In this problem, y = c1 cos 4x + c2 sin 4x is a two-parameter family of solutions of the second-order DE y'' + 16y = 0.
If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions. (If not possible, enter NOT.) y(0) = 0, y(π/8) = 4.

Answer:

  (0, -1), (1, 0), (2, 1)

Step-by-step explanation:

I find this easier to do after multiplying the equation by -1:

  y = x - 1

Pick any value for x, then subtract 1 from it to find the corresponding value of y.

She would offer to split the cookies evenly, so they each get 4.

If she offered Max less than 4, he would not accept and their dad would eat half, so each person would only get 2 cookies each.

If she offered Max more than 4, then she doesn't maximize the amount she would get.

Answer:

8.82 years.

Step-by-step explanation:

Since, the monthly payment formula is,

[tex]P=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]

Where, PV is the present value of the loan,

r is the rate per month,

n is number of months,

Here,

PV =  $ 25,000,

Annual rate = 5.2 % = 0.052 ⇒ Monthly rate, r = [tex]\frac{0.052}{12}[/tex]

( 1 year = 12 months )

P = $ 295,

By substituting the values,

[tex]295=\frac{25000(\frac{0.052}{12})}{1-(1+\frac{0.052}{12})^{-n}}[/tex]

By the graphing calculator,

We get,

[tex]n = 105.84[/tex]

Hence, the time ( in years ) = [tex]\frac{105.84}{12}=8.82[/tex]

Answer:

The future value of the annuity due to the nearest cent is $2956.

Step-by-step explanation:

Consider the provided information:

It is provided that monthly payment is $175, interest is 7% and time is 11 years.

The formula for the future value of the annuity due is:

[tex]FV of Annuity Due = (1+r)\times P[\frac{(1+r)^{n}-1}{r}][/tex]

Now, substitute P = 175, r = 0.07 and t = 11 in above formula.

[tex]FV of Annuity Due = (1+0.07)\times 175[\frac{(1+0.07)^{11}-1}{0.07}][/tex]

[tex]FV of Annuity Due = (1.07)\times 175[\frac{1.10485}{0.07}][/tex]

[tex]FV of Annuity Due = 187.5(15.7835)[/tex]

[tex]FV of Annuity Due = 2955.4789[/tex]

Hence, the future value of the annuity due to the nearest cent is $2956.

Answer:

The answer is - budget constraint

Step-by-step explanation:

The slope of the budget constraint is determined by the relative price of the two goods, which is calculated by taking the price of one good and dividing it by the price of the other good.  

A budget constraint happens when a consumer demonstrates limited consumption patterns by a certain income.

[tex]f(x)=\displaystyle\int_{-6}^x\sqrt{36-t^2}\,\mathrm dt[/tex]

The integrand is defined for [tex]36-t^2\ge0[/tex], or [tex]-6\le t\le6[/tex], so the domain should be the same, [tex]-6\le x\le6[/tex].

When [tex]x=-6[/tex], the integral is 0.

The integrand is non-negative for all [tex]x[/tex] in the domain, which means the value of [tex]f(x)[/tex] increases monotonically over this domain. When [tex]x=6[/tex], the integral gives the area of the semicircle centered at the origin with radius 6, which is [tex]\dfrac\pi26^2=18\pi[/tex], so the range is [tex]\boxed{0\le f(x)\le 18\pi}[/tex].

The range of the function f(x) is the integral from -6 to x of the square root of the quantity 36 minus t squared is [0, 6*π] because the total area of the semicircle is the maximum value.

The function f(x) is the integral from -6 to x of the square root of the quantity 36 minus t squared. This is a known geometrical shape, which is a semicircle with radius 6. To find the range of this function, we need to know the possible outcomes of this function. In general, for a semicircle of radius r, the values of the square root of the quantity r squared minus t squared will vary from 0 to r, both inclusive. So, if you consider the function from -6 to 6, the range would be [0, 6*π] because the total area of the semicircle is the maximum value.

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Answer:

(a) a=6 and b≠[tex]\frac{11}{4}[/tex]

(b)a≠6

(c) a=6 and b=[tex]\frac{11}{4}[/tex]

Step-by-step explanation:

writing equation in agumented matrix form

[tex]\begin{bmatrix}1 &1 & 0 &1 &b\\ 2 &3 & 1 &5 &6\\ 0& 0 & 1 &1 &4\\ 0& 2 & 2&a &1\end{bmatrix}[/tex]

now [tex]R_{2} =R_{2}-2\times R_{1}[/tex]

[tex]\begin{bmatrix}1 &1& 0 &1 &b\\ 0 &1& 1 &3 &6-2b\\ 0& 0 & 1 &1 &4\\ 0& 2 & 2&a &1\end{bmatrix}[/tex]

now [tex]R_{4} =R_{4}-2\times R_{2}[/tex]

[tex]\begin{bmatrix}1 &1& 0 &1 &b\\ 0 &1& 1 &3 &6-2b\\ 0& 0 & 1 &1 &4\\ 0& 0 & 0 &a-6 &4b-11\end{bmatrix}[/tex]

a) now for inconsistent

rank of augamented matrix ≠ rank of matrix

for that  a=6 and b≠[tex]\frac{11}{4}[/tex]

b) for consistent w/ a unique solution

rank of augamented matrix = rank of matrix

  a≠6

c) consistent w/ infinitely-many sol'ns

  rank of augamented matrix = rank of matrix < no. of variable

for that condition

 a=6 and b=[tex]\frac{11}{4}

then rank become 3 which is less than variable which is 4.

Answer:

Odd numbers that are multiple of 5 and are in between 22 and 66 are-

25, 35, 45, 55, 65

Let this set be represented by A

A= {25, 35, 45, 55, 65}

the above form represents the set in its roster form

The set notation for the odd numbers between 22 and 66 that are multiples of 5 is { x ∈ N | x is odd, 22 < x < 66, x ≡ 0 (mod 5) }.

The set notation for the odd numbers between 22 and 66 that are multiples of 5 can be written as:

{ x ∈ N | x is odd, 22 < x < 66, x ≡ 0 (mod 5) }

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Answer:

Option 1: triangle HFG is congruent to triangle KIJ

Step-by-step explanation:

F and I are same as they are on 90 degrees.

In figure 1, from I to K is the height of the triangle.

In figure 2, from F to H is the height of the triangle.

Therefore, IK is congruent to FH

In figure 1, I to J is the base of the triangle from 90 degrees.

In figure 2, F to G is the base of the triangle from 90 degrees.

Therefore, IJ is congruent to FG

Therefore, triangle HFG is congruent to triangle KIJ.

The first option is correct.

!!

Answer: 0.3679

Step-by-step explanation:

The formula for Poisson distribution  :-

[tex]P(x)=\dfrac{e^{-\lambda}\lambda^{x}}{x!}[/tex]

Let x be the number of breakdowns.

Given : The rate of breakdown per week :  0.5

Then , for 2 weeks period the number of breakdowns = [tex]\lambda=0.5\times2=1[/tex]

Then , the probability that there will be no breakdown on his car in the trip is given by :-

[tex]P(x)=\dfrac{e^{-1}1^{0}}{0!}=0.367879441171\approx0.3679[/tex]

Hence, the required probability : 0.3679

Answer:

2/13

Step-by-step explanation:

there are 4 jacks and 4 threes in a standard poker deck.

4+4 is 8

8/52=2/13

The probability of drawing either a Jack or a three from a standard deck of 52 cards is 2/13, because there are 8 such cards in a deck and the total number of cards in the deck is 52.

The question asks for the probability of drawing either a Jack or a three from a standard deck of 52 cards. To solve this, we need to count how many Jacks and threes there are in a deck. Since each suit (hearts, diamonds, clubs, and spades) includes one Jack and one three, there are 4 Jacks and 4 threes in a standard deck. Therefore, there are 8 cards that satisfy the condition (either a Jack or a three).

Since the total number of cards in the deck is 52, the probability of drawing either a Jack or a three is calculated as the number of favorable outcomes (drawing a Jack or a three) divided by the total number of outcomes (drawing any card from the 52-card deck). Thus, the probability is:

Probability = (Number of Jacks + Number of threes) / Total number of cards = (4 + 4) / 52 = 8 / 52 = 2 / 13

Therefore, the probability of drawing either a Jack or a three from a standard deck of 52 cards is 2/13.

Answer:

CF+PI=[tex]c_1e^{2x}+c_2e^{x}[/tex]+[tex]2e^{3t}[/tex]

Step-by-step explanation:

we have given y"-3y'=2y=[tex]4e^{3t}[/tex]

this differential equation solution have two part that CF and PI

CALCULATION OF CF :

[tex]m^2-3m+2=0[/tex]

[tex]m^2-2m-m+2=0[/tex]

[tex](m-1)(m-2)=0[/tex]

m=1 and m=2

so CF=[tex]c_1e^{2x}+c_2e^{x}[/tex]

CALCULATION OF PI :

PI =   [tex]\frac{4e^{3t}}{(m-1)(m-2)}[/tex]

at m= 3 in PI

[tex]PI=\frac{4e^{3t}}{2}=2e^{3t}[/tex]

so the complete solution is

CF+PI=[tex]c_1e^{2x}+c_2e^{x}[/tex]+[tex]2e^{3t}[/tex]

Answer:

[tex]y = 700 + 50x[/tex]

Step-by-step explanation:

Hello, great question. These types are questions are the beginning steps for learning more advanced Algebraic Equations.

If the company has a fixed cost (fixed being a keyword) of $700, then that cost will be a steady value before they even start to manufacture the bicycles. Afterwards they have to spend $50 on each bicycle they produce. Since we do not know the amount of bicycles that have been produced we can use the variable x to represent this.

[tex]y = 700 + 50x[/tex]

The equation above states that the company pays $700 plus $50 for every bike produced which comes out to a total of y.

A horizontal line is a line where all of the [tex]y[/tex] values are the same. In this case, [tex]\boxed{y=5}[/tex], so that is the equation.

A vertical line is where all of the [tex]x[/tex] values are the same.  Here, [tex]\boxed{x=7}[/tex], so that's the equation.

Answer:

see below

Step-by-step explanation:

A horizontal line has the same y value  and has a constant y value

y=5

A vertical line has the same x value  and has a constant x value

x=7

The function log311y can be expressed as the sum of two logarithms, log3(11) + log3(y), according to the product rule of logarithms.

The function log311y represents the logarithm base 3 of the product of the numbers 11 and y. Using the rules of logarithms, we can rewrite this expression as a sum of two logarithms.

According to the product rule of logarithms, the logarithm of a product is the sum of the logarithms of the component numbers. Applying this rule to our expression, log311y becomes:

log3(11) + log3(y)

This is the sum of the logarithm base 3 of 11 and the logarithm base 3 of y. In conclusion, the function log311y can be expressed as the sum of the separate logarithms: log3(11) + log3(y).

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The logarithm log3(11y) can be broken down into two simpler logarithms, log3(11) and log3(y), by using properties of logarithms. This is the sum of the two simpler logarithms.

To express the logarithm log3(11y) as the sum or difference of logarithms, we will utilize the properties of logs:

Applying these properties to the given logarithm, we find:

log3(11y) = log3(11) + log3(y)

Thus, the original logarithm has been expressed as a sum of two simpler logarithms.

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Answer: 0.33

Step-by-step explanation:

Let,

In this question we are using the Bayes' theorem,

where,

P(E1) = P(E2) = P(E3) = [tex]\frac{1}{3}[/tex]

As there is an equal probability assign for choosing a coin.

Given that,

it comes up heads

so, let A be the event that heads occurs

then,

P(A/E1) = 1

P(A/E2) = 0

P(A/E3) =  [tex]\frac{1}{2}[/tex]

Now, we have to calculate the probability that the opposite side of coin is tails.

that is,

P(E3/A) = ?

∴ P(E3/A) = [tex]\frac{P(E3)P(A/E3)}{P(E1)P(A/E1) + P(E2)P(A/E2) + P(E3)P(A/E3) }[/tex]

= [tex]\frac{(1/3)(1/2)}{(1/3)(1) + 0 + (1/2)(1/3)}[/tex]

= [tex]\frac{1}{6}[/tex] × [tex]\frac{6}{3}[/tex]

= [tex]\frac{1}{3}[/tex]

= 0.3333 ⇒ probability that the opposite face is tails.

Given a double-headed coin, a double-tailed coin, and a regular coin, the probability that the opposite face is tails after tossing a head is 33.33%, assuming we picked one coin randomly and tossed it to see a head.

The student is asking about a problem involving conditional probability, with the specific condition that one of the sides that came up is a head. We are given three coins: a double-headed coin, a double-tailed coin, and a regular coin. The aim is to calculate the probability that the opposite face is tails given that the tossed coin shows heads.

First, we need to consider the total number of heads that can come up when choosing any coin. This yields two heads from the double-headed coin, and one head from the regular coin, resulting in three possible heads. However, only the regular coin has a tail on the opposite side.

Consequently, the probability that the opposite face is tails given that a head has been tossed is 1 out of 3, or 33.33%.

The derivative of [tex]f(x,y,z)[/tex] at a point [tex]p_0=(x_0,y_0,z_0)[/tex] in the direction of a vector [tex]\vec a=a_x\,\vec\imath+a_y\,\vec\jmath+a_z\,\vec k[/tex] is

[tex]\nabla f(x_0,y_0,z_0)\cdot\dfrac{\vec a}{\|\vec a\|}[/tex]

We have

[tex]f(x,y,z)=3e^x\cos(yz)\implies\nabla f(x,y,z)=3e^x\cos(yz)\,\vec\imath-3ze^x\sin(yz)\,\vec\jmath-3ye^x\sin(yz)\,\vec k[/tex]

and

[tex]\vec a=-\vec\imath+2\,\vec\jmath+3\,\vec k\implies\|\vec a\|=\sqrt{(-1)^2+2^2+3^2}=\sqrt{14}[/tex]

Then the derivative at [tex]p_0[/tex] in the direction of [tex]\vec a[/tex] is

[tex]3\,\vec\imath\cdot\dfrac{-\vec\imath+2\,\vec\jmath+3\,\vec k}{\sqrt{14}}=-\dfrac3{\sqrt{14}}[/tex]

Answer with explanation:

Matrix A= (2 × 2) Matrix

Trace A= -9

Also,Determinant A= |A|=18

⇒Characteristics Polynomial is given by

Δ(A)=A² -A ×trace (A)+Determinant (A)

=A²+9 A+18

=(A+6)(A+3)

So, eigenvalues can be obtained by substituting :

 Δ(A)=0

(A+6)(A+3)=0

A= -6 ∧ A= -3

Two Eigenvalues are = -6, -3

Answer:

.188 pounds per day

Step-by-step explanation:

Given

Divide the amount of pounds gained by the total number of days

35/186 = .188

Answer

Approximately .188 pounds per day.

Answer:

Option b Tyree’s solution is inaccurate. In step 1, he substituted incorrectly.

Step-by-step explanation:

we know that

the formula to calculate the distance between two points is equal to

[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]

we have

[tex]A(-4,-2)\\B(-7,-7)[/tex]

step 1

substitute the values in the formula

[tex]d=\sqrt{(-7-(-2))^{2}+(-7-(-4))^{2}}[/tex]

step 2

Simplify

[tex]d=\sqrt{(-7+2)^{2}+(-7+4)^{2}}[/tex]

step 3

[tex]d=\sqrt{(-5)^{2}+(-3)^{2}}[/tex]

step 4

[tex]d=\sqrt{25+9}[/tex]

step 5

[tex]d=\sqrt{34}[/tex]

therefore

Tyree’s solution is inaccurate. In step 1, he substituted incorrectly.

Answer:

72 months approx.

Step-by-step explanation:

Monthly deposit = m = $140

r = 3.8% or 0.038

Amount needed in the account = A = $11300

The formula will be :

[tex]11300=140(\frac{(1+0.038/12)-1}{0.038/12} )[/tex]

[tex]11300=140(\frac{(1+0.038/12)-1}{0.003166})[/tex]

[tex]11300=44219.83((1.003166)^{m}-1)[/tex]

[tex]0.2555=(1.003166)^{m}-1[/tex]

[tex]1.2555=(1.003166)^{m}[/tex]

m=log1.2555/log1.003166

m =71.98 ≈ 72

Hence, it will take 72 months approx.

Final answer:

The probability that the second character will die given that the first character dies is 53.33%. This is known as conditional probability.

Explanation:

To find the probability that the second character will die given that the first character dies, we use the concept of conditional probability.

The formula for conditional probability is P(B|A) = P(A and B) / P(A), where P(B|A) is the probability of event B occurring given that event A has occurred, P(A and B) will be the probability of both events A and B occurring, and P(A) is the probability of event A occurring.

In this scenario, event A is the first character dying, and event B is the second character dying. The student has already stated there is a 70% chance that the first character will survive, which means there is a 30% (100% - 70%) chance that the first character will die.

They've also stated a 16% chance that both characters will die. Applying the formula gives us P(B|A) = P(A and B) / P(A) = 0.16 / 0.30 = 0.5333, or 53.33%.

Therefore, the probability that the second character will die given that the first character dies is 53.33%. This kind of probability is called conditional probability.

The net force is what remains of the pull when we subtract the friction force:

[tex]F = 64-36 = 28N[/tex]

Now, use the law

[tex]F=ma[/tex]

and solve it for the acceleration

[tex]a = \dfrac{F}{m}[/tex]

to get the result:

[tex]a = \dfrac{28}{14}=2[/tex]

Answer:

2 m/s²

Step-by-step explanation:

F = applied force in the horizontal direction = 64 N

f = frictional force acting between the box and the floor = 36 N

m = mass of the box = 14 kg

a = acceleration of the box = ?

Force equation along the horizontal direction is given as

F - f = ma

Inserting the values

64 - 36 = 14 a

28 = 14 a

a =  [tex]\frac{28}{14}[/tex]

a = 2 m/s²

[tex]HTT,HTH,HHH,TTT,THT,THH,TTH[/tex]

Answer:

A. number of decayed atoms = 73.197

Step-by-step explanation:

In order to find the answer we need to use the radioactive decay equation:

[tex]N(t)=N0*e^{kt}[/tex] where:

N0=initial radioactive atoms

t=time

k=radioactive decay constant

In our case, when t=0 we have 1,000,000 atoms, so:

[tex]1,000,000=N0*e^{k*0}[/tex]

[tex]1,000,000=N0[/tex]

Now we need to find 'k'. Using the provied information that after 365 days we have 973,635 radioactive atoms, we have:

[tex]973,635=1,000,000*e^{k*365}[/tex]

[tex]ln(973,635/1,000,000)/365=k[/tex]

[tex] -0.0000732=k[/tex]

A. atoms decayed in a day:

[tex]N(t)=1,000,000*e^{-0.0000732t}[/tex]

[tex]N(1)=1,000,000*e^{-0.0000732*1}[/tex]

[tex]N(1)= 999,926.803[/tex]

Number of atoms decayed in a day = 1,000,000 - 999,926.803 = 73.197

B. Because 'k' represents the probability of decay, then the probability that on a given day 51 radioactive atoms decayed is k=0.0000732.

Answer:

The mean is 5.309.

Step-by-step explanation:

Given group of data,

4.1, 8.9, 3.2, 1.9, 7.3, 6.3, 6.7, 8.6, 3.2, 2.3, 5.9,

Sum = 4.1+ 8.9 + 3.2 + 1.9 + 7.3 + 6.3 + 6.7 + 8.6 + 3.2 + 2.3 + 5.9 = 58.4,

Also, number of observations in the data = 11,

We know that,

[tex]Mean=\frac{\text{Sum of all observation}}{\text{Total observations}}[/tex]

Hence, the mean of given data = [tex]\frac{58.4}{11}=5.30909\approx 5.309[/tex]

The given sequence can be expressed as a sequence of partial sums by finding the differences between terms and adding them to the previous term. The closed formula for the nth partial sum is Sn = n/2(3n - 1), where Sn represents the nth partial sum.

To express the given sequence as a sequence of partial sums, we can find the differences between consecutive terms:

5 - 1 = 4

12 - 5 = 7

22 - 12 = 10

35 - 22 = 13

51 - 35 = 16

From these differences, we can observe that each term in the sequence is obtained by adding the difference to the previous term. Therefore, the sequence can be written as a sequence of partial sums:

1, 1+4, 1+4+7, 1+4+7+10, 1+4+7+10+13, ...

To find a closed formula for the nth partial sum, we can use the formula for the sum of an arithmetic series:

Sn = n/2(a1 + an), where Sn represents the nth partial sum, a1 is the first term, and an is the nth term.

For the given sequence, a1 = 1 and the difference between consecutive terms is 3, so the nth term can be represented as an = 1 + 3(n-1). Substituting these values into the formula, we get:

Sn = n/2(1 + 1 + 3(n-1)) = n/2(2 + 3(n-1)) = n/2(3n - 1).

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Answer:

The equation of the vertical line through (-6, -9) is 1x+0y=-6.

Step-by-step explanation:

The standard form of a line is

[tex]ax+by=c[/tex]

where a, b, and c are integers with no factor common to all three, and a>0.

If a vertical line passes through the point (a,b), then the equation of vertical line is x=a.

It is given that the vertical line passes through the point (-6,-9). Here a=-6 and b=-9, so the equation of the vertical line through (-6, -9) is

[tex]x=-6[/tex]

[tex]1x+0y=-6[/tex]

The standard form of the line is 1x+0y=-6. where the value of a,b c are 1, 0, -6 respectively.

Therefore the equation of the vertical line through (-6, -9) is 1x+0y=-6.

Answer:

  T(x, y) = T(0, -8)

Step-by-step explanation:

The first reflection can be represented as ...

  (x, y) ⇒ (-x, y)

__

The rotation about the origin is the transformation ...

  (x, y) ⇒ (-x, -y)

so the net effect of the first two transforms is ...

  (x, y) ⇒ (x, -y)

__

Then the reflection across y=4 alters the y-coordinate:

  (x, y) ⇒ (x, 8-y)

so the net effect of the three transforms is ...

  (x, y) ⇒ (x, 8+y)

__

In order to bring the figure back to place, we must translate it down 8 units using ...

  (x, y) ⇒ (x, y-8) . . . . net effect: (x, y) ⇒ (x, (8+y)-8) = (x, y)

The translation is by 0 units in the x-direction and -8 units in the y-direction.

Answer:

(A+B)(A+B)=A.A+B.A+A.B+B.B

Step-by-step explanation:

Given that matrices A and B are nxn matrices

We need to find (A+B)(A+B)

For understanding the multiplication of matrices let'take A is mxn and B is pxq matrices,we can multiple only when n=p,so our Ab matrices will be mxq.

We know that that in matrices AB is not equal to BA.

Now find  

(A+B)(A+B)=A.A+B.A+A.B+B.B

So from we can say that (A+B)(A+B) is not equal to A.A+2B.A+B.B because AB is not equal to BA in matrices.

So (A+B)(A+B)=A.A+B.A+A.B+B.B