Two different cultures of bacteria, culture A and culture B, begin growing at the same tim… — SAT

Two different cultures of bacteria, culture $A$ and culture $B$ , begin growing at the same time. The population of culture $A$ increases by $300\%$ every $8$ hours, and the population of culture $B$ increases by $700\%$ every $9$ hours. If both cultures start with the same initial population, after how many hours will the population of culture $B$ be $4$ times the population of culture $A$ ?

A
9.6
24Answer
C
2
D
48

Answer

The population of culture B will be 4 times the population of culture A after 24 hours.

The correct answer is the option stating 24 hours. The growth model for culture A is

P_A(t) = P_0 \cdot 4^{t/8}

and for culture B is

P_B(t) = P_0 \cdot 8^{t/9}

. Setting

P_B(t) = 4 P_A(t)

and converting all terms to base 2 yields

2^{t/3} = 2^{2 + t/4}

. Equating exponents gives

t/3 = 2 + t/4

, which simplifies to

t/12 = 2

and solves to

t = 24

hours.

Step-by-Step Solution

Determine the growth factors and set up the population equations for both cultures.

For culture

A

, a

300\%

increase means the population becomes

1 + 3.00 = 4

times its previous value every

8

hours, so

P_A(t) = P_0 \cdot 4^{t/8}

. For culture

B

, a

700\%

increase means the population becomes

1 + 7.00 = 8

times its previous value every

9

hours, so

P_B(t) = P_0 \cdot 8^{t/9}

Establishing the correct exponential growth models with the appropriate growth factors and time constants is necessary to relate the populations.

Set up the equation to find when the population of culture

B

4

times that of culture

A

P_B(t) = 4 P_A(t) \implies P_0 \cdot 8^{t/9} = 4 \cdot P_0 \cdot 4^{t/8} \implies 8^{t/9} = 4 \cdot 4^{t/8}

This sets up the equation that must be solved for

t

by dividing both sides by the non-zero initial population

P_0

Express all terms with a common base of

2

(2^3)^{t/9} = 2^2 \cdot (2^2)^{t/8} \implies 2^{t/3} = 2^2 \cdot 2^{t/4}

Expressing bases as powers of

2

allows the use of exponent rules to simplify the equation.

Apply the product rule of exponents to combine the terms on the right side.

2^{t/3} = 2^{2 + t/4}

The rule

2^a \cdot 2^b = 2^{a+b}

simplifies the multiplication of exponential terms with the same base.

Equate the exponents and solve the resulting linear equation for

t

t/3 = 2 + t/4 \implies t/3 - t/4 = 2 \implies t/12 = 2 \implies t = 24

Since the bases are equal, their exponents must be equal. Solving the linear equation yields the time in hours.

Key Concept

Solving exponential equations by converting to a common base and applying exponent laws.

Estimated Time:3m 0s

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