A bacteria culture grows with constant relative growth rate. The bacteria count

Question

A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and 25,600 after 6 hours.

What was the initial size of the culture?

Explanation / Answer

a) means that we are going to k as the following: A(t) = A0 * e^( k * t ) ----> A(2) = 400 A(2) = A0 * e^( k * 2 ) 400 = A0 * e^( 2k ) 400 / e^( 2k ) A(t) = A0 * e^( k * t ) ----> A(6) = 25,600 A(6) = A0 * e^( k * 6 ) 25,600 = A0 * e^( 6k ) 25,600 / e^( 6k ) = A0 A0 = A0 25,600 / e^( 6k ) = 400 / e^( 2k ) 25,600 / 400 = e^( 6k ) / e^( 2k ) 64 = e^( 4k ) ln( 64 ) = 4k ln( 64 ) / 4 = k A(2) = 400 & k = ln( 64 ) / 4 A(2) = A0 * e^( (ln( 64 ) / 4) * 2 ) 400 = A0 * e^( (ln( 64 ) / 2) ) 400 = A0 * e^( ( (1/2) * ln( 64 )) ) 400 = A0 * e^( ln( 64^(1/2) )) ) 400 = A0 * e^( ln( 8) ) 400 = A0 * 8 400 / 8 = A0 50 = A0

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