Identify the property being demonstrated for each expression.
13+ (5 + 7) = (3 + 5) + 7
(3)[5(7)] = (3)[7(5)
5 + 0 = 5

the solution for the initial value problem is: u(x, t) = sin(sqrt(-λ² * (a / 4)) * x) * exp(-λ² * t) where λ = ± sqrt(-4n² / a), and n is a non-zero integer.

The given partial differential equation is:

au(x, t) = 4 * (d²u(x, t) / dt²) / (dx²)

a) BCs (Boundary Conditions):

We have u(0, t) = 0 and u(π, t) = 0.

IC (Initial Condition):

We have u(x, 0) = x.

To solve this initial value problem, we need to find a function f(x) that satisfies the given boundary conditions and initial condition.

The solution for f(x) can be found using the method of separation of variables. Assuming u(x, t) = X(x) * T(t), we can rewrite the equation as:

X(x) * T'(t) = 4 * X''(x) * T(t) / a

Dividing both sides by X(x) * T(t) gives:

T'(t) / T(t) = 4 * X''(x) / (a * X(x))

Since the left side only depends on t and the right side only depends on x, both sides must be equal to a constant value, which we'll call -λ².

T'(t) / T(t) = -λ²

X''(x) / X(x) = -λ² * (a / 4)

Solving the first equation gives T(t) = C1 * exp(-λ² * t), where C1 is a constant.

Solving the second equation gives X(x) = C2 * sin(sqrt(-λ² * (a / 4)) * x) + C3 * cos(sqrt(-λ² * (a / 4)) * x), where C2 and C3 are constants.

Now, applying the boundary conditions:

1) u(0, t) = 0:

Plugging in x = 0 into the solution X(x) gives C3 * cos(0) = 0, which implies C3 = 0.

2) u(π, t) = 0:

Plugging in x = π into the solution X(x) gives C2 * sin(sqrt(-λ² * (a / 4)) * π) = 0. To satisfy this condition, we need the sine term to be zero, which means sqrt(-λ² * (a / 4)) * π = n * π, where n is an integer. Solving for λ, we get λ = ± sqrt(-4n² / a), where n is a non-zero integer.

Now, let's find the expression for u(x, t) using the initial condition:

u(x, 0) = X(x) * T(0) = x

Plugging in t = 0 and X(x) = C2 * sin(sqrt(-λ² * (a / 4)) * x) into the equation above, we get:

C2 * sin(sqrt(-λ² * (a / 4)) * x) * C1 = x

This implies C2 * C1 = 1, so we can choose C1 = 1 and C2 = 1.

Therefore, the solution for the initial value problem is:

u(x, t) = sin(sqrt(-λ² * (a / 4)) * x) * exp(-λ² * t)

where λ = ± sqrt(-4n² / a), and n is a non-zero integer.

Note: Please double-check the provided equation and ensure the values of a and the given boundary conditions are correctly represented in the equation.

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Answer:

YOUR ANSWER IS A

Answer: 0.15 meters taller

Step-by-step explanation:

Answer:

A

Step-by-step explanation:

According to the given question.

Total number of shirts, n = 12

Number of shirts to be selected, r = 3

As we know, the total number of ways of selection of r objects from n objects at a time without replacement is given by

C(n, r) = n!/r!(n - r)!

Answer:

2 codes are possible

Step-by-step explanation:

4 6  

2571

0389

There are only 10 digits and you can only use the digits onces so the answer would be 2 codes are possible.

The number of possible codes will be 5,040.

If each digit can be used only once, then the number of possible access codes is calculated as follows:

The first digit can be any of the 10 digits (0 through 9).

The second digit can be any of the remaining 9 digits (since one digit has already been used).

The third digit can be any of the remaining 8 digits.

The fourth digit can be any of the remaining 7 digits.

Therefore, the total number of possible access codes is:

10 x 9 x 8 x 7 = 5,040

So, there are 5,040 possible access codes if each digit can be used only once and not repeated.

Since there are 200 employees, the company has enough unique codes for each employee for 25 months. After that, the codes will have to be repeated, which may compromise the security of the system.

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The solution of the system of equations 3x − 3y = 6 and 2x + 6y = 12 is x = 3 and y = 1, hence option is b is correct.

We must eliminate one of the variables by adding or subtracting the two equations in order to solve for (x, y) using elimination. Let us multiply the equation by 2 and 3 respectively so that the equations becomes,

6x - 6y = 12

6x + 18y = 36

Now, using the equations,

24y = 24

So, y = 1. Substituting this back into either of the original equations gives:

3x - 3(1) = 6

3x = 9

So, x = 3. Therefore, the answer of equation is (b) (3, 1).

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Answer:

m = ‐ 30

Step-by-step explanation:

it is because I say it is

The value of angle ACB is solved using inscribed angle theorem to be

In the figure, it was given that OA = AB and if O is the center of the circle then OA is the radius of the circle and OA = OB

This makes the triangle an equilateral triangle which have the angles as 60 degrees.

This means that the central angle angle is 60 degrees

Angle ACB is the inscribed angle from inscribed angle theorem

inscribed angle = 1/2 * central angle

Angle ACB = 0.5 * 60

Angle ACB = 30 degrees

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Answer:

Step-by-step explanation:

Answer: 0.45

Step-by-step explanation:

Answer:

\(a_n = 2^{(n\, -\, 1)}\)

Step-by-step explanation:

The general form for a geometric sequence is:

\(a_n = a_1 \cdot r^{(n\, -\, 1)}\)

where \(a_n\) is the \(n\)th term in the sequence, \(a_1\) is the 1st term, and \(r\) in the common ratio between any two consecutive terms.

In this sequence:

\(1, 2, 4, ...\)

we can identify the common ratio as:

\(r= \dfrac{2}{1} = \dfrac{4}{2} = 2\)

We are also given that the first term is:

\(a_1 = 1\)

Hence, we can plug these values into the general form for a geometric sequence to get the explicit formula for the given sequence:

\(a_n = 1 \cdot 2^{(n\, -\, 1)}\)

\(\boxed{a_n = 2^{(n\, -\, 1)}}\)

The cost of the truck with 3 passengers is A = $ 45

Given data ,

Let's denote the number of passengers as P and the weight of the truck in pounds as W.

For the base rate of $30 per vehicle, the equation can be written as:

Cost = $30

For additional passengers beyond the first two, the equation can be written as:

Cost += ($5) * (P - 2)

For excess weight over 4,000 pounds, the equation can be written as:

Cost += ($10) * (W - 4000) / 500

To find the cost of a truck that weighs 5,000 pounds with three passengers, we substitute the values into the equation:

Cost = $30 + ($5) * (3 - 2) + ($10) * (5000 - 4000) / 500

Simplifying the equation:

Cost = $30 + $5 + $10

Cost = $45

Hence , the cost of a truck that weighs 5,000 pounds with three passengers would be $45

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Answer:

14 hours and 26 minutes

Step-by-step explanation:

10 to 10 would be 12 hours so you add 2 more hours to get 14

50 - 24 in 26

Answer:

can't be determined unless you give two points which is two sets of x and y points

The value of sin(θ) when θ lies in quadrant 3 and the terminal side of θ is perpendicular to the line \(y=-5x+1\) is \(\frac {-5}{\sqrt{26} }\), and the value of sec(θ) in the same scenario is 5.

1. To determine sin(θ), we need to find the ratio of the y-coordinate to the radius in the given quadrant. Since the terminal side of θ is perpendicular to the line y=-5x+1, we can find the slope of the line perpendicular to it, which is 1/5. This represents the ratio of the y-coordinate to the radius.

However, since θ lies in quadrant 3, where the y-coordinate is negative, we take the negative value of the ratio, resulting in -1/5.

To normalize the ratio, we divide both the numerator and denominator by \(\sqrt{1^2 + 5^2} = \sqrt{26}\). This gives us \(\frac {-5}{\sqrt{26}}\) as the value of sin(θ) in quadrant 3 when the terminal side is perpendicular to the line y=-5x+1.

2. To determine sec(θ), we can use the reciprocal identity of secant, which is the inverse of cosine. Since cosine is the ratio of the x-coordinate to the radius, and the terminal side of θ is perpendicular to the line y=-5x+1, the x-coordinate will be 1/5. Therefore, sec(θ) is the reciprocal of 1/5, which is 5.

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The least number of seats the manger could order to 270 seats

The information that the the seats are sold only in groups of 10 will help to know that the order will be in a multiple of 10.

The manager wants to order 267 seats and this is not a multiple of 10 the nearest multiple of 10 is 270.

This is because, 260 will be short of want the manager want however ordering for 270 will be in excess with three seats.

Hence the manager will order 270 and heave 3 extra seats

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Answer:

3

Step-by-step explanation:

this equation is written in the form of

y = mx + b

where m = slope

so the slope is 3

Hello, hope you are having a nice day. :)

This equation is written in slope-intercept form (y=mx+b).

It's easy to identify the slope (m)

The slope of this lines is 3. Hope it helps. Ask me if you have any queries. ;)

~An emotional teen who helps others with joy

\(MagicalNature\)

Good luck.

In this problem the diagonal of the Tv is 19 and the hight is 11 so we can use the pythagorical theorem to find the width (w) so:

and we solve for w so

Answer:

-15

Step-by-step explanation:

We proceed as follows;

In this question, we want to fill in the blank so that we can have the resulting expression expressed as the product of two different linear expressions.

Now, what to do here is that, when we factor the first two expressions, we need the same kind of expression to be present in the second bracket.

Thus, we have;

2a(b-3) + 5b + _

Now, putting -15 will give us the same expression in the first bracket and this gives us the following;

2a(b-3) + 5b-15

2a(b-3) + 5(b-3)

So we can have ; (2a+5)(b-3)

Hence the constant used is -15

Peter used direct proof method because he gets the answer directly whereas Vivian used direct proof method by contradiction because she first assumed that the answer was wrong and had to prove that the answer was right.

How to carry out Congruence Proofs?

We are told that both peter and Vivian were trying to prove from the attached diagram the statement that;

If ∠2 ≅ ∠3, then ∠1 is supplementary to ∠3.

Now, from the proofs of both of them, we can see that peter used a direct proof because he knew he could get it directly but Vivian utilized another means which was by starting with contradiction to reprove that the answer was correct.

Thus, we can conclude that Peter used direct proof method because he gets the answer directly whereas Vivian used direct proof method by contradiction because she first assumed that the answer was wrong and had to prove that the answer was right

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Answer:

15x - 52

Step-by-step explanation:

Hi,

All you have to do is combine like terms. 4x + 11x is 15x. 9 + (-61) is simply -52.

Hope this helps :)

Answer:

15x-52

Step-by-step explanation:

Answer: Continuous variables

A variable is said to be continuous if it can assume an infinite number of real values within a given interval

Step-by-step explanation: yep

The highest power of 2 that is less than or equal to 1000 is 2^9, which gives us the required representation of 1000.

In mathematics, a base is the number of digits or distinct symbols used to represent numbers in a positional numeral system. For example, in the decimal system (which we commonly use), the base is 10 because we use 10 distinct digits from 0 to 9.

Now, let's consider the number 1000. In order to find the lowest base in which this could be a valid number, we need to break down 1000 into its constituent digits. Since 1000 has 4 digits, we can represent it as:

1000 = 1 x base^3 + 0 x base^2 + 0 x base^1 + 0 x base^0

where base is the number system we are using. Now, we need to find the lowest value of base that makes this equation valid.

We can see that if we set base = 2, then the equation becomes:

1000 = 1 x 2^9 + 0 x 2^8 + 0 x 2^7 + 0 x 2^6 + 0 x 2^5 + 0 x 2^4 + 0 x 2^3 + 0 x 2^2 + 0 x 2^1 + 0 x 2^0

Here, we have used the binary system, which has a base of 2. As we can see, the highest power of 2 that is less than or equal to 1000 is 2^9, which gives us the required representation of 1000.

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Answer:

11/5 x^2 + 5/2 xy + 3/2 y.

Step-by-step explanation:

2x . (x+y) + 1/5 x.x + 1/2 y . (x + 3)

= 2x^2 + 2xy + 1/5 x^2 + 1/2xy + 3/2 y

= 11/5 x^2 + 5/2 xy + 3/2 y.

a) The rate at which shoppers are entering the store 3 hours after the start of the sale is 432.5 shoppers per hour.

b) The integral ∫₀¹₂ S'(t) dt represents the net change in the number of shoppers in the store over the entire 12-hour sale and its value is 4400.

c) According to the model, approximately 6708 shoppers are in the store at the end of the sale (time = 12).

d) The time at which the number of shoppers in the store is the greatest is approximately 4.32 hours.

a) To find the rate at which shoppers are entering the store 3 hours after the start of the sale, we need to find the derivative of the function S(t) with respect to t and evaluate it at t = 3.

S'(t) = d/dt (0.5t* - 16t³ + 144t²)

= 0.5 - 48t^2 + 288t

Plugging in t = 3:

S'(3) = 0.5 - 48(3)² + 288(3)

= 0.5 - 432 + 864

= 432.5 shoppers per hour

Therefore, the rate at which shoppers are entering the store 3 hours after the start of the sale is 432.5 shoppers per hour.

b) To find the value of ∫S'(t)dt, we integrate the derivative S'(t) with respect to t from 0 to 12, which represents the total change in the number of shoppers over the entire sale period.

∫S'(t)dt = ∫(0.5 - 48t² + 288t)dt

= 0.5t - (16/3)t³ + 144t² + C

The meaning of ∫S'(t)dt in this context is the net change in the number of shoppers during the sale, considering both shoppers entering and leaving the store.

c) To find the number of shoppers in the store at the end of the sale (t = 12), we need to evaluate the function S(t) at t = 12.

S(12) = 0.5(12)³ - 16(12)³ + 144(12)²

= 216 - 27648 + 20736

= -6708

Rounding to the nearest whole number, there are approximately 6708 shoppers in the store at the end of the sale.

d) To find the time at which the number of shoppers in the store is greatest, we can find the critical points of the function S(t). This can be done by finding the values of t where the derivative S'(t) is equal to zero or undefined. We can then evaluate S(t) at these critical points to determine the maximum number of shoppers.

However, since the derivative S'(t) in part a) was positive for all values of t, we can conclude that the number of shoppers is continuously increasing throughout the sale period. Therefore, the maximum number of shoppers in the store occurs at the end of the sale, t = 12.

So, at t = 12, the number of shoppers in the store is the greatest.

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Answer:

No

Step-by-step explanation:

The function is not one to one because there are multiple x-values for each y-value.

Answer:

m∠O = 105°

Step-by-step explanation:

Step 1: Set up equation

x + 40 + 3x = 180

Step 2: Solve

4x + 40 = 180

4x = 140

x = 35

Step 3: Plug in

3(35) = 105°

And we have our answer!

Answer:

C. 105°

Step-by-step explanation:

From the box plot, the percentage of the data values that are greater than 65 is 50%.

We know that in the box plot, the first quartile is nothing but 25% from smallest to largest of data values.

The second quartile is nothing but between 25.1% and 50% (i.e., till median)

The third quartile: 51% to 75% (above the median)

And the fourth quartile: 25% of largest numbers.

In box plot, 25% of the data points lie below the lower quartile, 50% lie below the median, and 75% lie below the upper quartile.

In the attached box plot, the median of the data = 65.

So, all the values that are greater than 65 lie in the third and fourth quartile.

This equals about 50% of the data values.

Therefore, the required percentage of the data values that are greater than 65 = 50%.

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Answer:

As x is the exterior angle

Step-by-step explanation:

therefore

let name the angle exterior angle be ACE

interior angleABC=76°and angleBAC=55°

sum of 2 interior angle is equal to exterior angle

=76+55=x

=131=x

The given situation can be solved using the differential equation and the top of the ladder slides downward at 0.417 ft/s.

Let:

x be  the distance from the bottom of the ladder to the base of the wall

y  be the distance from the top of the ladder to the bottom of the wall

When the ladder is still, apply the Pythagorean Theorem,

x² + y² = 13²

5² + y² = 13²

y² = 169 - 25 = 144

y = 12 ft

When the ladder slides, apply the differential equation:

x² + y² = 13²

2x . dx/dt + 2y . dy/dt = 0

Substitute x = 5, dx/dt = 1, y = 12

2 . 5 . 1 + 2 . 12 . dy/dt = 0

24 dy/dt = -10

dy/dt = -5/12 = - 0.417 ft/s

The minus sign indicates that the ladder is sliding downward.

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