Question

Suppose y left parenthesis t right parenthesis equals 35 e to the power of 3 t end exponent plus 57 represents the number of bacteria present at time t minutes. at what time will the population reach 200 bacteria? (note: answers are expressed in terms of the natural logarithm)

Answer 1

Answer: $t=\dfrac{\ln(4.09)}{3}$

Step-by-step explanation:

Given function: $y(t)=35e^{3t}+57$ which represents the number of bacteria present at time t minutes.

Put y(t) = 200 in the above equation , we get

$200=35e^{3t}+57\\\\\Rightarrow\ 35e^{3t}=143\\\\\Rightarrow\ e^{3t}=4.085714285\approx4.09$

Taking log on both sides, we get

$\ln(e^{3t})=\ln(4.09)\\\\\Rightarrow\ 3t=\ln(4.09)\\\\\Rightarrow\ t=\dfrac{\ln(4.09)}{3}$

Hence, the the population reach 200 bacteria at $t=\dfrac{\ln(4.09)}{3}$ minutes.

Answer 2

For this case we have the following equation:
 y (t) = 35 * exp (3t) + 57
 For 200 bacteria we have:
 200 = 35 * exp (3t) + 57
 Clearing t we have:
 exp (3t) = (200-57) / 35
 exp (3t) = 4.085714286
 ln (exp (3t)) = ln (4.085714286)
 3t = ln (4.085714286)
 t = ln (4.085714286) / 3
 t = 0.5 minutes
 Answer:
 the population will reach 200 bacteria at:
 t = 0.5 minutes
 or
 t = ln (4.09) / 3