Discrete Math I already saw the responses to this question but I want another way. Please don't copy and past it! Please show all work.
Dijkstra's algorithm　to find the length of the shortest path between vertices　a and z in the following weighted graph.　Please show your distinguished vertex set and distances for each iteration.

The probability of  P(A) = 15/16, P(B) = 1/3, P(BIA) = 1/3, and P(CAB) = 0.

The probability of an occurrence is a number used in science to describe how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.

(a) P(A) = P(1) + P(2) + P(3) + P(4) = 1/2 + 1/4 + 1/8 + 1/16 = 15/16
(b) P(B) = P(2) + P(4) + P(6) + ... = 1/4 + 1/16 + 1/64 + ... = 1/3
(c) P(BIA) = P(B ∩ A) / P(A) = (P(2) + P(4)) / P(A) = (1/4 + 1/16) / (15/16) = 5/15 = 1/3
(d) P(CAB) = P(C ∩ A ∩ B) / P(A ∩ B) = 0 / P(A ∩ B) = 0

Therefore, the probabilities of the events are: P(A) = 15/16, P(B) = 1/3, P(BIA) = 1/3, and P(CAB) = 0.

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The surface area of the prism is 224 ft².

Having identical endpoints in three dimensions, a prism is a solid object. This combination consists of flat faces, identical bases, and equal cross-sections.

We are given a figure of a prism having a square base.

We know that

Surface Area of Prism (A) = 2a² + 4ah

We are given the following information:

a = 4 ft

h = 12 ft

On substituting these values in the formula, we get

⇒A = 2(4)² + 4(4)(12)

⇒A = 2(16) + (16)(12)

⇒A = 32 + 192

⇒A = 224 ft²

Hence, the surface area of the prism is 224 ft².

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Since there are multiple questions so the question answered above is attached below

Answer:

x-coordinate is 3

Step-by-step explanation:

Q has y-coordinate of -4 => the distance from origin to y-coordinate is 4 units, which is one leg of the right triangle

Pythagorean theorem states that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse.

c^2 = a^2 + b^2

with c = 5, a = 4

5^2 = 4^2 + b^2

b^2 = 25 - 16 = 9

b = √9 = 3

so Q has coordinates (3,-4)

We are being informed that the sum of the two numbers is 12 and their difference is 6.

Thus, the two numbers whose sum is equal to 12 and their difference is equal to 6 are 9 and 3 respectively.

Let those two unknown numbers be (a) and (b)

a + b = 12

Their difference;

i.e.

a - b = 6

So, we can equate the two equations together:

i.e.

a + b = 12     --- (1)

a - b = 6        --- (2)

From equation (1); Let a be:

a = 12 - b

Now, Let's replace the value of (a) into equation (2)

12 - b - b = 6

12 - 2b = 6

-2b = 6 - 12

-2b = -6

2b = 6

b = 6/2

b = 3

If b = 3;

Then from equation, we have:

a + b = 12

replace b with 3 from the above equation, we have:

a + 3 = 12

a = 12 - 3

a = 9

Therefore, we can conclude that the sum of the two numbers are 9 and 3. respectively

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Using point-slope form, the coordinate of D(2, X) is 2 units away from point X(1, 1).

To find the coordinates of point D, we need to know the location of point X on the line YZ. We can use the slope-intercept form of the equation of a line to find the equation of the line passing through points Y and Z:

slope m = (y2 - y1) / (x2 - x1) = (0 - 2) / (3 - 2) = -2

Using the point-slope form of the equation of a line with point Y(2, 2) and slope m = -2:

y - y1 = m(x - x1)

y - 2 = -2(x - 2)

y - 2 = -2x + 4

y = -2x + 6

Now we can substitute x = 2 into the equation of the line to find the y-coordinate of point D:

y = -2x + 6

y = -2(2) + 6

y = 2

Therefore, the coordinates of point D are (2, 2).

So, D(2, 2) is the point that lies on line YZ and is 2 units away from point X(1, 1).

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Given that it includes a quadratic term ([tex]b^{2}[/tex]), this formula reduces to 4b(b) = 0, which is a complex quantity.

A function with the formula a=0, b=1, and c=2 is known as a quadratic function. The function is known as the quadratic term (abbreviated as ax2), the linear term (abbreviated as bx), and the constant term (abbreviated as c).

The quadratic formula may be used to determine a parabola's axis of symmetry, the amount of real zeros in the quadratic equation, and the noughts of any parabola. It also produces the zeros of the any parabola.

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The expression in terms of m and n is m^2+2/3n-3.

To rewrite the given expression in terms of m and n, we need to use the properties of logarithms to simplify the expression. The properties of logarithms that we will use are:

- log(ab) = log(a) + log(b)
- log(a/b) = log(a) - log(b)
- log(a^b) = b*log(a)

Using these properties, we can rewrite the given expression as follows:

\( \log \left(\frac{\left(M^{\log (M)} \cdot 10^{\log (N)}\right)}{1000 \sqrt[3]{N}}\right) = \log \left(M^{\log (M)} \cdot 10^{\log (N)}\right) - \log \left(1000 \sqrt[3]{N}\right) \)

\( = \log \left(M^{\log (M)}\right) + \log \left(10^{\log (N)}\right) - \log \left(1000\right) - \log \left(\sqrt[3]{N}\right) \)

\( = \log (M) \cdot \log (M) + \log (N) \cdot \log (10) - \log (10^3) - \frac{1}{3} \cdot \log (N) \)

Now, we can substitute m = log(M) and n = log(N) into the expression to get:

\( = m^2 + n - 3 - \frac{1}{3}n \)

\( = m^2 + \frac{2}{3}n - 3 \)

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Zero is an even number, but neither a prime number nor a composite number.

Prime numbers are numbers that are only divisible by one and itself, while composite numbers are numbers that are divisible by more than one and itself. Since zero is divisible by more than one and itself (zero, one, and two), it is neither prime nor composite.

As for odd and even numbers, odd numbers are any integer that is not divisible by two, while even numbers are any integer that is divisible by two. Since zero is divisible by two, it is an even number.

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If Lindsay he mows the lawn in 10 hours and with his brother Grant he does it in 2 hours, Grant would mow the lawn alone in 2.5 hours.

To find out how long it would take Grant to mow the lawn alone, we can use the formula for combined work:

1/t₁ + 1/t₂ = 1/ttotal

Where t₁ is the time it takes Lindsay to mow the lawn alone, t₂ is the time it takes Grant to mow the lawn alone, and ttotal is the time it takes both of them to mow the lawn together.

Plugging in the given values:

1/10 + 1/t₂ = 1/2

Multiplying both sides by 20t₂ to clear the fractions:

2t₂ + 20 = 10t₂

Rearranging and simplifying:
8t₂ = 20
t₂ = 20/8
t₂ = 2.5

So it would take Grant 2.5 hours to mow the lawn alone.

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Answer:

Let's call the weight of the book that was removed "x".

The total weight of the twelve books can be represented as 12 times the average weight of 2.75 pounds:

12(2.75) = 33

After the book is removed, there are only 11 books left, and their total weight can be represented as 11 times the new average weight of 2.70 pounds:

11(2.70) = 29.7

We can set up an equation using these two expressions and the weight of the book that was removed:

33 - x = 29.7

Solving for x, we get:

x = 33 - 29.7

x = 3.3

Therefore, the weight of the book that was removed was 3.3 pounds.

Answer: First we have to check the information they are giving in the problem. do we can see that the photocopier can copy 4 pages every 2 seconds.

so with this info we can calculate the ratio, or the number of copies that can be done in one second:

So we made a rule of 3 to solve that

So, if the photocopier can print 2 copys every second, the we can calculate the the time that takes to print 120 pages:

solving this rule of 3:

so is going to take 60 seconds to copy 120 pages.

Answer:

write relation between work and power

The solutions for the absolute value equation are x = 0 and x = -4.

To solve the absolute value equation |(5x + 10)/(2)| = 5, we need to remove the absolute value bars and create two separate equations, one positive and one negative. Then we can solve for x in each equation.

First, let's remove the absolute value bars and create two separate equations:

(5x + 10)/2 = 5 and (5x + 10)/2 = -5

Now we can solve for x in each equation:

(5x + 10)/2 = 5

5x + 10 = 10

5x = 0

x = 0

And:

(5x + 10)/2 = -5

5x + 10 = -10

5x = -20

x = -4

So the solutions to the equation |(5x + 10)/(2)| = 5 are x = 0 and x = -4.

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Answer: The amount is $43492.35 and the interest is $10992.35.

Step-by-step explanation:

To find amount we use formula:

A = P(1+r/n)^nt

A = total amount

P = principal or amount of money deposited,

r = annual interest rate

n = number of times compounded per year

t = time in years

In this example we have

P=32,000 , R=6% , N = 1 T= 5 YEARS

After plugging the given information we have

A=32000(1+0.06/1)^1.5

A=32500*1.06^5

A=32500*1.338226

A=$43492.35

To find interest we use formula A=P+I , since A= 43492.35  P =32500  we have:

A=P+I

$43492.35 = 32500+I

I=$43492.35-32500

I=$10992.35

The value of \( k \) that makes \( \vec{a} \) and \( \vec{b} \) orthogonal is \( k=1 \).

Two vectors are orthogonal if their dot product is zero. The dot product of two vectors \( \vec{a}=\langle a_1,a_2\rangle \) and \( \vec{b}=\langle b_1,b_2\rangle \) is given by \( \vec{a}\cdot\vec{b}=a_1b_1+a_2b_2 \).

In this case, we have \( \vec{a}=\langle 1,-3\rangle \) and \( \vec{b}=\langle 3, k\rangle \). So the dot product is:

\( \vec{a}\cdot\vec{b}=(1)(3)+(-3)(k)=3-3k \)

We want this dot product to be zero, so we can set it equal to zero and solve for \( k \):

\( 3-3k=0 \)

\( 3k=3 \)

\( k=1 \)

Therefore, the value of \( k \) that makes \( \vec{a} \) and \( \vec{b} \) orthogonal is \( k=1 \).

Answer: \( \boxed{k=1} \).

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To make the remainder 0, the final value of k must be -1.

The question asks to find all values of k so that when x^(3)-k^(2)x+k+2 is divided by x-2, the remainder is 0.

To find the values of k, we need to use synthetic division.

First, we can write the equation as follows:

Now we can continue with the synthetic division:

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The system performance is not affected by the thermodynamic losses of 1.5 °C

In a brine circulation MSF system, thermodynamic losses occur when heat is lost from the system, resulting in a decrease in the efficiency of the system. To compare the system performance if the thermodynamic losses are equal to 1.5 °C, we need to calculate the performance ratio (PR) of the system with and without the thermodynamic losses.

Without thermodynamic losses:
PR = (Production capacity) / (Heating steam flow rate)
= (1 kg/s) / ((116 °C - 40 °C) / (106 °C - 30 °C))
= 1 / (76 / 76)
= 1

With thermodynamic losses of 1.5 °C:
PR = (Production capacity) / (Heating steam flow rate)
= (1 kg/s) / ((116 °C - 40 °C - 1.5 °C) / (106 °C - 30 °C - 1.5 °C))
= 1 / (74.5 / 74.5)
= 1

The performance ratio of the system remains the same with and without the thermodynamic losses of 1.5 °C. This means that the system performance is not affected by the thermodynamic losses of 1.5 °C. However, it is important to note that thermodynamic losses can have a significant impact on the system performance if they are larger.

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1 at each vertex

An exterior angle of a polygon is an angle formed by a side of the polygon and the extension of an adjacent side. In the case of a triangle, each exterior angle is formed by one of the triangle's sides and the extension of an adjacent side.

When an exterior angle is formed at a vertex of a polygon, the measure of the exterior angle is equal to the sum of the measures of the two interior angles adjacent to it. In the case of a triangle, the sum of the measures of the two interior angles adjacent to the exterior angle is always 180 degrees (which is the sum of the measures of all three interior angles of a triangle).

Since each exterior angle of a triangle is formed by two interior angles, and the sum of the measures of those interior angles is always 180 degrees, there can only be one exterior angle at each vertex of a triangle. Therefore, a triangle has one exterior angle at each vertex.

The augmented matrix of a system of linear equations is given: , determine value(s) of k if 2 k - 2 if  the system has no solution there are no values of k that will make the system inconsistent.  The system has one solution for all values of k except k = 4.The system has infinitely many solutions if k = 0, a unique solution if k ≠ 0 and k ≠ 4, and no solutions if k = 4

To determine the value(s) of k for each case, we will perform row reduction on the augmented matrix and analyze the resulting echelon form.

1 3 1 | 0

2 k - 2 | 0

R2 - 2R1 -> R2

1 3 1 | 0

0 k - 4 | 0

Case (a): If k = 4, then the second row becomes all zeros except for the last entry, which means we have an inconsistent system with no solutions. If k ≠ 4, then we can use back-substitution to find the solution(s):

k - 4 = 0 => k = 4

Since this contradicts our assumption, there are no values of k that will make the system inconsistent.

Case (b): If k ≠ 4, then the echelon form shows that we have a leading coefficient in each row and the system has a unique solution. If k = 4, then the second row becomes all zeros except for the last entry, which means we have an inconsistent system with no solutions.

Case (c): If k = 0, then the second row reduces to 0 = 0, which means we have a free variable and infinitely many solutions. If k ≠ 0 and k ≠ 4, then the echelon form shows that we have a leading coefficient in each row and the system has a unique solution. If k = 4, then the system is inconsistent with no solutions.

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Answer:

You are distributing (multiplying) the three to all the terms in the parentheses.

Start:

3(x-2) = 4x + 2

Next:

3x - 6 = 4x +2

After that simplify:

-8 = x

Hope this helps!

Answer:

Step-by-step explanation:

Now we have to,

→ Find the required value of x.

The property we use,

→ Distributive property.

The equation is,

→ 3(x - 2) = 4x + 2

Then the value of x will be,

→ 3(x - 2) = 4x + 2

→ 3(x) - 3(2) = 4x + 2

→ 3x - 6 = 4x + 2

→ 3x - 4x = 2 + 6

→ -x = 8

→ [ x = -8 ]

Hence, the value of x is -8.

[tex]~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$5000\\ r=rate\to 2.5\%\to \frac{2.5}{100}\dotfill &0.025\\ t=years\dotfill &7 \end{cases} \\\\\\ A = 5000e^{0.025\cdot 7} \implies A=5000e^{0.175} A \approx 5956.23[/tex]

Answer:

The formula for calculating the value of an investment that grows continuously is:

A = Pe^(rt)

Where:

A is the final amount

P is the principal amount

e is Euler's number (approximately 2.71828)

r is the annual interest rate (as a decimal)

t is the time in years

In this case, P = 5000, r = 0.025 (2.5% expressed as a decimal), and t = 7. Plugging these values into the formula, we get:

A = 5000 * e^(0.025*7) = 5000 * e^0.175 = 5000 * 1.19128 = 5956.40

Therefore, Ryan's investment will be worth $5,956.40 after 7 years. Rounded to the nearest cent, the answer is $5,956.40.

The scale factor is 11.

Scale factor is a numerical value used to proportionally enlarge or reduce a size of an object or image. It is also used to compare two similar shapes or objects. When the scale factor is greater than 1, it indicates an increase in size and when the scale factor is less than 1, it indicates a decrease in size. Scale factors are usually expressed as a ratio, such as 1:2, which means that the object has been doubled in size.

To calculate it, divide the first number by the second number and multiply by the third number: (35/42) x 11 = 10.714. Therefore, the scale factor is 11.

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The zero vector [0, 0, 0], the length would be √(0^2 + 0^2 + 0^2) = 0.

Examples of 3 element vectors include:
- [1, 2, 3]
- [-1, 0, 5]
- [3, -2, 1]
- [0, 0, 0]
- [4, 4, 4]
- [-2, -2, -2]

There are infinitely many 3 element vectors that have a length of 0. A vector with a length of 0 is called the zero vector, and it has all of its components equal to 0. In the case of a 3 element vector, the zero vector would be [0, 0, 0].

Examples of 3 element vectors include:
- [1, 2, 3]
- [-1, 0, 5]
- [3, -2, 1]
- [0, 0, 0]
- [4, 4, 4]
- [-2, -2, -2]

Remember that the length of a vector is calculated using the formula √(x1^2 + x2^2 + x3^2), where x1, x2, and x3 are the components of the vector. So, for the zero vector [0, 0, 0], the length would be √(0^2 + 0^2 + 0^2) = 0.

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[tex]y =ax+b[/tex]

[tex]x = 0 \iff a(0) + b = 3 \implies b = 3[/tex]

[tex]x = 1 \iff a(1) + 3 = 5 \implies a = 2\\[/tex]

[tex]\implies \bf y = 2x + 3[/tex]

___________

[tex]x = 2 \iff 2(2) + 3 = 4 + 3 = 7[/tex]

[tex]x = 3 \iff 2(3) + 3 = 6 + 3 = 9[/tex]

The measure of angle D is \( 125^{\circ} 88^{\prime} 5^{\prime \prime} \).

The interior angles of a quadrilateral are measured as A \( 51^{\circ} 22^{\prime} 30^{\prime \prime} \), B \( 105^{\circ} 38^{\prime} 48^{\prime \prime} \), C \( 78^{\circ} 10^{\prime} 37^{\prime \prime} \), and D \( 124^{\prime} 46^{\prime \prime} \).

To find the measure of angle D, we can use the fact that the sum of the interior angles of a quadrilateral is 360 degrees.

Add the measures of angles A, B, and C:
\( 51^{\circ} 22^{\prime} 30^{\prime \prime} \) + \( 105^{\circ} 38^{\prime} 48^{\prime \prime} \) + \( 78^{\circ} 10^{\prime} 37^{\prime \prime} \) = \( 234^{\circ} 71^{\prime} 55^{\prime \prime} \)

Subtract the sum of angles A, B, and C from 360 degrees to find the measure of angle D:
360^{\circ} - \( 234^{\circ} 71^{\prime} 55^{\prime \prime} \) = \( 125^{\circ} 88^{\prime} 5^{\prime \prime} \)

Therefore, the measure of angle D is \( 125^{\circ} 88^{\prime} 5^{\prime \prime} \).

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Answer: x=-6

Step-by-step explanation:

8x+14=4x-10

-4x       -4x

4x+14=-10

-14       -14

4x=-24

/4    /4

x=-6

Given:-

[tex] \: [/tex]

Solution:-

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━

hope it helps ⸙

The slope of the line in simplest form is 1/3.

In Mathematics, the slope of any straight line can be determined by using this mathematical equation;

Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = rise/run

Substituting the given data points into the slope formula, we have the following;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (-2 + 5)/(9 - 0)

Slope, m = 3/9

Slope, m = 1/3

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Answer:

First Option, (A) $29,922.

Step-by-step explanation:

To calculate the future value of Heather's savings account after 5 years, we can use the formula for compound interest:

FV = P((1+i)^n - 1)/i

where:

FV = future value

P = principal (the initial amount Heather deposits)

i = interest rate per period (monthly in this case)

n = number of periods (months in this case)

P = $75 (the amount of Heather's monthly payments)

i = 3.6% / 12 = 0.003 (the monthly interest rate, calculated by dividing the annual interest rate by 12)

n = 5 x 12 = 60 (the total number of months in 5 years)

Substituting these values into the formula, we get:

FV = $75((1+0.003)^60 - 1)/0.003

FV = $75(1.21879)/0.003

FV = $29,922.02 (rounded to the nearest cent)

Therefore, Heather will have approximately $29,922.02 in her savings account after the 5 year period. The answer is option A: $29,922.