a scientist writes the equation ( n(h)=100e^{0.25h} ) to model the growth of a certain bacteria in a petri…

a scientist writes the equation ( n(h)=100e^{0.25h} ) to model the growth of a certain bacteria in a petri dish, where ( n ) represents the number of bacteria after ( h ) hours. after approximately how many hours will 450 bacteria be present? round your answer to the nearest whole number. 1 hour 6 hours 13 hours 15 hours

a scientist writes the equation ( n(h)=100e^{0.25h} ) to model the growth of a certain bacteria in a petri dish, where ( n ) represents the number of bacteria after ( h ) hours. after approximately how many hours will 450 bacteria be present? round your answer to the nearest whole number. 1 hour 6 hours 13 hours 15 hours

Answer

Explanation:

Step1: Substitute ( N(h) = 450 ) into the equation

Given ( N(h)=100e^{0.25h} ), substitute ( N(h) = 450 ). So, ( 450=100e^{0.25h} ). Divide both sides by 100: ( \frac{450}{100}=e^{0.25h} ), which simplifies to ( 4.5 = e^{0.25h} ).

Step2: Take the natural logarithm of both sides

Using the property ( \ln(e^{x})=x ), if ( 4.5 = e^{0.25h} ), then ( \ln(4.5)=\ln(e^{0.25h}) ). Since ( \ln(e^{0.25h}) = 0.25h ), we have ( \ln(4.5)=0.25h ). We know that ( \ln(4.5)\approx1.504 ). So, ( 1.504 = 0.25h ).

Step3: Solve for ( h )

Divide both sides of the equation ( 1.504 = 0.25h ) by ( 0.25 ). ( h=\frac{1.504}{0.25}=6.016\approx6 )

Answer:

6 hours