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LuckyWell [14K]
3 years ago
15

Newton’s law of cooling states that dx/dt= −k(x − A) where x is the temperature, t is time, A is the ambient temperature, and k

> 0 is a constant. Suppose that A A0 cos(ωt) for some constants A0 and ω. That is, the ambient temperature oscillates (for example night and day temperatures).a. Find the general solution.b. In the long term, will the initial conditions make much of a difference? Why or why not?
Mathematics
1 answer:
Vadim26 [7]3 years ago
7 0

Answer:

(a) The solution to the differential equation is x = A_0Coswt + Ce^(-kx)

(b) The initial condition t > 0 will not make much of a difference.

Step-by-step explanation:

Given the differential equation

dx/dt= −k(x − A); t > 0, A = A_0Coswt

(a) To solve the differential equation, first separate the variables.

dx/(x - A) = -kdt

Integrating both sides, we have

ln(x - A) = -kt + c

x - A = Ce^(-kt) (Where C = ce^(-kt))

x = A + Ce^(-kx)

Now, we put A = A_0Coswt

x = A_0Coswt + Ce^(-kx) (Where C is constant.)

And we have the solution.

(b) Since temperature t ≠ 0, the initial condition t > 0 will not make much of a difference because, Cos(wt) = Cos(-wt).

It is not any different from when t < 0.

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Marshall buys a sack of peaches for $5.98. The peaches weigh
kiruha [24]

Answer:

Step-by-step explanation:

a

6 0
3 years ago
Direct variations need help
Solnce55 [7]

Answer:

Part 11) The table represent a direct variation. The equation is y=18x

Part 12) The table represent a direct variation. The equation is y=0.4x

Step-by-step explanation:

we know that

A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form y/x=k or y=kx

In a proportional relationship the constant of proportionality k is equal to the slope m of the line and the line passes through the origin

Part 11)

For x=0.5, y=9

Find the value of k

k=y/x -----> k=9/0.5=18

For x=3, y=54

Find the value of k

k=y/x -----> k=54/3=18

For x=-2, y=-36

Find the value of k

k=y/x -----> k=-36/-2=18

For x=1, y=18

Find the value of k

k=y/x -----> k=18/1=18

For x=-8, y=-144

Find the value of k

k=y/x -----> k=-144/-8=18

The values of k is the same for each ordered pair

therefore

The table represent a direct variation

The linear equation is

y=18x

Part 12)

For x=-5, y=-2

Find the value of k

k=y/x -----> k=-2/-5=2/5=0.40

For x=3, y=1.2

Find the value of k

k=y/x -----> k=1.2/3=0.40

For x=-2, y=-0.8

Find the value of k

k=y/x -----> k=-0.8/-2=0.4

For x=10, y=4

Find the value of k

k=y/x -----> k=4/10=0.4

For x=20, y=8

Find the value of k

k=y/x -----> k=8/20=0.4

The values of k is the same for each ordered pair

therefore

The table represent a direct variation

The linear equation is

y=0.4x

8 0
2 years ago
At a hockey game, a vender sold a combined total of 235 sodas and hot dogs. The number of hot dogs sold was 59 less than the num
Neko [114]

Answer:

  • 147 sodas
  • 88 hot dogs

Step-by-step explanation:

This problem is similar to many others in which the sum of two quantities and their difference are given. The solution can be found easily when the equations for the relations are written in standard form.

<h3>Setup</h3>

Let s and h represent numbers of sodas and hot dogs sold, respectively. The given relations are ...

  • s +h = 235 . . . . . combined total
  • s -h = 59 . . . . . . difference in the quantities

<h3>Solution</h3>

Adding the two equations eliminates one variable.

  (s +h) +(s -h) = (235) +(59)

  2s = 294 . . . . simplify

  s = 147 . . . . . .divide by 2

  h = 147 -59 = 88 . . . . h is 59 less

147 sodas and 88 hot dogs were sold.

__

<em>Additional comment</em>

The solution to a "sum and difference" problem is always the same. One of the numbers is half the sum of those given, and the other is half their difference. ((235-59)/2 = 88)

5 0
1 year ago
Can someone help me with this worksheet please!!!!
Marat540 [252]

(1) The missing term in the sequence, a₁₂  = 0.8.

(2) The missing term in the sequence, a₈ = 102.5.

(3) The missing term in the sequence, a₈ = 111.

(4) The missing term in the sequence, a₁₂ = -19.

(5) The missing term in the sequence, a₁₂ = 94.

(6)  The missing term in the sequence, a₆ = 40.

(7)  The missing term in the sequence, a₃₆ = -52.

(8)  The missing term in the sequence, a₂₁ = -58.

<h3>Missing term of the sequence</h3>

The missing term in the sequence is determined as follows;

Tₙ = a + (n - 1)d

<h3>1.0 a₄ = 18.4 and a₅ = 16.2, a₁₂ = ?</h3>

T₄ = a + 3d

18.4 = a + 3d  ---(1)

T₅ = a + 4d

16.2 = a + 4d  ---(2)

subtract (1) from (2)

-2.2 = d

18.4 = a + 3(-2.2)

a = 25

a₁₂  = a + 11d

a₁₂  = 25 + 11(-2.2)

a₁₂  = 0.8

<h3>2.0 a₂ = 57.5 and a₅ = 80, a₈ = ?</h3>

a₂ = a + d

57.5 = a + d -- (1)

a₅ = a + 4d

80 = a + 4d  --- (2)

solve (1) and (2)

d = 7.5

a = 50

a₈ =  a + 7d

a₈ = 50 + 7(7.5)

a₈ = 102.5

<h3>3.0 a₁₀ = 141 and a₁₃ = 186, a₈ = ?</h3>

a₁₀ = a + 9d

141 = a + 9d --- (1)

a₁₃ = a + 12d

186 = a + 12d --- (2)

Subtract (1) from (2)

d = 15

a = 6

a₈ = a + 7d

a₈ = 6 + 7(15)

a₈ = 111

<h3>4.0 a₂₂ = -49 and a₂₅ = -58, a₁₂ = ?</h3>

a₂₂ = a + 21d

-49 = a + 21d ---- (1)

a₂₅ = a + 24d

-58 = a + 24d --- (2)

subtract (1) from (2)

d = -3

a = 14

a₁₂ = a + 11d

a₁₂ = 14 + 11(-3)

a₁₂ = -19

<h3>5.0 a₄ = -2 and a₈ = 46, a₁₂ = ?</h3>

a₄ = a + 3d

-2 = a + 3d --- (1)

a₈ = a + 7d

46 = a + 7d ---- (2)

Subtract (1) from (2)

d = 12

a = -38

a₁₂ = a + 11d

a₁₂ = -38 + 11(12)

a₁₂ = 94

<h3>6.0 a₉ = 64 and a₁₂ = 88, a₆ = ?</h3>

a₉ = a + 8d

64 = a + 8d --- (1)

a₁₂ = a + 11d

88 = a + 11d --- (2)

Subtract (1) from (2)

d = 8

a = 0

a₆ = a + 5d

a₆ = 0 + 5(8)

a₆ = 40

<h3>7.0 a₂₀ = -4 and a₂₃ = -13, a₃₆ = ?</h3>

a₂₀ = a + 19d

-4 = a + 19d ---- (1)

a₂₃ = a + 22d

-13 = a + 22d --- (2)

Subtract (1) from (2)

d = -3

a = 53

a₃₆ = a + 35d

a₃₆ = 53 + 35(-3)

a₃₆ = -52

<h3>8.0 a₂₈ = 5 and a₃₃ = 50, a₂₁ = ?</h3>

a₂₈ = a + 27d

5 = a + 27d ---- (1)

a₃₃ = a + 32d

50 = a + 32d --- (2)

Subtract (1) from (2)

d = 9

a = -238

a₂₁ = a + 20d

a₂₁ = -238 + 20(9)

a₂₁ = -58

Learn more about arithmetic progression here: brainly.com/question/6561461

#SPJ1

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