Use Laplace transformation to solve P.V.I y'+6y=e4t,
y(0)=2.

The area of the region that lies inside both the curves r = sin 2θ and r = sin θ is π/3 + (1/16)√3.

To find the area of the region that lies inside both the curves, we need to determine the limits of integration for the angle θ.

The curves r = sin 2θ and r = sin θ intersect at certain values of θ. To find these points of intersection, we can set the two equations equal to each other and solve for θ:

sin 2θ = sin θ

Using the trigonometric identity sin 2θ = 2sin θ cos θ, we can rewrite the equation as:

2sin θ cos θ = sin θ

Dividing both sides by sin θ (assuming sin θ ≠ 0), we have:

2cos θ = 1

cos θ = 1/2

θ = π/3, 5π/3

Now we have the limits of integration for θ, which are π/3 and 5π/3.

The formula for calculating the area in polar coordinates is given by:

A = (1/2) ∫[θ₁,θ₂] (r(θ))² dθ

In this case, the function r(θ) is given by r = sin 2θ. Therefore, the area is:

A = (1/2) ∫[π/3,5π/3] (sin 2θ)² dθ

To evaluate this integral, we can simplify the expression (sin 2θ)²:

(sin 2θ)² = sin² 2θ = (1/2)(1 - cos 4θ)

Now, the area formula becomes:

A = (1/2) ∫[π/3,5π/3] (1/2)(1 - cos 4θ) dθ

We can integrate term by term:

A = (1/4) ∫[π/3,5π/3] (1 - cos 4θ) dθ

Integrating, we get:

A = (1/4) [θ - (1/4)sin 4θ] |[π/3,5π/3]

Evaluating the integral limits:

A = (1/4) [(5π/3 - (1/4)sin (20π/3)) - (π/3 - (1/4)sin (4π/3))]

Simplifying the trigonometric terms:

A = (1/4) [(5π/3 + (1/4)sin (2π/3)) - (π/3 + (1/4)sin (4π/3))]

Finally, simplifying further:

A = (1/4) [(5π/3 + (1/4)√3) - (π/3 - (1/4)√3)]

A = (1/4) [(4π/3 + (1/4)√3)]

A = π/3 + (1/16)√3

Therefore, the area of the region that lies inside both the curves r = sin 2θ and r = sin θ is π/3 + (1/16)√3.

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(a) The distribution of IQ is approximately normal.

(b) The distribution of the sample mean IQ is approximately normal with a mean of 111 and a standard deviation of 3.0410.

(c) The probability that the sample mean IQ is less than 107 is 0.1056.

(d) The probability that the sample mean IQ is greater than 107 is 0.8944.

(e) The probability that the sample mean IQ is between 107 and 117 is 0.7881.

(a) The given information does not indicate any significant departure from normality, and with a large population, the Central Limit Theorem suggests that the distribution of IQ will be approximately normal.

(b) The mean of the sample mean IQ will be equal to the population mean, which is 111.

The standard deviation of the sample mean can be calculated by dividing the population standard deviation (22) by the square root of the sample size (55).

Therefore, the standard deviation of the sample mean IQ is 22 / √(55) = 3.0410.

(c) To calculate this probability, standardize the sample mean IQ value using the formula z = (x - μ) / (σ / √(n)),

where x = value to find the probability for, μ = population mean, σ = population standard deviation, and n = sample size.

In this case, find the probability for x = 107.

By plugging in the values, calculate the z-score and then use a standard normal distribution table or calculator to find the corresponding probability, which is 0.1056.

(d) Similar to part (c), use the same formula to standardize the value of 107 and calculate the z-score.

Then, find the probability of the sample mean IQ being greater than 107 by subtracting the probability found in part (c) from 1, which is 0.8944.

(e) To calculate this probability, find the individual probabilities for both values and then subtract the probability found in part (d) from the probability found in part (c).

The probability of the sample mean IQ being less than 107 is 0.1056, and the probability of the sample mean IQ being greater than 117 is 0.2119 (which can be found by subtracting the probability of being less than 117 from 1).

Therefore, the probability of the sample mean IQ being between 107 and 117 is 0.1056 + (1 - 0.2119) = 0.7881.

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Answer: A viable point on the graph is ✔ (8, 3,200).

The values of w must be✔ any whole number.

Answer:

A viable point on the graph is

✔ (8, 3,200)

.

The values of w must be

✔ any whole number

.Step-by-step explanation:

I did it on Edge and got it right

Answer:

4a + 2c = 1700

Step-by-step explanation:

1/4 or .75

6 divided by 4 equal 1/4 or .75

Answer: C

Step-by-step explanation:

Step-by-step explanation:

When performing operations on fractions, it is important to maintain the relationship between the numerator and the denominator. In general, if you do something to the numerator, you should also do the same to the denominator.

In your example, if you want to multiply the fraction 2/2 by 6/2, it is necessary to multiply both the numerator and the denominator by the same value. Here's why:

When you multiply fractions, you multiply the numerators together and the denominators together. So, in this case, the multiplication would be:

(2/2) * (6/2) = (2 * 6) / (2 * 2) = 12/4

If you had only multiplied the numerator (2) by 6, the result would have been:

(2 * 6) / 2 = 12/2

As you can see, these two results are different. The correct result is 12/4, which simplifies to 3/1 or simply 3. If you only multiplied the numerator, you would have obtained 12/2, which simplifies to 6.

So, it's necessary to apply the same operation (in this case, multiplication by 2) to both the numerator and the denominator in order to maintain the value of the fraction.

Answer:

The correct answers are:

• Option 1,

• Option 3 and

• Option 4

Step-by-step explanation:

From the two-way table, it is true that;

1.) 5 students study both French and Spanish.

2.) 30 students study French but not Spanish.

3.) 2 students study neither French nor Spanish

4.) 35 students study French

5.) 68 students study Spanish

6.) 63 students study Spanish but not French

This makes the correct answers options 1, 3 and 4

P.S: I believe your question is from the attached image

Answer:

A. 5 students study both French and Spanish.

C. 2 students study neither French nor Spanish.

D. 30 students study French, but not Spanish.

:)

Answer:

[tex]Area = 1560ft^2[/tex]

Step-by-step explanation:

Given

See attachment for playground

Required

Determine the area

The playground is a trapezoid. So;

[tex]Area = \frac{1}{2}(Sum\ parallel\ sides) * Height[/tex]

From the attachment, the parallel sides are: 68ft and 36ft

The height is: 30ft

So, the area is:

[tex]Area = \frac{1}{2}(68ft + 36ft) * 30ft[/tex]

[tex]Area = \frac{1}{2}(104ft) * 30ft[/tex]

[tex]Area = 52ft * 30ft[/tex]

[tex]Area = 1560ft^2[/tex]

Answer:

D

Step-by-step explanation:

when it's a power of the power we multiply the powers to get a single value for the power.

(6^(1/4))^4=6^(4*(1/4)) (4*(1/4)=1)

=6^1=6

so the answer is D

To find the value of w that satisfies the equation 2u v - 3w = 0, where u = (-6, 0, 0, 2) and v = (-3, 5, 1, 0), we can substitute the given values into the equation and solve for w.

Substituting the given values of u and v into the equation 2u v - 3w = 0, we have:

2(-6, 0, 0, 2)(-3, 5, 1, 0) - 3w = 0.

Expanding the scalar multiplication and performing the dot product, we get:

(-12, 0, 0, 4)(-3, 5, 1, 0) - 3w = 0,

(36 + 0 + 0 + 0) - 3w = 0,

36 - 3w = 0.

Simplifying the equation, we have:

36 = 3w,

w = 12.

Therefore, the value of w that satisfies the equation is 12. By substituting w = 12 into the equation 2u v - 3w = 0, we get:

2(-6, 0, 0, 2)(-3, 5, 1, 0) - 3(12) = 0,

(-12, 0, 0, 4)(-3, 5, 1, 0) - 36 = 0,

36 - 36 = 0,

0 = 0.

Hence, the value of w = 12 makes the equation true, satisfying the given condition.

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let $p$ and $q$ be the two distinct solutions to the equation$$\frac{4x-12}{x^2 2x-15}=x 2.$$if $p > q$. The value of $p - q$ is 4.

To find the value of $p - q$, we first need to solve the given equation and determine the values of $p$ and $q$.

The equation is:

$$\frac{4x-12}{x^2 - 2x - 15} = x^2.$$

Step 1: Factorize the denominator:

The denominator can be factored as $(x - 5)(x + 3)$.

Step 2: Simplify the equation:

$$\frac{4x-12}{(x - 5)(x + 3)} = x^2.$$

Step 3: Multiply both sides of the equation by $(x - 5)(x + 3)$ to eliminate the denominator:

$$(4x - 12) = x^2(x - 5)(x + 3).$$

Step 4: Expand and rearrange the equation:

$$4x - 12 = x^4 - 2x^3 - 15x^2 + 25x.$$

Step 5: Rearrange the equation and combine like terms:

$$x^4 - 2x^3 - 15x^2 + 21x - 12 = 0.$$

Step 6: Factorize the equation:

$$(x - 3)(x + 1)(x - 2)(x + 2) = 0.$$

From this, we get four possible solutions: $x = 3$, $x = -1$, $x = 2$, and $x = -2$.

However, we are interested in the two distinct solutions $p$ and $q$, where $p > q$. Therefore, the values of $p$ and $q$ are $p = 3$ and $q = -1$.

Finally, we can find the value of $p - q$:

$$p - q = 3 - (-1) = 3 + 1 = 4.$$

Hence, the value of $p - q$ is 4.

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Answer:

[tex](a)\ Area = 3765.32[/tex]

[tex](b)\ Area = 4773[/tex]

Step-by-step explanation:

Given

[tex]A_1 = 169in^2[/tex] --- area of each square

[tex]Shade = 4in[/tex]

See attachment for window

Solving (a): Area of the window

First, we calculate the dimension of each square

Let the length be L;

So:

[tex]L^2 = A_1[/tex]

[tex]L^2 = 169[/tex]

[tex]L = \sqrt{169[/tex]

[tex]L=13[/tex]

The length of two squares make up the radius of the semicircle.

So:

[tex]r = 2 * L[/tex]

[tex]r = 2*13[/tex]

[tex]r = 26[/tex]

The window is made up of a larger square and a semi-circle

Next, calculate the area of the larger square.

16 small squares made up the larger square.

So, the area is:

[tex]A_2 = 16 * 169[/tex]

[tex]A_2 = 2704[/tex]

The area of the semicircle is:

[tex]A_3 = \frac{\pi r^2}{2}[/tex]

[tex]A_3 = \frac{3.14 * 26^2}{2}[/tex]

[tex]A_3 = 1061.32[/tex]

So, the area of the window is:

[tex]Area = A_2 + A_3[/tex]

[tex]Area = 2704 + 1061.32[/tex]

[tex]Area = 3765.32[/tex]

Solving (b): Area of the shade

The shade extends 4 inches beyond the window.

This means that;

The bottom length is now; Initial length + 8

And the height is: Initial height + 4

In (a), the length of each square is calculated as: 13in

4 squares make up the length and the height.

So, the new dimension is:

[tex]Length = 4 * 13 + 8[/tex]

[tex]Length = 60[/tex]

[tex]Height = 4*13 + 4[/tex]

[tex]Height = 56[/tex]

The area is:

[tex]A_1 = 60 * 56 = 3360[/tex]

The radius of the semicircle becomes initial radius + 4

[tex]r = 26 + 4 = 30[/tex]

The area is:

[tex]A_2 = \frac{3.14 * 30^2}{2} = 1413[/tex]

The area of the shade is:

[tex]Area = A_1 + A_2[/tex]

[tex]Area = 3360 + 1413[/tex]

[tex]Area = 4773[/tex]

The Laplace transform of f(t) = et-³h(t – 3) is F(s) = 1/3 * 1/(s + 3) - 1/3 * 1/s, or F(s) = (s - 1)/(3s(s + 3)).

To find the Laplace transform F(s) = L{f(t)} of the function f(t) = et-³h(t – 3), we can use the properties and formulas of Laplace transforms. Let's break down the function and compute its transform step by step.

et-³: This term represents the exponential function with a power of t-³. The Laplace transform of et is given by 1/(s - a), where a is the exponent in the exponential function. Therefore, the Laplace transform of et-³ is 1/(s + 3).

h(t - 3): This term represents the Heaviside step function, h(t), shifted by 3 units to the right. The Laplace transform of the Heaviside step function h(t) is 1/s. When the function is shifted by a constant, the Laplace transform remains the same.

Combining these results, we have F(s) = L{f(t)} = 1/(s + 3) * 1/s.

To simplify the expression, we can use partial fraction decomposition:

1/(s + 3) * 1/s = A/(s + 3) + B/s.

Solving for A and B by equating numerators, we find A = 1/3 and B = -1/3.

Therefore, the Laplace transform of f(t) = et-³h(t – 3) is F(s) = 1/3 * 1/(s + 3) - 1/3 * 1/s, or F(s) = (s - 1)/(3s(s + 3)).

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Answer:

35.

Step-by-step explanation:

can i get brainliesttt

jus solved it.

Answer:

Step-by-step explanation:

An arithmetic sequence is when the difference of the terms is same

F)2, 5, 9, 14, ...

5 ≠ 4 ≠ 3, no

G)100, 50, 12.5, 1.6, ...

-10.9 ≠ -37.5 ≠ -50, no

H)3, 10, 17, 24,...

7 is the common difference, yes

j) -2,-1,-1/2,-1/4

1/4 ≠ 1/2 ≠ 1, no

Answer:

A

Step-by-step explanation:

idek sorry i think its A

1.) A=pi(r)^2

2.) V=Bh

3.) 3.1

4.) 20

5.)36

6.) 10

7.)18

Answer:

Step-by-step explanation:

1.) formula of circle =pi times r^2

2.)volume of cylinder =pi times r^2 times h

3.)value of pi rounded = 3.14

4.) diameter of can A =2r=2(10) = 20

diameter of Can A is 20

5.)diameter of can B =2r =2(18) =36

diameter of Can B is 36

6.)radius = 10

7.)radius =18

I did all of them.

Answer:

I think the answer would be Y= -2X+5 .

Hope it helps u ^^♥️

Answer:

t = 0, 1, 4, 8

h = 0, 12, 0 , 18

Step-by-step explanation:

t = 0 + 1 + 3 + 4

/ x1 - 5 (2) x + 4x + 9

h = 0, 12, 0, 18

After 18 seconds the ball attains...

x/3 = 1/4 - 4x 2(7) = 9/2 - 2 / 4 (0)

Answer:

t = 0, 1, 4, 8

h = 0, 12, 0 , 18  

t = 0 + 1 + 3 + 4

Step-by-step explanation:

/ x1 - 5 (2) x + 4x + 9

h = 0, 12, 0, 18

After 18 seconds the ball attains...

x/3 = 1/4 - 4x 2(7) = 9/2 - 2 / 4 (0)

Answer:

974

Step-by-step explanation:

Let assume that:

The set of student that took part in Calculus be = C

Those that took part in discrete mathematics be = D

Let those that took part in data structures be = DS; &

Those that took part in Programming language be = P

Thus;

{C} = 507

{D} = 292

{DS} = 312

{P} = 344

For intersections:

{C ∩ DS} = 14

{C ∩ P} = 213

{D ∩ DS} = 211

{D ∩ P}  =43

{C ∩ D} = 0

{DS ∩ P} = 0

{C ∩ D ∩ DS ∩ P} = 0

According to principle of inclusion-exclusion;

{C ∪ D ∪ DS ∪ P} = {C} + {D} + {DS} + {P} - {C ∩ D} - {C ∩ DS} - {C ∩ P} - {D ∩ DS} - {D ∩ P} - {DS ∩ P}

{C ∪ D ∪ DS ∪ P} = 507 + 292 + 312 + 344 - 14 - 213 - 211 - 43 - 0

{C ∪ D ∪ DS ∪ P} = 974

Hence, the no of students that took part in either course = 974

Answer:

£2308.5 per day

Step-by-step explanation:

Since in one day they have to deliver 384 catalogues and 1890 leaflets and each student can deliver either 16 catalogues or 90 leaflets in hour, the amount of students required to deliver 384 catalogues in one hour is 384/16 =  24 students.

Also, the number of students required to deliver 1890 leaflets in one hour is 1890/90 = 21 students.

So the total number of students required to make the delivery is thus 24 + 21 = 45 students. This is the minimum number of students required for the delivery.

Since all students hired are paid £51.30 per day, even if they don't work a full day, so the amount of wage paid for this minimum amount of students is thus minimum amount × wage = 45 × £51.30 per day = £2308.5 per day

Answer:

£307.80

Step-by-step explanation:

Wow seems the verified answer is wrong.

Who would've thought?

16*384=24

90*1890=21

21+24=45

45/8=5.62500

5.625 rounds to 6

51.30*6=£307.80

Thats your working out

You have to divide 45 by 8 because there are 8 hours in a day in which they can work.

Then round the number as you cant have a fraction of a person

Which you would then multiply by 51.30

Brainliest would be appreciated <33

Hope it helps!

Answer:

Percent- 30

decimal-0.3

Step-by-step explanation:

Hope this helps and have a wonderful day!!!

The value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is {0.2^2 p(1-p)}/{1.645^2}

Given a random variable X where X ~ N(p, p(1-p)/k) and 0<=p<=1, we need to find the value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9.

In general, if X ~ N(μ,σ²), then

P[|X-μ| < a] = 2Φ(a/σ) - 1

where Φ(z) is the standard normal cumulative distribution function.

Therefore, we can say that

P[|X-p| < 0.2] = 2Φ(0.2/√(p(1-p)/k)) - 1 ≥ 0.9

or 2Φ(0.2/√(p(1-p)/k)) ≥ 1.9

or Φ(0.2/√(p(1-p)/k)) ≥ 0.95

or 0.2/√(p(1-p)/k) ≥ Φ^(-1)(0.95)

where Φ^(-1)(z) is the inverse of the standard normal cumulative distribution function.

Therefore,  Φ^(-1)(0.95) = 1.6450.2/√(p(1-p)/k) ≥ 1.645

or k ≤ 0.2²p(1-p)/1.645²

From the above inequality, we get the maximum value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is given by the formula:

k ≤{0.2^2 p(1-p)}/{1.645^2}

Therefore, the value of k for which X will estimate to p within an accuracy of 0.2 with probability greater than 0.9 is {0.2^2 p(1-p)}/{1.645^2}

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The null hypothesis is that the actual numbers of games fit the distribution indicated by the expected proportions.

The following is the calculation of the t-statistics for the given data.

[tex]T=\frac{(O_i-E_i)} {\sqrt{E_i}} [/tex]where [tex]O_i[/tex] represents the observed frequency, and [tex]E_i[/tex] represents the expected frequency.t statistics for

4: [tex]\frac {(16-25)} {\sqrt {25(0.125)}} [/tex] = -3.2t statistics for

5: [tex]\frac {(21-25)} {\sqrt {25(0.25)}} [/tex] = -1.8t statistics for

6: [tex]\frac {(21-31)} {\sqrt {25(0.3125)}} [/tex] = -3.2t statistics for

7: [tex]\frac {(38-31)} {\sqrt {25(0.3125)}} [/tex] = 3.2

The critical value of t at the 0.05 level of significance is ± 2.132. Since the t-statistics of 3.2 > 2.132, we reject the null hypothesis. So, there is a significant difference between the actual number of games played and the expected number of games played in the baseball championship series at the 0.05 significance level. p-value = P (|t| > 3.2) = 0.002

The conclusion for the test statistic: Since the p-value (0.002) is less than the level of significance (0.05), we reject the null hypothesis and conclude that there is a significant difference between the actual numbers of games played and the distribution indicated by the expected proportions.

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Answer:

D. 109

Step-by-step explanation:

100 meters into yards is 109.361

the ratio is 1.094 so 109

Answer: 10

Step-by-step explanation:

Alexis Scored 4 more than twice the number of points Jessica scored.

Jessica scored 3

twice the number of 3 would be 3 x 2 which equals six

4 more than twice the number which is 6 would be 10, 4+6=10