#### Exponential Growth and Decay

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###### Exercises
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Exercises 1 Exponential functions that increase by a constant factor or percentage over equal intervals of x are called exponential growth functions. They can be written in the format shown below. y=a(1+r)x​ Here, a represents the initial quantity and r tells us how fast the function is growing as a percentage, written in decimal form. This is why r is known as the rate of growth . With this in mind, we can complete the exercise's sentence.In the exponential growth function y=a(1+r)x, the quantity r is called the rate of growth.
Exercises 2 Exponential functions that decrease by a constant factor or percentage over equal intervals of x are called exponential decay functions. They can be written in the format shown below. y=a(1−r)x​ Here, a represents the initial quantity and r tells us how fast the function is decaying as a percentage, written in decimal form. This is why r is known as the rate of decay.
Exercises 3 We can compare exponential growth and exponential decay functions by using a table.Exponential GrowthExponential Decay Special cases of an exponential function. Occurs when a quantity increases by the same factor over equal intervals of time.Occurs when a quantity decreases by the same factor over equal intervals of time. It is modeled by the function y=a(1+r)xIt is modeled by the function y=a(1−r)x In this equation r represents the rate of growth, given in decimal form.In this equation r represents the rate of decay, given in decimal form.
Exercises 4 First, recall that exponential growth happens when a quantity increases by the same factor over equal intervals of time, and exponential decay occurs when it decreases by the same factor over equal intervals of time instead. Let's consider the general form of an exponential function. f(x)=abx​ Notice that if a>0, the function will be increasing over time by the same factor only if b>1. On the other hand, it will be decreasing over time by the same factor if 0<b<1. We can summarize these ideas as shown below.An exponential function of the form f(x)=abx with a>0 exhibits exponential growth if b>1, and exponential decay if 0<b<1.
Exercises 5 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. y=350(1+0.75)t​ Our function represents exponential growth, with an initial amount a=350 and a rate of growth r=0.75. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=0.75⇔r=75%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. y=350(1+0.75)tAdd termsy=350(1.75)tt=5y=350(1.75)5 Evaluate right-hand side Calculate powery=350(16.4130859375)Multiply y=5744.580078125y≈5744.6
Exercises 6 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's consider the given function. y=10(1+0.4)t​ Our function represents exponential growth, with an initial amount a=10 and a rate of growth r=0.4. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=0.4⇔r=40%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. y=10(1+0.4)tt=5y=10(1+0.4)5 Evaluate right-hand side Add termsy=10(1.4)5Calculate powery=10(5.37824)Multiply y=53.7824y≈53.8
Exercises 7 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=25(1.2)tWrite as a sumy=25(1+0.2)t Our function represents exponential growth, with an initial amount a=25 and a rate of growth r=0.2. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=0.2⇔r=20%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. y=25(1.2)tt=5y=25(1.2)5 Evaluate right-hand side Calculate powery=25(2.48832)Multiply y=62.208y≈62.2
Exercises 8 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=12(1.05)t⇔y=12(1+0.05)t​ Our function represents exponential growth, with an initial amount a=12 and a rate of growth r=0.05. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=0.05⇔r=5%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. y=12(1.05)tt=5y=12(1.05)5 Evaluate right-hand side Calculate powery=12(1.2762815625)Multiply y=15.31537875y≈15.3
Exercises 9 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. f(t)=1500(1.074)tWrite as a sumf(t)=1500(1+0.074)t Our function represents exponential growth, with an initial amount a=1500 and a rate of growth r=0.074. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=0.074⇔r=7.4%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. f(t)=1500(1.074)tt=5f(5)=1500(1.074)5 Evaluate right-hand side Calculate powerf(5)=1500(1.428964392)Multiply f(5)=2143.446588f(5)≈2143.4
Exercises 10 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. h(t)=175(1.028)tWrite as a sumh(t)=175(1+0.028)t Our function represents exponential growth, with an initial amount a=175 and a rate of growth r=0.028. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=0.028⇔r=2.8%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. h(t)=175(1.028)tt=5h(5)=175(1.028)5 Evaluate right-hand side Calculate powerh(5)=175(1.148062610490368)Multiply h(5)=200.9109568358144h(5)≈200.9
Exercises 11 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is more than 1. Let's rewrite the given function to match one of the above formulas. g(t)=6.72(2)tWrite as a sumg(t)=6.72(1+1)t Our function represents exponential growth, with an initial amount a=6.72 and a rate of growth r=1. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=1⇔r=100%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. g(t)=6.72(2)tt=5g(5)=6.72(2)5 Evaluate right-hand side Calculate powerg(5)=6.72(32)Multiply g(5)=215.04g(5)≈215.0
Exercises 12 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. p(t)=1.8tWrite as a sump(t)=1(1+0.8)t Our function represents exponential growth, with an initial amount a=1 and a rate of growth r=0.8. To rewrite the rate of growth as a percent, we move the decimal point 2 places to the right. r=0.8⇔r=80%​ Finally, to evaluate the function when t=5, we will substitute 5 for t in the given formula. p(t)=1.8tt=5p(5)=1.85Calculate powerp(5)=18.89568p(5)≈18.9
Exercises 13 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. If the function represents growth then r>0, and if it represents decay then r<0.Writing the Equation To write the function, we first need to define the variables. Let y be the sales, and let x be the number of years after the initial value. In this case, the initial value is \$10000. Since the price increases 65% each year, we have that r=0.65. y=10000[1+0.65]x⇕y=10000(1.65)x​
Exercises 14 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. To write the equation, we first need to define the variables. Let y be the annual salary, and let x be the number of years after the initial value. In this case, the initial value is a annual salary of \$35000. Since the salary increases 4% each year, we have that r=0.04. y=35000(1+0.04)x⇕y=35000(1.04)x​
Exercises 15 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. If the function represents growth then r>0, and if it represents decay then r<0.Writing the Equation To write the function, we first need to define the variables. Let y be the population, and let x be the number of years after the initial value. In this case, the initial value is a population of 210000. Since the price increases 12.5% each year, we have that r=0.125. y=210000[1+0.125]x⇕y=210000(1.125)x​
Exercises 16 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. If the function represents growth then r>0, and if it represents decay then r<0.Writing the Equation To write the function, we first need to define the variables. Let y be the price of the item, and let x be the number of years after the initial value. In this case, the initial value is a price of \$4.5. Since the price increases 3.5% each year, we have that r=0.035. y=4.5[1+0.035]x⇕y=4.5(1.035)x​
Exercises 17
Exercises 18
Exercises 19 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. y=575(1−0.6)t​ Our function represents exponential decay, with an initial amount a=575 and a rate of decay r=0.6. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.6⇔r=60%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. y=575(1−0.6)tSubtract termy=575(0.4)tt=3y=575(0.4)3 Evaluate right-hand side Calculate powery=575(0.064)Multiply y=36.8
Exercises 20 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's consider the given function. y=8(1−0.15)t​ Our function represents exponential decay, with an initial amount a=8 and a rate of decay r=0.15. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.15⇔r=15%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. y=8(1−0.15)tt=3y=8(1−0.15)3 Evaluate right-hand side Subtract termy=8(0.85)3Calculate powery=8(0.614125)Multiply y=4.913y≈4.9
Exercises 21 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. g(t)=240(0.75)tWrite as a sumg(t)=240(1−0.25)t Our function represents exponential decay, with an initial amount a=240 and a rate of decay r=0.25. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.25⇔r=25%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. g(t)=240(0.75)tt=3g(3)=240(0.75)3 Evaluate right-hand side Calculate powerg(3)=240(0.421875)Multiply g(3)=101.25g(3)=101.3
Exercises 22 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. f(t)=475(0.5)t ⇔ f(t)=475(1−0.5)t​ Our function represents exponential decay, with an initial amount a=475 and a rate of decay r=0.5. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.5⇔r=50%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. f(t)=475(0.5)tt=3f(3)=475(0.5)3 Evaluate right-hand side Calculate powerf(3)=475(0.125)Multiply f(3)=59.375f(3)≈59.4
Exercises 23 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. w(t)=700(0.995)tWrite as a sumw(t)=700(1−0.005)t Our function represents exponential decay, with an initial amount a=700 and a rate of decay r=0.005. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.005⇔r=0.5%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. w(t)=700(0.995)tt=3w(3)=700(0.995)3 Evaluate right-hand side Calculate powerw(3)=700(0.985074875)Multiply w(3)=689.5524125w(3)≈689.6
Exercises 24 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. h(t)=1250(0.865)tWrite as a sumh(t)=1250(1−0.135)t Our function represents exponential decay, with an initial amount a=1500 and a rate of decay r=0.135. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.135⇔r=13.5%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. h(t)=1250(0.865)tt=3h(3)=1250(0.865)3 Evaluate right-hand side Calculate powerh(3)=1250(0.647214625)Multiply h(3)=809.01828125h(3)≈809
Exercises 25 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=(87​)tCalculate quotienty=(0.875)tWrite as a differencey=1(1−0.125)t Our function represents exponential decay, with an initial amount a=1 and a rate of decay r=0.125. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.125⇔r=12.5%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. y=(87​)tt=2y=(87​)3 Evaluate right-hand side (ba​)m=bmam​y=512343​Calculate quotient y=0.669921875y≈0.7
Exercises 26 Let's start by recalling the general formulas for exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a>0 is the initial amount and r>0 is the rate of growth or decay written in decimal form. Moreover, for exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=0.5(43​)tCalculate quotienty=0.5(0.75)tWrite as a differencey=0.5(1−0.25)t Our function represents exponential decay, with an initial amount a=0.5 and a rate of decay r=0.25. To rewrite the rate of decay as a percent, we move the decimal point 2 places to the right. r=0.25⇔r=25%​ Finally, to evaluate the function when t=3, we will substitute 3 for t in the given formula. y=0.5(43​)tt=3y=0.5(43​)3 Evaluate right-hand side (ba​)m=bmam​y=0.5(6427​)Calculate quotienty=0.5(0.421875)Multiply y=0.2109375y≈0.2
Exercises 27 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. If the function represents growth then r>0, and if it represents decay then r<0.Writing the Equation To write the function, we first need to define the variables. Let y be the population, and let x be the number of years after the initial value. In this case, the initial value is a population of 100000. Since the population decreases 2% each year, we have that r=-0.02. y=100000[1+(-0.02)]x⇕y=100000(0.98)x​
Exercises 28 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. If the function represents growth then r>0, and if it represents decay then r<0.Writing the Equation To write the function, we first need to define the variables. Let y be the price of the sound system, and let x be the number of years after the initial value. In this case, the initial value is a price of \$900. Since the price decreases 9% each year, we have that r=-0.09. y=900[1+(-0.09)]x⇕y=900(0.91)x​
Exercises 29 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. If the function represents growth then r>0, and if it represents decay then r<0.Writing the Equation To write the function, we first need to define the variables. Let y be the price of the stock, and let x be the number of years after the initial value. In this case, the initial value is a price of \$100. Since the price decreases 9.5% each year, we have that r=-0.095. y=100[1+(-0.095)]x⇕y=100(0.905)x​
Exercises 30 To write an exponential function to model the given situation, let's first recall the general form of an exponential equation. y=abx​ In this formula, a is the initial value and b=1+r, where r is the rate of change. To write the equation, we first need to define the variables. Let y be the company profit, and let x be the number of years after the initial value. In this case, the initial profit of the company is \$20000. Since the profit decreases by 13.4% each year, we have that r=-0.134. y=20000[1+(-0.134)]x⇕y=20000(0.866)x​
Exercises 31 We will start by writing the function. The bacteria population b(t) after t hours can be modeled by a exponential growth function. b(t)=a(1+r)t​ Here, a is the initial value, and r is the rate of growth. In the question, the rate of growth is given as 150%, or 1.5. The initial bacteria population is 10. Then, the function becomes as follows. b(t)=10(1+1.5)t⇒b(t)=10(2.5)t​ We see that the growth factor is equal to 2.5, not just 1.5 as written in the given solution. Correct:Incorrect:​  b(t)=10(2.5)t✓  b(t)=10(1.5)t×​ To find the number of bacteria after 8 hours, we will substitute 8 for t. b(t)=10(2.5)tt=8b(t)=10(2.5)8 Simplify right-hand side Calculate powerb(8)=10(1525.87890…)Multiplyb(8)=15258.78906…Round to nearest integer b(8)=15259 After 8 hours, there are about 15259 bacteria.
Exercises 32 We are given that the value of the car decreases by 14% annually so we will use exponential decay function. The function below models the value v(t) of the car after t years. v(t)=a(1−r)t​ Here, a is the initial value, and r is the rate of decay. In the question, the rate of decay is given as 14%, or 0.14. The initial value of the car is 25000. Then, the function becomes as follows. v(t)=25000(1−0.14)t⇒v(t)=25000(0.86)t​ We see that the decay factor is equal to 0.86, not 1.14 as written in the given solution. Correct:Incorrect:​  v(t)=25000(0.86)t✓  v(t)=25000(1.14)t×​ To find the value of the car in 2015, we will substitute 5 for t. v(t)=25000(0.86)tt=5b(5)=25000(0.86)5 Simplify right-hand side Calculate powerv(5)=25000(0.47042…)Multiplyv(5)=11760.67544Round to nearest integer v(5)=11761 In 2015, the value of the car is about \$11761.
Exercises 33 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 1, y is multiplied by 0.2, which is less than 1. This tells us that the table represents an exponential decay function.
Exercises 34 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 1, y is multiplied by different numbers each time. This tells us that the table represents neither an exponential growth function nor an exponential decay function.
Exercises 35 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 1, y is multiplied by different numbers each time. This tells us that the table represents neither an exponential growth function nor an exponential decay function.
Exercises 36 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 1, y is multiplied by 3, which is greater than 1. This tells us that the table represents an exponential growth function.
Exercises 37 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 5, y is multiplied by 4, which is greater than 1. This tells us that the table represents an exponential growth function.
Exercises 38 Let's take a look at how the x-variable increases as well as analyze the behavior of the y-variable.As x increases by 2, y is multiplied by 0.16667, which is less than 1. This tells us that the table represents an exponential decay function.
Exercises 39
Exercises 40
Exercises 41 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=4(0.8)tWrite as a differencey=4(1−0.2)t Our function matches the format of exponential decay. Here a=4 and r=0.2. y=4(1−0.2)t​ This means that the rate of decay, also known as the percent of change, is 20%.
Exercises 42 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=15(1.1)t⇔y=15(1+0.1)t​ Our function matches the format of exponential growth. Here a=15 and r=0.1. Let's move the decimal point two places to the right to write the rate of growth, also known as the percent of change, in percentage. r=0.1⇔r=10%​
Exercises 43 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=30(0.95)tWrite as a differencey=30(1−0.05)t Our function matches the format of exponential decay. Here a=30 and r=0.05. y=30(1−0.05)t​ This means that the rate of decay, also known as the percent of change, is 5%.
Exercises 44 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. y=5(1.08)tWrite as a sumy=5(1+0.08)t Our function matches the format of exponential growth. Here a=5 and r=0.08. y=5(1+0.08)t​ This means that the rate of growth, also known as the percent of change, is 8%.
Exercises 45 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. r(t)=0.4(1.06)tWrite as a differencer(t)=0.4(1+0.06)t Our function matches the format of exponential growth. Here a=0.4 and r=0.06. y=0.4(1+0.06)t​ This means that the rate of growth, also known as the percent of change, is 6%.
Exercises 46 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​s(t)=a(1+r)t​Exponential Decay​s(t)=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. s(t)=0.65(0.48)t⇕s(t)=0.65(1−0.52)t​ Our function matches the format of exponential decay. Here a=0.65 and r=0.52. We can express the rate of decay, also known as the percent of change, as a percent by moving the decimal point two places to the right. r=0.52⇔r=52%​
Exercises 47 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. g(t)=2(45​)tCalculate quotientg(t)=2(1.25)tWrite as a sumg(t)=2(1+0.25)t Our function matches the format of exponential growth. Here a=2 and r=0.25. y=2(1+0.25)t​ This means that the rate of growth, also known as the percent of change, is 25%.
Exercises 48 Let's start by recalling the general formula of exponential growth and exponential decay functions. Exponential Growth​y=a(1+r)t​Exponential Decay​y=a(1−r)t​ In both cases, a and r are positive numbers. For exponential decay, r is less than 1. Let's rewrite the given function to match one of the above formulas. m(t)=(54​)tCalculate quotientm(t)=(0.8)tWrite as a differencem(t)=(1−0.2)t Our function matches the format of exponential decay. Here a=1 and r=0.2. y=1(1−0.2)t​ This means that the rate of decay, also known as the percent of change, is 20%.
Exercises 49 We want to rewrite the given exponential function in the form y=a(1+r)t or y=a(1−r)t. To do so, we will use the Quotient of Powers Property. y=(0.9)t−4am−n=anam​y=0.940.9t​Calculate powery=0.65610.9t​ba​=b1​⋅ay=0.65611​⋅0.9tCalculate quotienty=1.52⋅0.9tWrite as a differencey=1.52(1−0.1)t Note that the obtained formula involves subtraction. Therefore, it represents exponential decay. Finally, we will multiply r=0.1 by 100 to calculate the percent rate of change. 0.1×100=10% decay​
Exercises 50 We want to rewrite the given exponential function in the form y=a(1+r)t or y=a(1−r)t. To do so, we will use the Product of Powers Property. y=(1.4)t+8am+n=am⋅any=(1.4)8⋅(1.4)tUse a calculatory≈14.76(1.4)tWrite as a sumy≈14.76(1+0.4)t Note that the obtained formula involves addition. Therefore, it represents exponential growth.
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