyer can now paint the fence in 66 hours. A. How long would it take Huck and om to paint the fence together? he amount of fence that Huck can paint per hour is, and the amount that om can paint per hour is. So, together they can paint + of the fence per hour. Suppose the 66 66 entire job of painting the fence takes hh hours. hen, the amount of the fence that is painted is hh +. Since the entire fence is painted, we need to solve the equation 66 hh + =. 66 hh + = 66 hh = + = 3333 66 + = 3333 66 So, together they can paint the fence in 3333 hours, which is hours and 4444 minutes. B. om demands a half-hour break while Huck continues to paint, and they finish the job together. How long does it take them to paint the fence? Suppose the entire job of painting the fence takes hh hours. hen, Huck paints at a rate of of the fence per hour for hh hours, so he paints hh of the fence. om paints at a rate of of the fence per hour for hh hour, so he paints hh of the fence. ogether, 66 66 they paint the whole fence; so, we need to solve the following equation for hh: hh + 66 hh = hh + 66 hh = hh + hh = 66 6666 hh + hh = 6666 66 + = 6666 hh = 6666. hus, it takes 6666 hours, which is hours minutes, to paint the fence with om taking a hour break. 272 his file derived from ALG II--E-1.3.0-07.2015
C. Suppose that they have to finish the fence in 33 hours. What s the longest break that om can take? Suppose the entire job of painting the fence takes 77 hours, and om stops painting for bb hours for his break. hen, Huck paints at a rate of of the fence per hour for 77 hours, so he paints 77 of the fence. om paints at a rate of of the fence per hour for 66 77 bb hours, so he paints 66 77 bb of the fence. ogether, they paint the whole fence; so, we need to solve the following equation for bb: 77 + 66 77 bb = 77 + 77 bb 66 = 6666 77 + 77 bb = 6666 66 4444 + 3333 = 6666 4444 + 3333 6666 = bb =. hus, if om takes a break for hours, which is hour and 4444 minutes, the fence will be painted in 33 hours. 273 his file derived from ALG II--E-1.3.0-07.2015