Step: 1

Exponential growth can be modeled by the equation y = C (1 + r )^{t}, where C is the initial amount, r is the growth rate and t is the time.

Correct Answer is : y = C(1 + r )^{t}

Step: 1

Let y be the profit.

Step: 2

The initial profit in the business is $12143.

Step: 3

The rate of increase of profit, r is 2% or 0.02.

Step: 4

The number of years, t is 7.

Step: 5

[Write exponential growth model.]

Step: 6

[Substitute C = 12143, r = 0.02 and t = 7.]

Step: 7

[Add.]

Step: 8

So, the equation for the profit is y = 12143(1.02)^{7}.

Correct Answer is : y = 12143(1.02)^{7}

Step: 1

Exponential decay can be modeled by the equation y = C(1 - r )^{t}, where C is the initial value, r is the decay rate where 0 < 1 - r < 1, and t , the time.

Step: 2

In the model y = 3(1.24)^{t}, 1 - r = 1.24 and 1.24 > 1

Step: 3

The model y = 3(1.24)^{t} is not an exponential decay model.

Step: 4

[Compare with exponential decay model.]

Step: 5

In the model y = 8(0.67)^{t}, 1 - r = 0.67 and 0 < 0.67 < 1

Step: 6

So, the model y = 8(0.67)^{t} is an exponential decay model.

Correct Answer is : y = 8(0.67)^{t}

Step: 1

Let y be the value of the bike.

Step: 2

Let t be the number of years of ownership.

Step: 3

The initial value of the bike C is $4300.

Step: 4

The decay rate r is 13% or 0.13.

Step: 5

[Write exponential decay model.]

Step: 6

= 4300(1 - 0.13)^{t}

[Replace C with 4300 and r with 0.13.]

Step: 7

= 4300(0.87)^{t}

[Subtract 0.13 from 1.]

Step: 8

The exponential decay model is y = 4300(0.87)^{t}.

Correct Answer is : y = 4300(0.87)^{t}

Step: 1

Let y be Arthur's decreasing annual profit in the business.

Step: 2

The number of years that the profit decreased is t .

Step: 3

His initial profit C is $47000.

Step: 4

The decay rate r is 2% or 0.02.

Step: 5

The exponential decay can be modeled by the equation y = C(1 - r )^{t}.

Step: 6

[Replace C and r with the values $47000 and 0.02.]

Step: 7

[Simplify.]

Step: 8

The exponential decay model of Arthur's decreasing annual profit in the business is y = 47000(0.98)^{t}.

Correct Answer is : y = 47000(0.98)^{t}

Step: 1

The initial value of the machinery, C is $29000.

Step: 2

The decay rate, r is 3% = 0.03.

Step: 3

The exponential decay can be modeled by the equation y = C(1 - r )^{t}.

Step: 4

[Replace C with 29000, and r with 0.03.]

Step: 5

[Subtract 0.03 from 1.]

Step: 6

The exponential decay model that represents the depreciation of the machinery is y = 29000(0.97)^{t}.

Correct Answer is : y = 29000(0.97)^{t}

Step: 1

The initial value of the refrigerator C is $4000.

Step: 2

The decay rate r is 5% or 0.05.

Step: 3

The exponential decay can be modeled by the equation y = C(1 - r )^{t}.

Step: 4

[Replace C with 4000, and r with 0.05.]

Step: 5

[Subtract.]

Step: 6

The exponential decay model that represents the situation is y = 4000(0.95)^{t}.

Correct Answer is : y = 4000(0.95)^{t}

Step: 1

The standard form of an exponential function is y = ab ^{x} where a is a constant ≠ 0, b is the base, b > 0 and b ≠ 1, and x is a real number.

[Definition.]

Correct Answer is : y = ab ^{x}, a is a constant ≠ 0, b > 0, b ≠ 1,and x is a real number.

Step: 1

The exponential function y = a b ^{x}, where a is a constant > 0, b > 1, and x is a real number models an exponential growth.

[Definition.]

Correct Answer is : y = ab ^{x}, a is a constant > 0, b > 1, and x is a real number

Step: 1

The exponential function y = ab ^{x}, where a is a constant > 0, 0 < b < 1, and x is a real number models an exponential decay.

[Definition.]

Correct Answer is : y = ab ^{x}, a is a constant > 0, 0 < b < 1, and x is a real number

Step: 1

Let y be the value of the bike.

Step: 2

Let t be the number of years of ownership.

Step: 3

The initial value of the bike C is $5300.

Step: 4

The decay rate r is 4% or 0.04

Step: 5

Step: 6

= 5300(1 - 0.04)^{t}

Step: 7

= 5300(0.96)^{t}

Step: 8

The exponential decay model is 5300(0.96)^{t}.

Correct Answer is : 5300(0.96)^{t}

Step: 1

[Standard exponential growth function.]

Step: 2

Here the base, b = 8

Step: 3

Step: 4

105 = a (8)^{0}, a = 105

[Replace y with 105, b with 8 and x with zero.]

Step: 5

So, the initial value of y is a = 105.

Step: 6

So, y = 105(8^{x}) is the required expoential growth function.

Correct Answer is : y = 105(8^{x}), x is a real number

Step: 1

Let y be the length of the hair during the first six weeks and t be the number of days.

Step: 2

[Write exponential growth model.]

Step: 3

= 0.23(1 + 0.1)^{t}

[Replace C = 0.23 and r = 10% = 0.1.]

Step: 4

= 0.23(1.1)^{t}

[Add.]

Step: 5

The model for the length of the hair in first six weeks is y = 0.23(1.1)^{t}.

Correct Answer is : y = 0.23(1.1)^{t}

Step: 1

The initial value of the machinery, "C" is $29000.

Step: 2

The decay rate, "r " is 4% = 0.04

Step: 3

The exponential decay can be modeled by the equation y = C(1 - r )^{t}.

Step: 4

[Substitute 29000 for C, and 0.04 for r .]

Step: 5

[Subtract 0.04 from 1.]

Step: 6

The exponential decay model that represents the depreciation of the machinery is y = 29000(0.96)^{t}

Correct Answer is : y = 29000(0.96)^{t}

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