55. Let A(t) be the area of a tissue culture at time I and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to √A(t). So a reasonable model for the growth of tissue is obtained by assuming that the rate of growth of the area is jointly proportional to √A(t) and M - A(t). (a) Formulate a differential equation and use it to show that the tissue grows fastest when A(t) = M. (b) Solve the differential equation to find an expression for A(t). Use a computer to perform the integration.
55. Let A(t) be the area of a tissue culture at time I and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to √A(t). So a reasonable model for the growth of tissue is obtained by assuming that the rate of growth of the area is jointly proportional to √A(t) and M - A(t). (a) Formulate a differential equation and use it to show that the tissue grows fastest when A(t) = M. (b) Solve the differential equation to find an expression for A(t). Use a computer to perform the integration.
55. Let A(t) be the area of a tissue culture at time I and let M be the final area of the tissue when growth is complete. Most cell divisions occur on the periphery of the tissue and the number of cells on the periphery is proportional to √A(t). So a reasonable model for the growth of tissue is obtained by assuming that the rate of growth of the area is jointly proportional to √A(t) and M - A(t). (a) Formulate a differential equation and use it to show that the tissue grows fastest when A(t) = M. (b) Solve the differential equation to find an expression for A(t). Use a computer to perform the integration.
Hello, I am stuck on part b of this separable differential equation in Calculus II. I am not sure if how I solved the differntial equation was correct. The first picture shows the problem, the second picture shows my work.
Transcribed Image Text:55. Let A(t) be the area of a tissue culture at time t and let M be
the final area of the tissue when growth is complete. Most
cell divisions occur on the periphery of the tissue and the
number of cells on the periphery is proportional to √A(t).
So a reasonable model for the growth of tissue is obtained
by assuming that the rate of growth of the area is jointly
proportional to √√A(t) and M - A(t).
(a) Formulate a differential equation and use it to show that
the tissue grows fastest when A(t) = M.
(b) Solve the differential equation to find an expression for
A(t). Use a computer to perform the integration.
Transcribed Image Text:at
12 =K (MA²-A^)
d
FILMA-A
OFM-35A
214 2
м
AA
2
WA
3A=M
3/2
A = M MARPT
3
b) dA-KJA (M-A)
at
dA
JJA(M-A)
fredt
In (√A+√M) - In (1√A -√ml)
e
In JA+M
(
√A-
√A+√m
1
JA-SM
te
JAKE
JA TJM = (TARE (JA-JM)
Jankt
√A = (√ME (√A -√M) -√m
JA = CENA - Comico Sm-Für
>JA-CA- - CoJake Jim-Fi
JA (1-Cemiter) = - Ce²=kojm-m
2
(JA) ((1+ (√) JM-
Az M (1 + Cermice) 2
(1-1)2
A
2
A(c) = M (1+/+ m
A
2
JMKE
(1- (JA+UM) VALLE)2
प्
LE
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
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