Name: Period: For full credit, show all step by step work required to support your answers on your own paper. 1. The temperature outside a house during a 4-hour period is given by t F t 8 1cos 1, t 4 F t is measured in degrees Fahrenheit and t is measured in hours. a) Sketch the graph of F., where b) Find the average temperature, to the nearest degree Fahrenheit, between t 6 and t 14. c) An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit. For what values of t was the air conditioner cooling the house? d) The cost of cooling the house accumulates at the rate of $.5 per hour for each degree the outside temperature exceeds 78 degrees Fahrenheit. What was the total cost, to the nearest cent, to cool the house for this 4-hour period?. The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function R of time t. The table shows the rate as measured every 3 hours for a 4-hour period. a) Use a midpoint Riemann sum with 4 subdivisions of equal length 4 to approximate R t dt. Using correct units explain the meaning of your answer in terms of water flow. b) Is there some time t, t 4 answer. c) The rate of water flow, such that R' t? Justify your 1 R t can be approximated by Qt 79 768 3t t. Use Q t to approximate the average rate of water flow during the 4-hour time period. Indicate units of measure. 1
3. The rate at which people enter an amusement park on a given day is modeled by the function E 156 defined by E t. The rate at which people leave the same amusement park on t 4t16 the same day is modeled by the function L defined by Lt t 989 38t 37. Both Et and L t are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 t 3 the hours during which the park is open. At time t 9, there are no people in the park. a) How many people have entered the park by 5: PM ( t 17 )? Round your answer to the nearest whole number. b) The price of admission to the park is $15 until 5: PM ( t 17 ). After 5: PM, the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number. t c) Let H t ExLx dx for 9 t 3. The value of 17 is 375. Find the value of H '17, and explain the meaning of H 17 and '17 9 of the amusement park. H to the nearest whole number H in the context d) At what time t, for 9 t 3, does the model predict that the number of people in the park is a maximum?. t P t, of a pollutant in a lake changes at the rate P' t 1 3e gallons per day, where t is measured in days. There are 5 gallons of the pollutant in the lake at time t. The lake is considered to be safe when it contains 4 gallons or less of pollutant. 4. The number of gallons, a) Is the amount of pollutant increasing at time t 9? Why or why not? b) For what value of t will the number of gallons of pollutant be at its minimum? Justify your answer. c) Is the lake safe when the number of gallons of pollutant is at its minimum? Justify your answer. d) An investigator used the tangent line approximation to P t at t as a model for the amount of pollutant in the lake. At what time t does this model predict that the lake becomes safe?
5. The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a twice-differentiable and strictly increasing function R of time t. The graph of R and a table of selected values of Rt, for the time interval t 9 minutes, are shown. a) Use data from the table to find an approximation for R ' 45. Show the computations that lead to your answer. Indicate units of measure. b) The rate of fuel consumption is increasing fastest at time t 45 minutes. What is the value of R '' 45? Explain your reasoning. 9 c) Approximate the value of R t dt using a left Riemann sum with the five subintervals indicated by the data in the table. Is this numerical approximation less than the value of 9 R t dt? Explain your reasoning. b d) For b 9 minutes, explain the meaning of R t dt in terms of fuel consumption for the b 1 plane. Explain the meaning of b R t dt in terms of fuel consumption for the plane. Indicate units of measure in both answers. 6. A tank contains 15 gallons of heating oil at time t. During the time interval t 1 1 hours, heating oil is pumped into the tank at the rate H t gallons per hour. 1ln t 1 During the same interval, heating oil is removed from the tank at the rate per hour. R t t 1sin gallons 47 a) How many gallons of heating oil are pumped into the tank during the time interval t 1 hours? b) Is the level of heating oil in the tank rising or falling at time t 6 hours? Give reason for your answer. c) How many gallons of heating oil are in the tank at time t 1 hours? d) At what time t, for t 1, is the volume of heating oil in the tank the least? Show the analysis that leads to your conclusion. 3
7. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by t F t 8 4sin for 3 t, where F t is measured in cars per minute and t is measured in minutes. a) To the nearest whole number, how many cars pass through the intersection over the 3-minute period? b) Is the traffic flow increasing or decreasing at t 7? Give reason for your answer. c) What is the average value of the traffic flow over the time interval 1 t 15? Indicate units of measure. d) What is t average rate of change of the traffic flow over the time interval 1 t 15? Indicate units of measure. 8. For t 31, the rate of change of the number of mosquitoes on Tropical Island at time t days t is modeled by R t 5 tcos mosquitoes per day. There are 1 mosquitoes on Tropical 5 Island at time t. a) Show that the number of mosquitoes is increasing at time t 6. b) At time t 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer. c) According to the model, how many mosquitoes will be on the island at time t 31? Round your answer to the nearest whole number. d) To the nearest whole number, what is the maximum number of mosquitoes for t 31? Show the analysis that leads to your conclusion. 4
9. The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by 4 t R t 5sin 5. A pumping station adds sand to the beach at a rate modeled by the function S, given by 15t S t. 1 3t Both Rt and S t have units of cubic yards per hour and t is measured in hours for t 6. At time t, the beach contains 5 cubic yards of sand. a) How much sand will the tide remove from the bench during this 6-hour period? Indicate units of measure. b) Write an expression for Y t, the total number of cubic yards of sand on the beach at time t. c) Find the rate at which the total amount of sand on the beach is changing at time. d) For t 6, at what time t is the amount of sand on the beach at a minimum? What is the minimum value? Justify your answers. 1. A water tank at Camp Newton holds 1 gallons of water at time t. During the time t interval t 18 hours, water is pumped into the tank at the rate W t 95 tsin gallons per 6 t hour. During the same time interval, water is removed from the tank at the rate R t 75sin 3 gallons per hour. a) Is the amount of water in the tank increasing at time t 15? Why or why not? b) To the nearest whole number, how many gallons of water are in the tank at time t 18? c) At what time, t, for t 18, is the amount of water in the tank, but water continues to be removed at the rate until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k. 5
11. At an intersection in Thomasville, Oregon, cars turn t left at the rate L t 6 tsin cars per hour overt 3 the time interval t 18 hours. The graph of y Lt is shown above. a) To the nearest whole number, find the total number of cars turning left at the intersection over the time interval hours. b) Traffic engineers will consider turn restrictions when Lt 15 cars per hour. Find all values of t for which L t 15 and compute the average value of L over this time interval. Indicate units of measure. c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than,. In every two-hour time interval, 5 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion. 6
Topic 5: Accumulation & Rates Year & Question GCA Comments 1998 AB5BC5 1 Yes Draw graph, average value, accumulation 1999 AB3BC3 From table: Riemann sum, Rolle s theorem MVT, Yes average value AB4 ex No solution methods: accumulation, max/min. ABBC 3 Yes Amusement Park: Inout, amounts, max/min, values ABBC B 4 Yes Pollutants: max/min, amounts, accumulation 3 AB3 5 From graph and table: difference quotient, max/min, Yes Riemann sum, interpret integrals. 3 AB B 6 Yes Heating oil: Inout, inc/dec, accumulation, max/min 4 AB1BC1 7 Yes Traffic flow: average value, average rate of change, accumulation, inc/dec. 4 AB B 8 Yes Mosquitoes: values, inc/dec, accumulation, max/min 5 AB 9 Yes Sandy Beach: Inout, amounts, max/min, FTC 5 ABBC B 1 Yes Water tank: Inout, amount, max/min, inc/dec 6 ABBC 11 Yes Thomasville: Average value and application 7