A researcher models the population of a certain species of bacteria in a culture. The popu… — SAT

A researcher models the population of a certain species of bacteria in a culture. The population $N(t)$ of bacteria $t$ hours after the culture is established is modeled by the function

N(t) = N_0 \cdot b^{\frac{t}{3}}

where $N_0$ is the initial population of the bacteria and $b$ is a constant. If the population of the bacteria increases by $44\%$ every 2 hours, what is the value of $b$ ?

A
1.200
B
1.275
C
1.660
1.728Answer

Answer

1.728

The correct answer is

1.728

. Since the population increases by

44\%

every 2 hours, the population at

t = 2

1.44

times the initial population, meaning

N(2) = 1.44 \cdot N_0

. Substituting

t = 2

into the model

N(t) = N_0 \cdot b^{\frac{t}{3}}

yields

N_0 \cdot b^{\frac{2}{3}} = 1.44 \cdot N_0

. Dividing both sides by

N_0

gives

b^{\frac{2}{3}} = 1.44

. Raising both sides to the power of

\frac{3}{2}

isolates

b

b = (1.44)^{\frac{3}{2}}

. Since

1.44 = (1.2)^2

, we can simplify this expression using exponent rules:

b = (1.2^2)^{\frac{3}{2}} = 1.2^3 = 1.728

Step-by-Step Solution

Write the equation relating the population at time

t = 2

hours to the initial population at

t = 0

hours using the given percentage increase.

N(2) = 1.44 \cdot N_0

An increase of

44\%

means the population becomes

100\% + 44\% = 144\%

of its initial value, which corresponds to multiplying by a factor of

1.44

Substitute the function definition

N(t) = N_0 \cdot b^{\frac{t}{3}}

into the equation for

t = 2

N_0 \cdot b^{\frac{2}{3}} = 1.44 \cdot N_0

This allows us to set up an equation to solve for the constant

b

Divide both sides of the equation by

N_0

and isolate

b

b^{\frac{2}{3}} = 1.44

Since the initial population

N_0

is positive, we can divide both sides by

N_0

to isolate the exponential base term.

Solve for

b

by raising both sides of the equation to the power of

\frac{3}{2}

b = (1.44)^{\frac{3}{2}}

To solve for

b

, we multiply the exponent

\frac{2}{3}

by its reciprocal

\frac{3}{2}

Evaluate the expression

(1.44)^{\frac{3}{2}}

using exponent properties.

b = (1.2^2)^{\frac{3}{2}} = 1.2^3 = 1.728

Expressing

1.44

1.2^2

allows us to simplify the fractional exponent using the power of a power rule,

(x^a)^b = x^{ab}

Key Concept

Solving exponential equations by applying exponent rules and interpreting exponential growth factors in context.

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