Question

1. (5 points) a bacterial culture (in millions) is modeled by the function b(t)=12 + 4ln(t), where t is measured in hours after the culture is started. find the instantaneous rate of change of the culture size at t = 4 hours. round your answer to the nearest whole number. a. 4 million cells per hour b. 1 million cells per hour c. 12 million cells per hour d. 16 million cells per hour e. none of the above

Explanation:

Step1: Find the derivative of \(B(t)\)

The derivative of a constant \(C\) is \(0\), and the derivative of \(\ln(t)\) is \(\frac{1}{t}\). Using the sum - rule and constant - multiple rule of differentiation, if \(B(t)=12 + 4\ln(t)\), then \(B^\prime(t)=\frac{d}{dt}(12)+4\frac{d}{dt}(\ln(t))\). Since \(\frac{d}{dt}(12) = 0\) and \(\frac{d}{dt}(\ln(t))=\frac{1}{t}\), we have \(B^\prime(t)=0 + 4\times\frac{1}{t}=\frac{4}{t}\).

Step2: Evaluate the derivative at \(t = 4\)

Substitute \(t = 4\) into \(B^\prime(t)\). So \(B^\prime(4)=\frac{4}{4}=1\).

Answer:

B. 1 million cells per hour