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Vitek1552 [10]
3 years ago
14

The fish population in a pond with carrying capacity 1200 is modeled by the following logistic equation where N(t) denotes the n

umber of fish at time t in years.
(dN)/(dt) = (0.4)/(1200)N(1200-N)
Starting at the time at which the number of fish reached 150, the owner of the pond removed (harvested) fish at a constant rate of 50 fish per year. We take t = 0 to be the time at which the owner started to harvest fish from the pond.

A) Modify the differential equation to model the population of fish from the time it reached 150.
dN/dt =______

B) How far below the original carrying capacity will the number of fish be in the long run? (Give your answer correct to the nearest whole fish.)
Number of fish= _______
Mathematics
1 answer:
MArishka [77]3 years ago
8 0

9514 1404 393

Answer:

  (dN)/(dt) = (0.4)/(1200)N(1200-N) -50

  142 fish

Step-by-step explanation:

A) The differential equation is modified by adding a -50 fish per year constant term:

  (dN)/(dt) = (0.4)/(1200)N(1200-N) -50

__

B) The steady-state value of the fish population will be when N reaches the value that makes dN/dt = 0.

  (0.4/1200)(N)(1200-N) -50 = 0

  N(N-1200) = -(50)(1200)/0.4) . . . . rewrite so N^2 has a positive coefficient

  N^2 -1200N + 600^2 = -150,000 +600^2 . . . . complete the square

  (N -600)^2 = 210,000 . . . . . simplify

  N = 600 + √210,000 ≈ 1058

This steady-state number of fish is ...

  1200 - 1058 = 142 . . . . below the original carrying capacity

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4) solve it a) 3x-2÷5=2 ​
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Answer:

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The table shows the number of medical clinics and the number of bowling alleys in several cities.
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Answer:

Step-by-step explanation:

a) to create a scatter plot for the data, let data for clinics be x and that one for Bowling alleys be y then plot x values with corresponding y values on a graph tool.

<u>x                     y</u>

14                   12

1                      3

6                     6

5                     7

10                    7

6                      8

11                      9

8                      15

6                       7

16                     12

The scatter plot is attached below

b) a cluster is a group of data that is close together. From the plot, the ordered pairs forming a cluster are;

(6,6), (6,7), (5,7), and (6,8)

c) An outlier is a point in a plot which lies very far from the other values. In this case, point its  point (8,15)

d)The linear regression equation is one written in the form of

Y=a+bX where ;

Y=the dependent variable

X=the explanatory variable

a=the intercept, which is the value of y when x=o

b=slope of the line

The formula to apply here is ;

a=(∑y)(∑x²)-(∑x)(∑xy) / n(∑x²)-(∑x)² where n is number of samples

b= n(∑xy)-(∑x)(∑y) / n(∑x²)-(∑x)²

Form a table like below

<u>x                     y               xy               x²                 y²</u>

14                   12              168            196              144

1                      3                 3               1                   9

6                     6                 36            36                 36

5                     7                 35             25                49

10                    7                 70             100              49

6                      8                 48             36              64

11                      9                99             121               81

8                      15               120             64            225

6                       7                42               36              49

16                     12                192            256            144

83                     86              813           871               850------------sum

∑x=83   ∑y=86  ∑xy=813  ∑x²=871  ∑y²=850 and n=10

Applying the formula

a=(∑y)(∑x²)-(∑x)(∑xy) / n(∑x²)-(∑x)²

a=[(86 * 871) - (83*813) ]/ 10 (871-83²)

a=[74906 - 67479] / 10(-6018)

a=[7427] / -60180

a= - 0.1234

b= n(∑xy)-(∑x)(∑y) / n(∑x²)-(∑x)²

b=10(813)-(83*86) / 10 (871)-(83²)

b=[8130-7138] / 8710 -6889

b= 992/1821

b=0.5448

The liner regression equation will be;

Y=a+bX

Y= -0.1234 + 0.5448 X

e) correlation coefficient r is given by;

r= n∑xy - (∑x)(∑y) / √n(∑x²)-(∑x)² × √n(∑y²)-(∑y)²

r=10*813 - (83*86) / √10 *871 -83² ×√10*850 - 86²

r=8130-7138 / √8710-6889 × √8500-7396

r= 992 / (√1821 × √1104)

r=992/ 42.67×33.23

r=992/1417.9

r=0.6996

f) r² is important because it shows you the proportion of the variance of one variable that is predictable from the other variable.

r=0.6996

r²=0.6996² =0.4894

8 0
3 years ago
Read 2 more answers
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