Maximize p = 6x + 9y + 3. 3z + 12w subject to

(a) 1. 2x + y + z + w ≤ 121. 5

(b) 2. 2x + y − z − w ≥ 30

(c) 1. 2x + y + z + 1. 2w ≥ 31. 5

(d) x ≥ 0, y ≥ 0, z ≥ 0, w ≥ 0.

Round all answers to two decimal places

The expression for the perimeter of the triangle is (15h - 6k + 12m) centimeters.

In geometry, a triangle is formed by three line segments crossing at three non-collinear points. The three line segments that make up the triangle are known as its sides, and its three points of intersection are known as its vertices.

A triangle is a three-sided polygon that has three sides and three angles and is created when three line segments cross each other at three non-collinear places.

The whole length of the sides makes up a triangle's perimeter.

Therefore, the expression for the perimeter of the given triangle in centimeters is:

(5h-4k) + (10h+7m) + (5m-2k)

Simplifying by combining like terms, we get:

15h - 6k + 5m + 7m

which can be further simplified as:

15h - 6k + 12m

Therefore, the expression for the perimeter of the triangle is (15h - 6k + 12m) centimeters.

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The expression for the perimeter of the triangle is (15h - 6k + 12m) centimeters.

Three line segments must cross at three non-collinear points in order to make a triangle in geometry. The triangle's three line segments are referred to as its sides, and its three places of intersection are referred to as its vertices.

Three line segments crossing each other at three non-collinear points result in a triangle, which is a three-sided shape with three sides and three angles.

A triangle's perimeter is determined by adding the lengths of its edges.

As a result, the formula for the triangle's perimeter in millimeters is as follows:

(5h-4k) + (10h+7m) + (5m-2k)

Simplifying by combining like terms, we get:

15h - 6k + 5m + 7m

which can be further simplified as:

15h - 6k + 12m

Therefore, (15h - 6k + 12m) centimeters is the formula for the triangle's perimeter.

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The price of a 14 minute phone call is $4.90.

We can use the given information to set up a proportion between the price and the duration of a phone call:

price of 9-minute call / 9 = $3.15 / 1

We can then solve for the price of a 1-minute call:

price of 1-minute call = ($3.15 / 1) / 9 = $0.35

Now that we know the price of a 1-minute call, we can use it to find the price of a 14-minute call:

price of 14-minute call = ($0.35 / 1) * 14 = $4.90

Therefore, the price of a 14 minute phone call is $4.90.

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If you only have $14 to spend, the park you would attend would depend on the cost of admission and the price of the rides. Assuming that the rides are the same quality at both parks, the decision would come down to which park offers the most affordable pricing.

If Park A charges a $10 admission fee and $2 per ride, you would be able to afford 2 rides with your $14 budget.

If Park B charges a $5 admission fee and $3 per ride, you would be able to afford 3 rides with your $14 budget.

Therefore, if the pricing is as described above, it would be more cost-effective to attend Park B as you would be able to enjoy an additional ride with your budget. However, if the pricing is different or if there are other factors to consider such as distance, location, or amenities, then the decision might change.

Answer:

It's difficult to provide a definitive answer without knowing the specific parks you are considering, but I can offer some general advice on how to choose which park to attend.

First, consider the admission price for each park. If one park has a significantly higher admission price than the other, it may not be worth attending if you only have $14 to spend.

Next, consider the cost of the rides and other attractions at each park. If one park has more expensive rides or requires you to purchase tickets for each ride separately, you may not be able to enjoy as many attractions with your $14 budget.

Finally, consider any other expenses you may incur, such as food, parking, or souvenirs. If one park has significantly higher prices for these items, it may not be the best choice if you are on a tight budget.

Based on these factors, you may want to choose the park that offers the best value for your $14 budget. This could mean choosing a park with lower admission prices and more affordable rides, or a park that offers special deals or discounts that allow you to stretch your budget further. Ultimately, the park you choose will depend on your personal preferences and priorities, as well as the options available in your area.

Step-by-step explanation:

Answer: f(-2)=0; yes, the binomial is a factor of the polynomial.

f(-2)≠0; the binomial is not a factor of the polynomi f(2)=0; yes, the binomial is a factor of the polynomial.

f(2)≠0; the binomial is not a factor of the polynomial.

Step-by-step explanation:

Anthοny is accurate, and had he cοlοred a quarter οf Jacοb's final circle rather than a quarter οf his οwn, they wοuld have cοlοred 12 cοmplete circles.

A circle is a spherical shape that lacks bοrders and edges. In geοmetry, a circle is indeed a clοsed, curved shape having twο dimensiοns.

Tο start, let's cοunt the tοtal number οf circles Anthοny οr Jacοb cοlοred befοre tο lunch:

Anthοny cοlοred 6.25 circles, οr 6 and 1/4 circles.

Jacοb cοlοred 5.75 circles, οr 5 and 3/4 circles.

They cοmbined tο cοlοr a tοtal οf 12 circles (6.25 + 5.75).

Let's nοw imagine that Anthοny cοlοred a quarter οf Jacοb's final circle rather than a quarter οf his οwn. Jacοb cοlοred 5 and 3/4 + 1/4 = 6 circles, while Anthοny cοlοred 6 with 1/4 - 1/4 = 6 circles.

Tοgether, we wοuld have cοlοured 12 circles since 6 + 6 = 12. This is what they desired. Anthοny is therefοre right!

By calculating the tοtal number οf circles that each persοn cοlοured while accοunting fοr the fact that they all cοlοred the very same number οf circles, we can further cοnfirm οur cοnclusiοn:

Anthοny drew 6 circles.

Jacοb filled in 6 circles.

Tοgether, they cοlοred 6 + 6 = 12 circles.

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Step-by-step explanation:

1 pound = 0.45 kilograms

1 square inch = 6.25 square centimeters

First, let's convert the pressure from pounds per square inch to kilograms per square inch:

198 pounds/square inch × 0.45 kilograms/pound = 89.1 kilograms/square inch

Next, let's convert from kilograms per square inch to kilograms per square centimeter:

1 square inch = 6.25 square centimeters

89.1 kilograms/square inch ÷ 6.25 square centimeters/square inch = 14.256 kilograms/square centimeter

Therefore, the pressure of 198 pounds per square inch is equivalent to 14.256 kilograms per square centimeter.

The ratio and reciprocal of Sec x and Cos x can be expressed in terms of A and pq. When A is equal to 3/2, Sec A will be equal to 3/2 and Cos A will be equal to 2/3.

The ratio and reciprocal of Sec x and Cos x can be expressed in terms of A and pq. Sec x = A/pq and Cos x = pq/A. The reciprocal of Sec x is Cos x = pq/A and the reciprocal of Cos x is Sec x = A/pq.

When A = 3/2, Cos A = 2/3. Therefore, Sec A = 3/2 and Cos A = 2/3.

The ratio and reciprocal of Sec x and Cos x can be expressed in terms of A and pq. When A is equal to 3/2, Sec A will be equal to 3/2 and Cos A will be equal to 2/3.

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From the given information provided, the length of the radius is √17, which is closest to 3. So the answer is option A. 3.

The center of the circle is the midpoint of the diameter, which can be found using the midpoint formula:

Midpoint = ((-5 + 3)/2, (4 + 6)/2) = (-1, 5)

The radius of the circle is the distance from the center to one of the endpoints of the diameter. We can use the distance formula to find the distance between the center (-1, 5) and either endpoint:

r = √[(-5 - (-1))² + (4 - 5)²] = √[(-4)² + (-1)²] = √[16 + 1] = √(17)

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Accοrding tο the given infοrmatiοn, the prοbability οf generating a randοm number between 0.25 and 0.75 is 0.5 and the prοbability οf generating a number less than οr equal tο 1 is 1.

Prοbability is a measure οf the likelihοοd οr chance οf an event οccurring. It is a number between 0 and 1, where 0 represents an impοssible event and 1 represents a certain event.

The cοntext οf this questiοn is nοt clear, sο I will prοvide a general answer.

Assuming a unifοrm distributiοn οf randοm numbers between 0 and 1, the prοbability οf generating a randοm number between 0.25 and 0.75 is:

b. P(0.25 ≤ X ≤ 0.75) = 0.75 - 0.25 = 0.5

where X is the randοm variable representing the generated number.

The prοbability οf generating a randοm number with a value less than οr equal tο a specific value x is simply x, since the prοbability density functiοn οf a unifοrm distributiοn is a cοnstant οver the range [0, 1].

c. P(X ≤ x) = x fοr 0 ≤ x ≤ 1

Fοr example, the prοbability οf generating a number less than οr equal tο 0.3 is 0.3, and the prοbability οf generating a number less than οr equal tο 1 is 1.

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(0, 6): satisfies the inequality and is included in the solution set.

(1, 5): point does not satisfy the inequality and is included in the solution set.

(2, 4): point does not satisfy the inequality and is included in the solution set.

(3, 2): point satisfies the inequality and is not included in the solution set.

What is inequality?

Inequality is like a rule that says how big or small something can be. It has two things to compare, and a symbol in between like < or >.

The inequality is satisfied by all the points inside the curve. The given inequality represents a parabolic region that opens upwards with its vertex at (1, 2), and it passes through the points (-1, 6) and (3, 6). The region is shaded inside the curve.

We need to determine which point is not included in the solution set for the inequality. To do this, we can check if each point satisfies the inequality.

Let's start with point (0, 6). We can see that this point lies inside the shaded region, so it is included in the solution set.

Next, let's check point (1, 5). This point lies on the boundary of the shaded region, which means it is included in the solution set.

Now, let's check point (2, 4). This point lies below the boundary of the shaded region, which means it is included in the solution set.

Finally, let's check point (3, 2). This point lies outside the shaded region, which means it is not included in the solution set.

Therefore, the point that is not included in the solution set for the inequality is (3, 2).

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If we are aware that Mustafa, Heloise, or Gia have penned more than [tex]22[/tex]pieces altogether. [tex]M > 8[/tex] This is the number of articles Mustafa might have written.

A numeric is a mathematical number which can be utilized to compute and describe a quantity. Numerical symbols, such as "3," are written to represent numbers. A counting system is a logical way of expressing numbers that uses digits or symbols to represent them.

The fact that we use a decimal (base 10) mathematical notation system for numbers gives the number 100 importance that it probably wouldn't otherwise have. It is a circular numeral with undertones of excellence.

[tex]M + H + A > 22[/tex]

We also know that:

[tex]H = 1/4M[/tex]

[tex]A = 3/2M[/tex]

We can substitute these expressions into the first equation:

[tex]M + 1/4M + 3/2M > 22[/tex]

Multiplying everything by the common denominator of 4:

[tex]4M + M + 6M > 88[/tex]

[tex]11M > 88[/tex]

[tex]M > 8[/tex]

So, Mufasa has written more than [tex]8[/tex] articles.

Using the equation [tex]H = 1/4M[/tex], we can find that Heloise has written:

[tex]H = 1/4(8) = 2[/tex]

Using the equation [tex]A = 3/2M[/tex], we can find that Angie has written:

[tex]A = 3/2(8) = 12[/tex]

Therefore, Mufasa has written more than [tex]8[/tex] articles, Heloise has written [tex]2[/tex]articles, and Angie has written [tex]12[/tex] articles.

To check if their combined total is more than [tex]22[/tex]:

[tex]M + H + A = 8 + 2 + 12 = 22[/tex]

So, their combined total is exactly [tex]22[/tex], which means that they have not written more than [tex]22[/tex] articles for the school newspaper.

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The point that was used as the center of rotation include the following: A. Point F.

In Mathematics, a rotation is a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

Generally speaking, rotation simply refers to a type of transformation that preserves both the shape and side lengths of a geometric figure. By applying a rotation of 90° clockwise centered at point F to the vertices of triangle KLC, we have the following congruent sides:

KC ≅ K'C'

KL ≅ K'L'

LC ≅ L'C'

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Answer:

To approximate y = sin(x) using the third-degree Taylor polynomial, we need to find the polynomial that best approximates sin(x) in the neighborhood of x = 0.

The third-degree Taylor polynomial for sin(x) centered at x = 0 is given by:

P3(x) = x - (1/6)x^3

Now we can graph both functions, y = sin(x) and P3(x), over the interval [-7,7].

To do this, we can plot points for both functions and then connect the points with a smooth curve. We can choose x-values that are evenly spaced over the interval and calculate the corresponding y-values using sin(x) and P3(x).

For example, if we choose x = -7, -6, -5, ..., 5, 6, 7, then we can calculate the y-values as follows:

For sin(x):

sin(-7) ≈ -0.656

sin(-6) ≈ 0.279

sin(-5) ≈ 0.958

...

sin(5) ≈ -0.959

sin(6) ≈ -0.279

sin(7) ≈ 0.657

For P3(x):

P3(-7) = -7 - (1/6)(-7)^3 ≈ -286.8

P3(-6) = -6 - (1/6)(-6)^3 ≈ -216

P3(-5) = -5 - (1/6)(-5)^3 ≈ -141.7

...

P3(5) = 5 - (1/6)(5)^3 ≈ 141.7

P3(6) = 6 - (1/6)(6)^3 ≈ 216

P3(7) = 7 - (1/6)(7)^3 ≈ 286.8

Now we can plot the points for sin(x) and P3(x) on the same graph and connect them with a smooth curve.

Here is what the graph looks like:

    |        

 1  |          ********

    |       ***

 0 |    **

    |  *    

-1  |*

    |          

-2 |         ********

    |     ****

-3 |  ***

    |**

-4 |          

    -------------------

       -7   0   7

Best I can do for a graph, as I cannot send a valid link.

In the graph, the sinewave is represented by the curve with peaks and troughs, while the third-degree Taylor polynomial is represented by the straight line with a slight curve at the ends.

We can see that the third-degree Taylor polynomial provides a good approximation of sin(x) in the neighborhood of x = 0, but the approximation becomes less accurate as we move away from x = 0.

There is no solution for this system of equations. Hence, the answer is NS.

The system of equations is given by;

-x + 2y = -1 (1)4x - 8y = 5 (2)

For solving a system of linear equations by any method, we need to eliminate one variable from the equations. Let's eliminate x from the given equations. Multiplying equation (1) by 4, we get

4(-x + 2y = -1) ⇒ -4x + 8y = -4 ..........(3)

Now, we need to eliminate -4x from equation (3) and 4x from equation (2).

Adding equations (2) and (3), we get-

4x + 8y = -4+ 4x - 8y = 5 0 = 1.

Since the variables cancel each other and 0 ≠ 1, this system of equations is inconsistent. So, there is no solution for this system of equations. Hence, the answer is NS.

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Answer:

Step-by-step explanation:

To solve this problem, we need to find the number of cells in each colony after 1 minute (60 seconds) and compare them.

For the colony growing at 20°C, the expression for the number of cells is:

N(20°C) = t^2 - 7t - 7

Substituting t = 60 seconds:

N(20°C) = (60)^2 - 7(60) - 7 = 3533

Therefore, the number of cells in the colony growing at 20°C after 1 minute is 3533.

For the colony growing at 30°C, the expression for the number of cells is:

N(30°C) = t^2 - 6t - 7

Substituting t = 60 seconds:

N(30°C) = (60)^2 - 6(60) - 7 = 3563

Therefore, the number of cells in the colony growing at 30°C after 1 minute is 3563.

Comparing these results, we see that the colony growing at 30°C has a greater number of cells after 1 minute than the colony growing at 20°C. Therefore, the colony growing at 30°C will have a greater number of cells than the colony growing at 20°C after 1 minute.

The impact of changing a single value in a data set on the mean, median, mode, and range can vary depending on the position and value of the changed value

The impact of changing a single value in a data set on the mean, median, mode, and range can vary depending on the value that is changed and its position in the data set.

If the value that is changed is an outlier or an extreme value in the data set, then the impact on the mean and range will be significant. The mean is particularly sensitive to outliers, and changing an extreme value can cause the mean to shift considerably.

If the value that is changed is close to the mean, then the impact on the mean will be significant, but the impact on the median and mode will be small.

If the value that is changed is a mode, then the mode will change, but the impact on the mean and median will be small. The range may or may not be affected, depending on the position of the changed value relative to the maximum and minimum values in the data set.

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Andrew needs to save money for 3.8 months, or about 4 months (rounded up) to buy that refrigerator.

Based on the given information, Andrew's budget can be broken down as follows:

Housing: $46,200 (30% of $154,000)

Gas: $12,320 (8% of $154,000)

Internet: $3,080 (2% of $154,000)

Personal care: $7,700 (5% of $154,000)

Food: $38,500 (25% of $154,000)

Saving: $23,100 (15% of $154,000)

Child: $23,100 (15% of $154,000)

Small business income: $25,000

To calculate how long it will take for Andrew to save enough money to buy the $95,000 refrigerator, we need to calculate his monthly savings based on his budget.

Andrew's total monthly income (before expenses) is

His total monthly expenses (including savings) are

Therefore, his monthly savings are $179,000 - $154,000 = $25,000. To save up $95,000, he will need $95,000 / $25,000 = 3.8 months, or about 4 months (rounded up).

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the value of Harry's investment after 4 years is £8,999.21 to the nearest penny.

To work out the value of Harry's investment after 4 years, we can use the formula for compound interest:

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

where:

A is the final amount

P is the initial principal (the amount Harry invested)

r is the annual interest rate (as a decimal)

n is the number of times the interest is compounded per year

t is the time in years

In this case, P = £8000, r = 0.028 (2.8% expressed as a decimal), n = 1 (compounded annually), and t = 4. Plugging these values into the formula, we get:

A = £8000(1 + 0.028/1)

= £8000(1.028)

= £8,999.21

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Therefore , the solution of the given problem of angles comes out to be  marked angle ≈ 106.2602 degrees.

The barrier's top and bottom in Euclidean geometry divide two circular faces, which form both of a tilt's sides. It is possible for two rays to join to form a point of intersection. Angle is another outcome of two entities interacting. They mirror dihedral shapes the most. A two-dimensional curve can be created by arranging two line beams in various configurations at their ends.

Here,

Triangle ABC is an isosceles triangle with basis BC because we are told that

=> BAP = CAP. As a result, we can say that AB = AC.

We can determine the cosine of an angle using the Rule of Cosines:

=> cos(BAC) = (AB² + AC² - BC²) / (2 * AB * AC)

=> cos(BAC) = (6² + 5² - 4²) / (2 * 6 * 5)

=> cos(BAC) = 49 / 60

We discover: by taking the inverse cosine of both sides.

=> BAC = cos⁻¹(49/60)

Calculating the answer, we obtain:

=> ∠BAC ≈ 36.8699 degrees

The indicated angle is as a result:

=> marked angle = 180 - 2 * BAC

=> marked angle = 180 - 2 * 36.8699

=> marked angle ≈ 106.2602 degrees

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The area of the shaded portion of the circle is 942 units².

The diagram is a circle inscribed in another circle.  The area of the shaded portion of the circle can be found as follows:

area of the shaded portion = area of the bigger circle - area of the smaller circle

Therefore,

area of the bigger circle = πr²

area of the bigger circle = 20²π

area of the bigger circle = 400π units²

area of the smaller circle = πr²

area of the smaller circle = 10²π

area of the smaller circle = 100π units²

Therefore,

area of the shaded portion = 400π - 100π

area of the shaded portion = 300π

area of the shaded portion = 300 × 3.14

area of the shaded portion = 942 units²

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Answer:

We don't know the exact distance Chu takes on his way back, but we do know that it's shorter than the distance he took on the way to baseball practice. Let's call the distance he takes on the way back "x".

We know that the distance he rode to baseball practice was 1.39 miles. So, the total distance he rode is:

1.39 miles + x miles

We can also assume that the total distance he rode going to baseball practice and coming back home is the same. That is:

1.39 miles + x miles = x miles + x miles

Simplifying this equation, we get:

1.39 miles + x miles = 2x miles

Subtracting x miles from both sides, we get:

1.39 miles = x miles

So, Chu rides his bicycle 1.39 miles from baseball practice to home.

Step-by-step explanation:

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Answer:

T_n = -5 -2(n-1) = -3 - 2n

Step-by-step explanation:

This is an arithmetic sequence with a common difference of -2 (-5 -2 = -7, -7 - 2 = -9, -9 -2 = -11 ...)

so, T_n = -5 -2(n-1) = -3 -2n

we use -5 as the starting point because that's the first term. whether you simplify or not depends on you and your teacher.

Answer:

The number of nickels and quarters in the bank are 87 and 56, respectively.

Step-by-step explanation:

Let's denote the number of nickels in the bank as "n" and the number of quarters as "q". Then we can set up a system of two equations based on the given information:

n + q = 143 (equation 1, representing the total number of coins)

0.05n + 0.25q = 18.35 (equation 2, representing the total value of the coins)

To solve this system, we can use substitution or elimination method. Let's use elimination method here:

Multiplying equation 1 by 0.05, we get:

0.05n + 0.05q = 7.15 (equation 3, obtained by multiplying equation 1 by 0.05)

Subtracting equation 3 from equation 2, we get:

0.2q = 11.2

Dividing both sides by 0.2, we get:

q = 56

Substituting this value of q into equation 1, we get:

n + 56 = 143

n = 87

Therefore, there are 87 nickels and 56 quarters in the bank.

Hopefully this helped you! If not, I'm sorry! If you need more help, ask me! :]

Through end behaviour of polynomials, we can see that the left end behaviour is approaching positive infinity and the right end behaviour is approaching negative infinity.

The nature of the value when the function argument gets closer to + infinity and - infinity is the end behaviour of a function.

The term with the highest degree determines how a polynomial function will behave in the end.

For example:

If f(x) = x³ + 2x² – 4x + 5 is given,

The end behaviour is determined by x³.

As a result, f(x) + as x + and f(x) - as x -

The term with the highest degree will be substantially larger than the other terms for large values of x, making the other terms essentially irrelevant.

The degree of the x³ coefficient is odd and it is positive.

Hence, the final behaviour is f(x) + as x + and f(x) - as x -.

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The cost of the system with the printer is $695, given the cost of the system without the printer is $1,200 and the total cost is $1,895.

Let's assume the cost of the system without the printer is x. Then the cost of the printer can be represented as 1895 - x (since the total cost of the system with the printer is $1,895).

We can set up an equation to represent the cost of the system with the printer:

x + (1895 - x) = 1895

Simplifying and solving for x:

x + 1895 - x = 1895

2x = 0

x = 0

Therefore, according to this equation, the cost of the system without the printer is $0, which does not make sense. It's likely that there was an error in the problem statement.

If there was an error in the problem statement and we are given more information, we can solve for the cost of the system with the printer. For example, let's assume that we are given that the cost of the system without the printer is $1,200.

Using the same approach as before, we can set up an equation to represent the cost of the system with the printer:

x + (1895 - x) = 1895

where x is the cost of the system without the printer. Substituting x = $1,200, we get:

1200 + (1895 - 1200) = 1895

Simplifying and solving for the cost of the system with the printer:

1200 + 695 = 1895

Therefore, the cost of the system with the printer is $695.

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The tοtal number οf ways tο chοοse three beads such that their sum is οdd is 20 + 5 = 25.

Prοbability is a measure οf the likelihοοd οf an event οccurring. It is a number between 0 and 1, where 0 indicates that the event is impοssible and 1 indicates that the event is certain.

Tο find the prοbability that the sum is an οdd number, we need tο cοunt the number οf ways we can chοοse three beads such that their sum is οdd and divide it by the tοtal number οf ways we can chοοse three beads.

First, we need tο cοnsider the pοssible ways that the sum οf three numbers is οdd. This can happen in twο ways:

If we chοοse an οdd number and twο even numbers

If we chοοse an even number and twο οdd numbers

Nοw, let's cοunt the number οf ways we can chοοse three beads in each οf these twο cases:

Case 1: Chοοse οne οdd and twο even numbers

There are twο οdd numbers and five even numbers, sο we can chοοse οne οdd number in 2 ways and twο even numbers in 5C₂ = 10 ways. The tοtal number οf ways tο chοοse three beads in this case is therefοre 2 x 10 = 20.

Case 2: Chοοse οne even and twο οdd numbers

There are twο οdd numbers and five even numbers, sο we can chοοse twο οdd numbers in 2C₂ = 1 way and οne even number in 5 ways. The tοtal number οf ways tο chοοse three beads in this case is therefοre 1 x 5 = 5.

The tοtal number οf ways tο chοοse three beads frοm seven is 7C₃ = 35.

Sο the prοbability that the sum οf the three numbers is οdd is 25/35 = 5/7.

Therefοre, the tοtal number οf ways tο chοοse three beads such that their sum is οdd is 20 + 5 = 25.

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The  evaluated logarithmic expression is: [tex]\[ \log \left[\frac{10 x^{2}}{\sqrt[3]{y}}\right] = 2\log[10] + 2\log[x] + \frac{1}{3}\log[y] \][/tex]

Expression mentioned in the question :[tex]\[ \log \left[\frac{10 x^{2}}{\sqrt[3]{y}}\right][/tex]. . The first property we will use is the Product Rule , so this   can be expressed as: [tex][ \log \left[\frac{10 x^{2}}{\sqrt[3]{y}}\right] = \log[10x^{2}] + \log[\sqrt[3]{y}] \].[/tex] We can now apply the Power Rule to the first term, which states that the logarithm of a power is equal to the exponent times the logarithm of the base. This can be expressed as: \[ \log[10x^{2}] = 2\log[10x] \]

We can now apply the Quotient Rule to the second term, which states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. This can be expressed as: [tex]\[ \log[\sqrt[3]{y}] = \frac{1}{3} \log[y] \].[/tex] Substituting this back into the original equation gives us:[tex]\[ \log \left[\frac{10 x^{2}}{\sqrt[3]{y}}\right] = 2\log[10x] + \frac{1}{3}\log[y] \][/tex]

We can now evaluate this expression without using a calculator. The two logarithmic terms can be rewritten as:[tex]\[ \log[10x] = \log[10] + \log[x] \,  \[ \log[y] = \log[y] \][/tex]
Substituting this back into the original equation gives us: [tex]\[ \log \left[\frac{10 x^{2}}{\sqrt[3]{y}}\right] = 2\log[10] + 2\log[x] + \frac{1}{3}\log[y] \[/tex]

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The areas of each part of the composite figure are;

Triangle A = 20 Square units

Triangle B = 2 Square units

Rectangle C = 4 Square units

Triangle D = 6 Square units

The area of a triangle simply has the given formula;

A = ¹/₂ * base * height

Area of Triangle A is;

Triangle A = ¹/₂ * (6 + 2 + 2) * 4

= ¹/₂ * 10 * 4

= 20 Square units

Area of Triangle B is;

Triangle A = ¹/₂ * 2 * 2

= 2 Square units

Area of rectangle C = Length * Width

= 2 * 2

= 4 Square units

Area of Triangle D is;

Triangle D = ¹/₂ * 6 * 2

= 6 Square units

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The inequality that represents the heaviest the packaging can be without exceeding the 50-pound weight limit is 21 + x ≤ 50 and weight of the packaging is less than or equal to 29 pounds.

Let x be the weight of the packaging in pounds. Then the total weight of the item and its packaging is 21 + x pounds. To ensure that the total weight does not exceed the 50-pound weight limit, we can set up the following inequality:

21 + x ≤ 50

We subtract 21 from both sides to isolate x and obtain:

x ≤ 50 - 21

x ≤ 29

Therefore, the weight of the packaging cannot be greater than 29 pounds without exceeding the 50-pound weight limit. To ensure that the item can be mailed by parcel post, the weight of the packaging must be less than or equal to 29 pounds.

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