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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Answers

1st is 40/100 = 4/10 =40%
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Answer:

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Top Right: 52/100 = 13/25 = 52%

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Step-by-step explanation:

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[tex]P(Sum = 9) = \frac{1}{9}[/tex]

Step-by-step explanation:

Given

Outcomes of rolling two dice

Required

P(Sum = 9)

From the given outcomes, we have:

[tex]n(S) = 36[/tex] -- the sample size

[tex]Sum = 9:=\{(3,6),(4,5),(5,4),(6,3)\}[/tex]

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The probability is:

[tex]P(Sum = 9) = \frac{n(Sum = 9)}{n(S)}[/tex]

[tex]P(Sum = 9) = \frac{4}{36}[/tex]

Simplify

[tex]P(Sum = 9) = \frac{1}{9}[/tex]

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Step-by-step explanation:

Answer:

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Step-by-step explanation:

77.0075 = 770075/10000

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Answers

Answer:

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Answer:

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Step-by-step explanation:

The measure of a vertex angle outside a circle = ½(difference of intercepted arcs)

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m<S = ½(UV - TW)

m<S = 34°

UV = (3x - 2)°

TW = (x + 6)°

Plug in the values into the equation

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Step-by-step explanation:

Answer:

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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

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Step-by-step explanation:

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The answer would be 108 ft^2

Give each side and angle measure rounded to the nearest whole number.

Answers

Answer:

Step-by-step explanation:

By applying sine rule in the given triangle,

cos(25°) = [tex]\frac{\text{AC}}{AB}[/tex]

cos(25°) = [tex]\frac{b}{400}[/tex]

b = 400[cos(25°)]

b = 362.52

b = 363 ft

By applying sine rule in the given triangle,

sin(25°) = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]

sin(25°) = [tex]\frac{a}{400}[/tex]

a = 400[sin(25°)]

a = 169.05

a = 169 ft

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Answer: No es porque el número podría ser 35

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