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Every hour, a research assistant removes half the mass of a bacteria culture originally weighing 4,800 micrograms to observe under a microscope. The research assistant will stop this process once the mass is equal to 300 micrograms.

Which equation represents this situation, and after how many hours will the research assistant stop removing half the mass of the bacteria culture?

2 Answers

5 votes

Answer:


4,800\left((1)/(2)\right)^(t) \, = \, 300; \ 3 \ \text{hours}

Explanation:

Every hour, the research assistant is removing half the mass from the original 4,800 micrograms of the bacteria culture. So, the mass remaining after t hours can be represented by an exponential expression, a(b)t, where a is the initial mass of the bacteria culture and b is the base of the exponent that represents the decay factor. Since half of the mass of bacteria is being removed every hour, half of the mass remains. So, b =
(1)/(2). The exponential expression for this situation is
4,800\left((1)/(2)\right)^(t).

The research assistant will stop removing half the mass from the original bacteria culture after the original bacteria culture has a mass of 300 micrograms. Hence, set the expression for the mass remaining after t hours equal to 300.


4,800\left((1)/(2)\right)^(t) \, = \, 300

In order to solve this equation for t, the base needs to be isolated. Divide both sides of the inequality by 4,800.


\begin{array}{rclC40C40} 4,800\left((1)/(2)\right)^(t) &=& 300\\ \left((1)/(2)\right)^(t) &=& (1)/(16) \end{array}

Next, rewrite the base of
(1)/(16) as a power of
(1)/(2). Then, set the exponents equal to each other to solve for t.


\begin{array}{rclC40C40C30} \left((1)/(2)\right)^(t) &=& (1)/(16) \\ \left((1)/(2)\right)^(t) &=& \left((1)/(2)\right)^(4) \\ t &=& 4 \end{array}

So, after 4 hours the research assistant will stop removing half the mass of the bacteria culture.

User Max Schmidt
by
4.9k points
11 votes

Answer:

Explanation:

4,800 mcg × 0.5^t = 300 mcg

0.5^t = 1/16 = (1/2)⁴

t = 4

User Somi Meer
by
5.3k points