Separable Differential Equations
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- Edmund Fisher
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3 1 Solve the following differential equations: 1. There are 120 grams of a radioactive substance whose half-life is 74 ears. Determine how much of the substance will remain after 50 ears. 2. A bacteria culture containing 2,400 cells 3 hours ago has now grown to 5,200 cells. Assuming the rate of growth is proportional to the number present, determine the time (from now) at which the population will reach 10, A colon of bacteria with originall 100 cells has now grown to 400 in 2 hours. Assuming the rate of growth is proportional to the present number, find the number of bacteria in 5 hours from now.
4 4. There were 100 grams of a radioactive substance ten ears ago, now there are onl 32 grams. Find the substance s half-life A bank account contains $1,200 and pas an annual interest rate of 5.25% compounded continuousl. Determine the time at which the mone doubles. 6. An object is 3400 ears old, find the percentage of its original Carbon-14 content it should have now (Carbon-14 has a half-life of 5730 ears).
5 7. Intensit of light beam passing through an absorbing medium decreases at a rate proportional to the intensit at an given depth. Suppose at the surface of the water, the intensit of a light bulb is 20 candelas and 14 candelas under a ard of water. Find the light intensit under 20 feet of water The population of a countr is growing at a rate proportional to its population. If the growth rate per ear is 5% of the current population, in how man ears will the population double? 9. Suppose the prices for real estate grow eponentiall. If a house was worth $100,000 five ears ago, and is now worth $250,000, find the ear (from now) in which the price will eceed $500,000.
6 4 11. When a capacitor is being discharged, the equation describing the charge on one plate of the capacitor is where q is the charge (C), R is the resistance of the circuit (Ω), C is the capacitance of the capacitor (F), and t is time in seconds. If a 5 mf capacitor with an initial charge of 3μC is discharged through a 100 resistor, find the time when 90% of the charge has been drained. 12. A RL circuit has a resistance of 2 Ω an inductance of 5 H, and an initial current of 8 A. Find (a) the current in the circuit at an time t and (b) its current after 10 seconds. The equation describing the current in the circuit is where L is the inductance (H), I is the current (A), R is the resistance (Ω), and t is time in seconds.
7 Calculus Maimus WS 7.3: Separable Diff EQ Name Date Period Worksheet 7.3 Separable Differential Equations Show all work. No Calculator unless specified. Multiple Choice 1. (OK, so ou can use our calculator right awa on a non-calculator worksheet. Use it on this one.)a sample of Kk-1234 (an isotope of Kulmakorpium) loses 99% of its radioactive matter in 199 hours. What is the half-life of Kk-1234? (A) 4 hours (B) 6 hours (C) 30 hours (D) hours (E) 143 hours 2. In which of the following models is directl proportional to? dt kt I. = e + C II. III. = Ce = kt 28 kt 3t+ 1 1 IV. = 3 2 (A) I onl (B) II onl (C) I and II onl (D) II and III onl (E) II, III, and IV (F) all of them Page 1 of 6
8 Calculus Maimus WS 7.3: Separable Diff EQ 3. (Use our calculator on this one, too, but get the eact answer first.) The rate at which acreage is being consumed b a plot of kudzu is proportional to the number of acres alrea consumed at time t. If there are 2 acres consumed when t = 1 and 3 acres consumed when t = 5, how man acres will be consumed when t = 8? (A) (B) (C) (D) (E) Free Response For problems 4 13, find the general solution to the following differential equations, then find the particular solution using the initial condition. 4. d =, ( ) 1 = 2 5. d =, ( ) 4 = 3 6. d 2 = 2 =, ( ) 7. 2 d =, ( 0) = 3 8. ( 5)( 2) d = + +, ( ) 0 = 1 9. cos 2 d =, ( ) 0 = 0 Page 2 of 6
9 Calculus Maimus WS 7.3: Separable Diff EQ d + sin 10. = ( cos ) e, ( ) 0 = d = e, ( ) 0 = d =, ( ) 1 = d 4 ln =, ( e ) = 1 For problems 14 17, find the solution of the differential equation dt conditions. = k that satisfies the given k =, ( ) 0 = k =, ( ) 0 = ( 0) = 50, ( 5) = ( 1) = 55, ( ) 10 = 30 (divide one b the other) Page 3 of 6
10 Calculus Maimus WS 7.3: Separable Diff EQ 18. AP 2010B-5 (No Calculator) + 1 Consider the differential equation =. d (a) On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated, and for 1 1 0, 1. < <, sketch the solution curve that passes through the point ( ) (b) While the slope field in part (a) is drawn at onl twelve points, it is defined at ever point in the -plane for which 0. Describe all points in the -plane, 0, for which 1 d =. (c) Find the particular solution f ( ) f ( 0) = 2. = to the given differential equation with the initial condition Page 4 of 6
11 Calculus Maimus WS 7.3: Separable Diff EQ 19. AP Consider the differential equation d 1+ =, where 0. (a) On the aes provided, sketch a slope field for the given differential equation at the eight points indicated. (b) Find the particular solution f ( ) f ( 1) = 1and state its domain. = to the differential equation with the initial condition Page 5 of 6
12 Calculus Maimus WS 7.3: Separable Diff EQ 20. AP Consider the differential equation =. d (a) On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (b) Let f ( ) f ( 1) = 1. Write an equation for the line tangent to the graph of f at ( 1, 1) approimate f ( 1.1). = be the particular solution to the differential equation with the initial condition and use it to (c) Find the particular solution f ( ) f ( 1) = 1. = to the given differential equation with the initial condition Page 6 of 6
Name Class. 5. Find the particular solution to given the general solution y C cos x and the. x 2 y
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Objective 2 Inverse Functions Illustrate the idea of inverse functions. f() = 2 + f() = Two one-to-one functions are inverses of each other if (f g)() = of g, and (g f)() = for all in the domain of f.
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