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eduard
3 years ago
7

Find the solution r(t)r(t) of the differential equation with the given initial condition: r′(t)=⟨sin9t,sin6t,9t⟩,r(0)=⟨4,6,3⟩

Mathematics
1 answer:
klemol [59]3 years ago
4 0

\mathbf r'(t)=\langle\sin9t,\sin6t,9t\rangle


\mathbf r(t)=\displaystyle\int\mathbf r'(t)\,\mathrm dt


\mathbf r(t)=\left\langle\displaystyle\int\sin9t\,\mathrm dt,\int\sin6t\,\mathrm dt,\int9t\,\mathrm dt\right\rangle


\displaystyle\int\sin9t\,\mathrm dt=\frac19\cos9t+C_1

\displaystyle\int\sin6t\,\mathrm dt=\frac16\cos6t+C_2

\displaystyle\int9t\,\mathrm dt=\frac92t^2+C_3


With the initial condition \mathbf r(0)=\langle4,6,3\rangle, we find


\dfrac19\cos0+C_1=4\implies C_1=\dfrac{35}9

\dfrac16\cos0+C_2=6\implies C_2=\dfrac{35}6

\dfrac92\cdot0^2+C_3=3\implies C_3=3


So the particular solution to the IVP is


\mathbf r(t)=\left\langle\dfrac19\cos9t+\dfrac{35}9,\dfrac16\cos6t+\dfrac{35}6,\dfrac92t^2+3\right\rangle

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Write the equation for g (x)
belka [17]

Answer:

g(x) = (-1/25)x + (203/25)

Step-by-step explanation:

The general equation for a line is slope-intercept form is:

y = mx + b

In this form, "m" represents the slope and "b" represents the y-intercept.

We know that perpendicular lines have opposite-signed, reciprocal slopes of the original line. Therefore, if the slope of f(x) is m = 25, the slope of g(x) must be m = (-1/25).

To find the y-intercept, we can use the newfound slope and the values from the given point to isolate "b".

g(x) = mx + b                                    <----- General equation

g(x) = (-1/25)x + b                             <----- Plug (-1/25) in "m"

8 = (-1/25)(3) + b                               <----- Plug in "x" and "y" from point

8 = (-3/25) + b                                  <----- Multiply (1/25) and 3

200/25 = (-3/25) + b                       <----- Covert 8 to a fraction

203/25 = b                                      <----- Add (3/25) to both sides

Now that we know both the values of the slope and y-intercept, we can construct the equation of g(x).

g(x) = (-1/25)x + (203/25)

4 0
2 years ago
We discovered that 650 GSS respondents in 2006 watched television for an average of 2.98 hrs/day, with a standard deviation of 2
Aliun [14]

Answer:

a) 2.09167

b) 0.2

c) 130 correspondents

d) 5.38 hours per day

e) 69%

Step-by-step explanation:

The formula for calculating a z-score is is z = (x-μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation.

Mean = 2.98hrs/day, Standard Deviation = 2.4

a) What is the Z score for a person who watches more than 8 hrs/day.

z = 8 - 2.98/2.4

= 2.09167

b) What proportion of people watch 5 hrs/day or more television?

z = 5 - 2.98/2.4

z = 0.84167

Determining the Probabilityvalue from Z-Table:

P(x<5) = 0.80001

The proportion of people watch 5 hrs/day or more television

P(x>5) = 1 - P(x<5) = 0.19999

Approximately = 0.2

c) How many does this correspond to in the sample?

From the question above, we are told that: there are 650 GSS respondents in 2006 watched television

Hence the proportion that corresponds to Question b is

0.2 × 650

= 130 correspondents

Therefore, 130 correspondents watch 5 hrs a day or more television

d) What number of television hours per day corresponds to a Z = +1.

z score formula is given as:

z = (x-μ)/σ

Z = +1

x = unknown

Mean = 2.98

Standard deviation = 2.4

Hence:

1 = x - 2.98/2.4

Cross Multiply

1 × 2.4 = x - 2.98

2.4 = x - 2.98

2.4 + 2.98 + x

= 5.38 hours per day.

e)What is the percentage of people who watch between 1 and 6 hours of television per day?

For x = 1

z = 1 - 2.98/2.4

z = -0.825

Probability value from Z-Table:

P(x= 1) = 0.20469

For x = 6

z = 6 - 2.98/2.4

z = 1.25833

Probability value from Z-Table:

P(x = 6) = 0.89586

The percentage of people who watch between 1 and 6 hours of television per day is calculated as:

P(x = 6) - P(x = 1)

0.89586 - 0.20469

= 0.69117

Converting to percentage,

0.69117 × 100%

= 69.117%

To the nearest whole number = 69%

Therefore, 69% of the people who watch between 1 and 6 hours of television per day

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3 years ago
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Answer:

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(2,-5). y=mx+b or -5=0 × 2+b, or solving for b: b=-5-(0)(2). b=-5.

y=-5

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