A scientist estimates that the population of a certain bacteria culture doubles every 4 ho… — ACT

A scientist estimates that the population of a certain bacteria culture doubles every 4 hours. If the culture starts with $3.0 \times 10^5$ bacteria, the population after 24 hours can be written in scientific notation as $a \times 10^7$ . What is the value of $a$ ?

Answer: 1.92

Answer

The value of the coefficient

a

1.92

The correct answer is

1.92

. In 24 hours, a population that doubles every 4 hours undergoes 6 doubling cycles. The growth factor is

2^6 = 64

. Multiplying the initial population of

3.0 \times 10^5

by 64 yields

192 \times 10^5

. Converting this value into standard scientific notation gives

1.92 \times 10^7

. Comparing this expression to the format

a \times 10^7

shows that the coefficient

a

equals

1.92

Step-by-Step Solution

Determine the number of doubling periods.

6 doubling periods

Since the population doubles every 4 hours, over a span of 24 hours it will double

24 \div 4 = 6

times.

Calculate the growth multiplier.

Doubling 6 times is represented by the exponential expression

2^6

, which evaluates to 64.

Multiply the initial population by the growth factor.

192 \times 10^5

Multiply the coefficient of the initial population by the growth multiplier:

3.0 \times 64 = 192

, yielding a total population of

192 \times 10^5

Convert the resulting value to the required scientific notation format.

1.92 \times 10^7

To express

192 \times 10^5

in the standard scientific notation format

a \times 10^7

, divide 192 by 100 to get 1.92, and multiply the power of 10 by

10^2

(raising the exponent from 5 to 7).

Key Concept

Evaluating exponential growth and converting numbers to standard scientific notation format

Estimated Time:1m 30s

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