The population of a colony of bacteria is modeled by the function P(t) = 250 \cdot 8^{\fra… — SAT

The population of a colony of bacteria is modeled by the function $P(t) = 250 \cdot 8^{\frac{t}{6} + 1}$ , where $P(t)$ is the estimated number of bacteria $t$ hours after the start of an experiment. The function can be rewritten in the equivalent form $P(t) = A \cdot 2^{\frac{m}{k}}$ , where $m$ is the number of 20-minute intervals that have elapsed since the start of the experiment, and $A$ and $k$ are constants. What is the value of $A + k$ ?

A
506
B
2,002
2,006Answer
D
2,018

Answer

2,006

The correct answer is 2,006. By writing the base 8 as 2^3 and applying the exponent rules, the function becomes 250 * 2^(t/2 + 3) = 2000 * 2^(t/2). Since m represents 20-minute intervals and there are 3 intervals in an hour, we have t = m/3. Substituting this into the function yields 2000 * 2^(m/6), which gives A = 2000 and k = 6. The sum is 2000 + 6 = 2006.

Step-by-Step Solution

Express the base 8 in terms of base 2 to match the target form.

P(t) = 250 \cdot (2^3)^{\frac{t}{6} + 1}

Converting to base 2 allows simplification of the exponent to match the required base of 2 in the target equation.

Apply the exponent power rule

(b^y)^z = b^{yz}

to simplify the expression.

P(t) = 250 \cdot 2^{3\left(\frac{t}{6} + 1\right)} = 250 \cdot 2^{\frac{t}{2} + 3}

Multiplying the exponents simplifies the expression into a single power of base 2.

Isolate the constant coefficient by separating the terms in the exponent using the rule

b^{x+y} = b^x \cdot b^y

P(t) = 250 \cdot 2^3 \cdot 2^{\frac{t}{2}} = 250 \cdot 8 \cdot 2^{\frac{t}{2}} = 2000 \cdot 2^{\frac{t}{2}}

This separates the initial coefficient multiplier from the variable growth term.

Convert the time variable from hours (

t

) to 20-minute intervals (

m

Since there are 3 twenty-minute intervals in one hour,

m = 3t

, which means

t = \frac{m}{3}

. Substituting this into the equation yields

P(t) = 2000 \cdot 2^{\frac{m/3}{2}} = 2000 \cdot 2^{\frac{m}{6}}

This expresses the population model in terms of the new time unit variable,

m

, as required.

Identify the constants

A

and

k

and calculate their sum.

A = 2000

k = 6

, so

A + k = 2000 + 6 = 2006

Comparing the final model to

A \cdot 2^{\frac{m}{k}}

directly gives the values of the constants.

Key Concept

Rewriting and interpreting exponential functions by manipulating bases and changing variables.

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