Let c be a positive number. A differential equation of the form dy/dt=ky^1+c where k is a positive constant, is called a doomsday equation because the exponent in the expression ky^1+c is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term My^1.01. If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

Answers

Answer 1

Answer:

The doomsday is 146 days

Step-by-step explanation:

Given

[tex]\frac{dy}{dt} = ky^{1 +c}[/tex]

First, we calculate the solution that satisfies the initial solution

Multiply both sides by

[tex]\frac{dt}{y^{1+c}}[/tex]

[tex]\frac{dt}{y^{1+c}} * \frac{dy}{dt} = ky^{1 +c} * \frac{dt}{y^{1+c}}[/tex]

[tex]\frac{dy}{y^{1+c}} = k\ dt[/tex]

Take integral of both sides

[tex]\int \frac{dy}{y^{1+c}} = \int k\ dt[/tex]

[tex]\int y^{-1-c}\ dy = \int k\ dt[/tex]

[tex]\int y^{-1-c}\ dy = k\int\ dt[/tex]

Integrate

[tex]\frac{y^{-1-c+1}}{-1-c+1} = kt+C[/tex]

[tex]-\frac{y^{-c}}{c} = kt+C[/tex]

To find c; let t= 0

[tex]-\frac{y_0^{-c}}{c} = k*0+C[/tex]

[tex]-\frac{y_0^{-c}}{c} = C[/tex]

[tex]C =-\frac{y_0^{-c}}{c}[/tex]

Substitute [tex]C =-\frac{y_0^{-c}}{c}[/tex] in [tex]-\frac{y^{-c}}{c} = kt+C[/tex]

[tex]-\frac{y^{-c}}{c} = kt-\frac{y_0^{-c}}{c}[/tex]

Multiply through by -c

[tex]y^{-c} = -ckt+y_0^{-c}[/tex]

Take exponents of [tex]-c^{-1[/tex]

[tex]y^{-c*-c^{-1}} = [-ckt+y_0^{-c}]^{-c^{-1}[/tex]

[tex]y = [-ckt+y_0^{-c}]^{-c^{-1}[/tex]

[tex]y = [-ckt+y_0^{-c}]^{-\frac{1}{c}}[/tex]

i.e.

[tex]y(t) = [-ckt+y_0^{-c}]^{-\frac{1}{c}}[/tex]

Next:

[tex]t= 3[/tex] i.e. 3 months

[tex]y_0 = 2[/tex] --- initial number of breeds

So, we have:

[tex]y(3) = [-ck * 3+2^{-c}]^{-\frac{1}{c}}[/tex]

-----------------------------------------------------------------------------

We have the growth term to be: [tex]ky^{1.01}[/tex]

This implies that:

[tex]ky^{1.01} = ky^{1+c}[/tex]

By comparison:

[tex]1.01 = 1 + c[/tex]

[tex]c = 1.01 - 1 = 0.01[/tex]

[tex]y(3) = 16[/tex] --- 16 rabbits after 3 months:

-----------------------------------------------------------------------------

[tex]y(3) = [-ck * 3+2^{-c}]^{-\frac{1}{c}}[/tex]

[tex]16 = [-0.01 * 3 * k + 2^{-0.01}]^{\frac{-1}{0.01}}[/tex]

[tex]16 = [-0.03 * k + 2^{-0.01}]^{-100}[/tex]

[tex]16 = [-0.03 k + 0.9931]^{-100}[/tex]

Take -1/100th root of both sides

[tex]16^{-1/100} = -0.03k + 0.9931[/tex]

[tex]0.9727 = -0.03k + 0.9931[/tex]

[tex]0.03k= - 0.9727 + 0.9931[/tex]

[tex]0.03k= 0.0204[/tex]

[tex]k= \frac{0.0204}{0.03}[/tex]

[tex]k= 0.68[/tex]

Recall that:

[tex]-\frac{y^{-c}}{c} = kt+C[/tex]

This implies that:

[tex]\frac{y_0^{-c}}{c} = kT[/tex]

Make T the subject

[tex]T = \frac{y_0^{-c}}{kc}[/tex]

Substitute: [tex]k= 0.68[/tex], [tex]c = 0.01[/tex] and [tex]y_0 = 2[/tex]

[tex]T = \frac{2^{-0.01}}{0.68 * 0.01}[/tex]

[tex]T = \frac{2^{-0.01}}{0.0068}[/tex]

[tex]T = \frac{0.9931}{0.0068}[/tex]

[tex]T = 146.04[/tex]

The doomsday is 146 days


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Step-by-step explanation:

1. Let the small number be x

Then, the bug number will be: x + 25

Therefore, x + x + 25 = 131

2x + 25 = 131

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2. Let the small number be x

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Step-by-step explanation:

Given

Outcomes of rolling two dice

Required

P(Sum = 9)

From the given outcomes, we have:

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Simplify

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A. Associative Property of Addition

B. Commutative Property of Addition

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Answer:

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Step-by-step explanation:

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Answer:

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cos(25°) = [tex]\frac{b}{400}[/tex]

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b = 362.52

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The Math Club makes 35 bars of soap a week and sells these at $20 each. Before the soap can be sold, 6 bars were eaten by a polar bear. If they sell the remaining bars of soap, how much will they make this week?

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EDGE

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Step-by-step explanation:

Answer:

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Step-by-step explanation:

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Step-by-step explanation:

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Answer:

multiply each term in one polynomial by each term in the other polynomial.

add those answers together, and simplify if needed.

Step-by-step explanation:

Use FOIL !

1. multiply each term in one polynomial by each term in the other polynomial

2. add those answers together, and simplify if needed

Multiply polynomials the same way you multiply monomials and bionomials. You can also use symbolab to check if your answer is correct

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Answers

Answer:

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d1 = 44 / tan(60) = 25.403 m

For point2:

d2 = 44 / tan(30) = 76.210 m

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The speed is then equal to 0.050807/(1/12) kph = 0.61 kph.

I hope my answer has come to your help. Thank you for posting your question here in Brainly. We hope to answer more of your questions and inquiries soon. Have a nice day ahead!

Step-by-step explanation:

Sam scored 98, 25, 105, 62 and 65 run in 5 matches. What was the
average score per match?

Answers

Answer:

75

Step-by-step explanation:

am scored 92,25,105,62,65 in 5 matches

The average score per match is

= total of runs in 5 matches/ 5

= 92+25+105+62+65/5

=355/5

= 71

Answer: the average score per match is 71 maybe try looking around before wasting points but im glad to help

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Answers

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C

Step-by-step explanation:

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C). pentagon is the correct answer.

Step-by-step explanation:

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Answer:

We need to put this equation into either form of a line, pererably, slope intercept form!

Step-by-step explanation:

To do this, out goal is to get y alone.

1. We subtract x on both sides.

2 We divide 6 on both sides.

3. Our answer is y= -x/6 ( negative x divided by 6)

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6. From (0,0) point, you go down -1 and move 1 to the right each time!

The graph will be a line with a slope of -1/6 that passes through the points (0,0), (6,-1), and (-6,1).

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Graph is a mathematical representation of a network and it describes the relationship between lines and points.

To graph this equation, start by rearranging it to solve for y.

y = -x/6

Next, plot a few points to fill in the graph.

when x is 0,

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Finally, connect the plotted points with a straight line.

The final graph will be a line with a slope of -1/6 that passes through the points (0,0), (6,-1), and (-6,1).

Hence, graph will be a line with a slope of -1/6 that passes through the points (0,0), (6,-1), and (-6,1).

To learn more on Graph click:

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i will give a crown if you are the first one to answer and if you get it right and if you just get it right then i will i give you a full star rating and a thanks

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Answer:

10x+8

Step-by-step explanation:

Answer:

10x + 8 is the answer

Step-by-step explanation:

To simplify the equation we must use the Distributive Property meaning that

we must distribute 2 to each term

                                                     5x(2) + 4(2)

                                                        10x + 8

            There You Go, Have a Wonderful Day, And Hope This Helps

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Answer:

x = 6.7

Step-by-step explanation:

We know that 3.5 + x = 10.2, so if we do inverse operations we can find out the value of x.

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we can check our work by substituing it back in the equation for x

6.7 + 3.5 = 10.2

so now we know that x = 6.7. I hope this helps :)

Answer: x=6.7

Step-by-step explanation:

x+3.5=10.2

-3.5    -3.5

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Answer:

G

Step-by-step explanation:

Answer:

now they can have brain

Step-by-step explanation:

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