Exponential and Logarithms

[Pompeii]

Hey i have this question from my maths assignment that's been bugging me for weeks since i first got the assignment, and i know what i have to do i just dont know HOW to do it. If someone could help me differentiate it in the beginning i'd really appreciate it so i can get on from there

I have been given the values for A,B and C. Where A = 90, B = 0.97 and C = 900

A typical fish farming process might involve placing small fish (fingerlings) in a dam, waiting for them to grow, and then harvesting all of the remaining fish.

People who rely on growing fish as a source of food are faced with the problem of working out when is the best time to harvest the fish, because although the fish grow bigger as time passes, unfortunately some of them die before they are harvested for food.

Data was collected and analysed for a particular species of fish. The following information was obtained:

Fish Length
The following formula gives a relationship between the fish length (L) in centimetres and time elapsed (t) in months since the fingerlings were placed in the dam:
L = A (1 – B^t )


Fish Weight
The following weight-length data is available for the species concerned: (which works out to be, W = 0.0141L^3.0164)

Length (cm)	10.1	25.0	32.6	35.4	43.8	45.5	55.7
Weight (g)	15	236	520	660	1250	1425	2590

Length of Life
For every ‘C’ fingerlings of this species of fish placed in the dam, the number, n, still alive after t months is given by the formula:
N = C x B^t


Based on this information, find the best time to harvest all of the fish remaining in the dam in order to get the maximum weight of live fish.


I know that the T total fish will be equal to N times W and ive tried differentiating the equation but i get nowhere

T = N * W
T = 900*0.97^t * 0.0141*90(1-0.97^t)^3.0164
T' = W' N + N W'
T' = ?

Thanks again for any help

 

[Deleted member 4993]

Hey i have this question from my maths assignment that's been bugging me for weeks since i first got the assignment, and i know what i have to do i just dont know HOW to do it. If someone could help me differentiate it in the beginning i'd really appreciate it so i can get on from there

I have been given the values for A,B and C. Where A = 90, B = 0.97 and C = 900

A typical fish farming process might involve placing small fish (fingerlings) in a dam, waiting for them to grow, and then harvesting all of the remaining fish.

People who rely on growing fish as a source of food are faced with the problem of working out when is the best time to harvest the fish, because although the fish grow bigger as time passes, unfortunately some of them die before they are harvested for food.

Data was collected and analysed for a particular species of fish. The following information was obtained:

Fish Length
The following formula gives a relationship between the fish length (L) in centimetres and time elapsed (t) in months since the fingerlings were placed in the dam:
L = A (1 – B^t )


Fish Weight
The following weight-length data is available for the species concerned: (which works out to be, W = 0.0141L^3.0164)

Length (cm)	10.1	25.0	32.6	35.4	43.8	45.5	55.7
Weight (g)	15	236	520	660	1250	1425	2590

Length of Life
For every ‘C’ fingerlings of this species of fish placed in the dam, the number, n, still alive after t months is given by the formula:
N = C x B^t


Based on this information, find the best time to harvest all of the fish remaining in the dam in order to get the maximum weight of live fish.


I know that the T total fish will be equal to N times W and ive tried differentiating the equation but i get nowhere

T = N * W
T = 900*0.97^t * 0.0141*90(1-0.97^t)^3.0164
T' = W' N + N W'
T' = ?

Thanks again for any help

You'll need to use chain rule of differentiation:

T = 900*0.97^t * 0.0141*90(1-0.97^t)^3.0164

T = 1142.1 * 0.97^t * (1 - 0.97^t)^3.0164

substitute:

u = 0.97^t

u' = du/dt = 0.97^t * ln(0.97) = -0.03046 * 0.97^t

v = (1 - u)^3.0164 = (1 - u)^m

v' = 3.0164 * (1 - 0.97^t)^2.0164 * (-u')

T = 1142.1 * u * v

T' = ?? ....Now continue......

 

[Pompeii]

You'll need to use chain rule of differentiation:

T = 900*0.97^t * 0.0141*90(1-0.97^t)^3.0164

T = 1142.1 * 0.97^t * (1 - 0.97^t)^3.0164

substitute:

u = 0.97^t

u' = du/dt = 0.97^t * ln(0.97) = -0.03046 * 0.97^t

v = (1 - u)^3.0164 = (1 - u)^m

v' = 3.0164 * (1 - 0.97^t)^2.0164 * (-u')

T = 1142.1 * u * v

T' = ?? ....Now continue......

Thankyou

So

T' = 1142.1 * (-0.03046 * 0.97^t) * (3.0164 * (1 - 0.97^t)^2.0164 * -(-0.03046 * 0.97^t))

Is that right? or do i disregard the 1142.1 in the beginning because it has no variable

 

[stapel]

So

T' = 1142.1 * (-0.03046 * 0.97^t) * (3.0164 * (1 - 0.97^t)^2.0164 * -(-0.03046 * 0.97^t))

No. Apply the Product Rule for derivatives.

do i disregard the 1142.1 in the beginning because it has no variable

Why would you ignore part of the answer, merely because it is numerical instead of variable? Why would you ignore only that number? :shock: