Find the area of the triangle below.
20 cm
5 cm
13 cm

The probability of having every bin filled with at least one ball if n balls are distributed randomly into m bins is given by the formula 1 - (m-1/m)^n.

Let's break it down step-by-step:

Step 1: Probability of a ball being put into a specific Bin Since there are m bins, the probability of a ball being put into a specific bin is 1/m. Therefore, the probability of a ball not being put into that bin is (m-1)/m.

Step 2: Probability of a ball not being put into a specific Bin Since there are m bins, the probability of a ball not being put into a specific bin is (m-1)/m. Therefore, the probability of a ball being put into that bin is 1/m.

Step 3: Probability of every bin being filled with at least one ball Using the above two probabilities, the probability of a specific bin not being filled with a ball is (m-1)/m. Therefore, the probability of all m bins not being filled with a ball is (m-1/m)^n. Finally, the probability of every bin being filled with at least one ball is 1 - (m-1/m)^n.

Therefore, the probability of having every bin filled with at least one ball if n balls are distributed randomly in to m bins is given by the formula 1 - (m-1/m)^n.

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Let's find the magnitude of each component:

Let:

 [tex](x1,y1)=(4,4)[/tex]

[tex](x2,y2)=(-12,8)[/tex]

     [tex]u=ax+by[/tex]

  [tex]||u||=\sqrt{a^2+b^2}[/tex]

So, let's find a and b:

[tex]a=|x2-x1|=|-12-4|=|-16|=16[/tex]

       [tex]b=|y2-y1|=|8-4|=|4|=4[/tex]

so:

[tex]||u||=\sqrt{16^2+4^2}[/tex]

   [tex]||u||=\sqrt{272}[/tex]

 [tex]||u||\thickapprox16.492[/tex]

And the direction is:

         [tex]\theta=180-\text{tan}^{-1}\huge \text(\dfrac{b}{a}\huge \text)[/tex]

        [tex]\theta=180-\text{tan}^{-1}\huge \text(\dfrac{4}{16}\huge \text)[/tex]

[tex]\theta\thickapprox180-\text{tan}^{-1}\huge \text(\dfrac{4}{16}\huge \text)\thickapprox165.964[/tex]

We need more information, specifically the median value and the other data points, to calculate the exact number of ants in the ant farm.

To answer your question, let's first break down the given information and terms:
1. The student's ant farm is being included in the data.
2. The median is mentioned but not provided.

The number of ants in the student's ant farm:
Step 1: Determine whether the total number of data points is odd or even.
Since the median changes when the ant farm is included, we can deduce that the total number of data points must be odd.

This is because, if the number of data points was even, adding one more data point (the ant farm) would not change the median significantly.
Step 2: Identify the relationship between the median and the data points.
With an odd number of data points, the median has to be one of the data points.

This is because the median is the middle value when the data points are arranged in ascending order, and with an odd number of points, there must be a single middle value.
Step 3: Determine the number of ants in the student's ant farm.
Since we know that the median has to be one of the data points and it is not given, we cannot directly determine the number of ants in the student's ant farm.

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Answer:

The numerator is x²+6x+5.

We need to first factorise the expression

The coefficient of x² is 1, and the constant term of the quadratic expression is 5.

the product of coefficient of x² and constant term=5

∴ we break the middle term, 6x, such that,  the product of the coeffcients of  the two terms is 5, while their sum is 6

∴ 6x=5x+x

Now, rewriting the expression,

x²+5x+x+5

Taking x as common factor in x² and 5x,

x(x+5)+x+5

Again, taking x+5 as the common factor,

(x+5)(x+1)

Now, replacing x²+6x+5 with (x+5)(x+6) in the expression,

(x+5)(x+1)/(x+2)

Hence the value of the expression is (x+5)(x+1)/(x+2)

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To find the volume of the solid generated by revolving about the y-axis the region under the curve y = 5e^-2x in the first quadrant, we can use the method of cylindrical shells.

The formula for the volume generated by a cylindrical shell is V = 2πrh Δx, where r is the radius of the shell, h is the height of the shell, and Δx is the thickness of the shell.

In this case, we are rotating about the y-axis, so our cylindrical shells will have a height of y and a thickness of Δy. The radius of each shell will be the distance from the y-axis to the curve, which is x.

We can solve for x in terms of y by rearranging the equation y = 5e^-2x:

y/5 = e^-2x

ln(y/5) = -2x

x = -ln(y/5)/2

So the volume of each cylindrical shell is:

V = 2πx * y * Δy

Substituting for x:

V = 2π(-ln(y/5)/2) * y * Δy

V = -πln(y/5) * y * Δy

To find the total volume, we need to integrate this expression from y = 0 to y = infinity:

V = ∫[0,∞] -πln(y/5) * y dy

Using integration by parts:

u = ln(y/5), dv = y dy

du = 1/y dy, v = 1/2 y^2

∫[0,∞] -πln(y/5) * y dy = [-π(1/2y^2 ln(y/5) - 1/4y^2)] [0,∞]

= [π/4]

Therefore, the volume of the solid generated by revolving about the y-axis in the region under the curve y = 5e^-2x in the first quadrant is π/4, which is a finite value.

Hence, the answer is 0.7854 (rounded to four decimal places).

The minimum number of possible successes of the experiment which can be expected, is calculated out to be 6.

Since the probability of success is the same for men as it is for women, we know that the probability of success for a man in this trial is also 0.60.

This defines a random binomial variable with n=10 trials and p=0.60 probability of success.

The minimum number of possible successes is zero. It is possible for none of the 10 men to respond positively to the treatment, although the probability of this occurring is relatively low (0.4¹⁰ = 0.0001048576 or about 0.01%).

The maximum number of possible successes is 10. It is possible for all 10 men to respond positively to the treatment, although the probability of this occurring is also relatively low (0.60¹⁰ = 0.0060466176 or about 0.60%).

To find the expected number of successes, we can use the formula for the expected value of a binomial random variable:

E(X) = n x p

E(X) = 10 x 0.60

E(X) = 6.0

So the expected number of successes is 6.0.

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To find out how far Jacky walks every day, we need to divide the total distance he walks every week (220 1/2 m) by the number of days he walks.

Assuming he walks every day, there are 7 days in a week. So,

220 1/2 m ÷ 7 = 31 1/2 m

Therefore, Jacky walks 31 1/2 meters every day.

The expression that represents the maximum weight in tons of all the whales in the oceans is 150y tons.

A variable in mathematics is a symbol or letter that designates a value that is subject to variation or change. In algebraic equations and expressions, variables are used to represent unknown numbers or quantities with a range of possible values. Typically, variables are represented by letters like x, y, and z or a, b, and c. The variable is frequently placed on one side of the equal sign in an equation, and the value that the variable equals is denoted by the expression on the other side of the equal sign.

Given that, the weight of one single whale = 150 tons.

Now,

Maximum weight = 150y tons

Hence, the expression that represents the maximum weight in tons of all the whales in the oceans is 150y tons.

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Let's first find the area of the window:

Area of window = 4 * m * 4 * m = 16 * m^2

Now, let's find the area of the curtain:

Area of curtain = 3 * m * 2.5 * m = 7.5 * m^2

We're given that the area of the window is 2 square meters less than the area of the curtain, so we can set up the following equation:

16 * m^2 = 7.5 * m^2 - 2

Simplifying this equation, we get:

8.5 * m^2 = 2

m^2 = 2 / 8.5

m^2 = 0.2353

Taking the square root of both sides, we get:

m = 0.4851

Therefore, the possible areas of the window are:

16 * m^2 = 16 * (0.4851)^2 = 3.7 square meters (rounded to one decimal place).

A residence assistant (ra) at a local university devises a plan to spot check the dorm for illegal drinking, the technique they using is Systematic Sampling.

Systematic sampling is a form of probability sampling technique in which sample members are chosen from a wider population using a defined, periodic interval but a random beginning point. By dividing the population size by the required sample size, this interval, also known as the sampling interval, is computed.

If the periodic interval is predetermined and the beginning point is random, systematic sampling is still regarded as random even though the sample population has been chosen in advance.

Systematic sampling, when done effectively on a big population of a defined size, can assist researchers, particularly those in marketing and sales, in obtaining representative results on a sizable group of individuals without having to contact every single one of them.

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The given statement "if an iterative method for solving a nonlinear equation gains more than one bit of accuracy per iteration, then it is said to have a superlinear convergence rate" is true.

Explanation:

In numerical analysis, iterative methods are used to solve nonlinear equations. Iterative methods, unlike direct methods, are used to solve equations without knowing the exact solution, and they rely on an iterative process to obtain a sufficiently accurate result.

The rate of convergence of the iterative method determines how quickly the iterative method converges to the desired solution. The rate of convergence is one of the most critical performance metrics for iterative methods.The rate of convergence of an iterative method can be classified as linear or superlinear. An iterative method is said to converge linearly if the number of accurate digits in the solution is approximately proportional to the number of iterations. A method is said to converge superlinearly if the number of accurate digits in the solution grows faster than the number of iterations. When a nonlinear equation is solved using an iterative method, if the accuracy gained by the iterative method is greater than one bit per iteration, the method is said to have a superlinear convergence rate. Therefore, the given statement is true.

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No, there cannot be a right triangle with sides of lengths 1, 2, and 3. This is because of the Pythagorean theorem.

The theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, this can be written as:

[tex]c^2 = a^2 + b^2[/tex]

where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

If we substitute the values of 1, 2, and 3 for a, b, and c, respectively, we get:

[tex]3^2 = 1^2 + 2^2[/tex]

which simplifies to:

9 = 1 + 4

This is not true, so the sides 1, 2, and 3 do not form a right triangle.

However, there exist right triangles with consecutive natural numbers as their side lengths. One such example is the Pythagorean triple (3, 4, 5), where:

[tex]3^2 + 4^2 = 9 + 16 = 25 = 5^2[/tex]

This is a right triangle where the lengths of the sides are consecutive natural numbers. There are infinitely many such Pythagorean triples, and they can be generated using the formula:

[tex]a = 2mn, b = m^2 - n^2, c = m^2 + n^2[/tex]

where m and n are any two positive integers with m > n.

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Answer:

Could you please provide more information so I can help you with your problem?

The weight of the cable Dustin is using would be 58.67 pounds.

As per the given statement, Dustin is using a cable to lift heavy equipment.

His company has a table of the weights of various lengths of cable.

If Dustin needs to use 88 feet of cable, Pounds will this cable weigh:

Let us find the solution to the problem given in the question.

The length of the cable Dustin is using is 88 feet.

Using the table given by his company, the weight of the 88 feet long cable can be determined.

Using the terms given in the question, the solution is as follows:

Dustin needs to use 88 feet of cable, and the weight of this cable can be found from the table given by the company.

If we look at the table, it is evident that there is a weight of 2 pounds for every 3 feet of cable, so the weight of the 88 feet of cable Dustin is using would be:

Pounds of 3 feet of cable = 2 pounds of 3 feet of cable.

Pounds of 1 foot of cable = (2/3) pounds of 1 foot of cable

So, the weight of 88 feet of cable = (2/3) × 88 = 58.67 pounds.

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Question: Dustin is using cable to lift heavy equipment. his company has a table of the weights of various lengths of cable. if dustin needs to use 88 feet of cable, how many pounds will this cable weigh?

Length of cable in feet: 5 15 25 35

Weight of cable in pounds: 24.4 73.2 122.0 170.0

Answer: 120 %

Step-by-step explanation:

part/whole = percent/100

Let x be the percentage we are trying to find, then we can write:

5.40/4.50 = x/100

To solve for x, we can cross-multiply:

4.50x = 5.40 * 100

Dividing both sides by 4.50, we get:

x = (5.40 * 100) / 4.50 = 120

Therefore, $5.40 is 120% of $4.50.

The statement "As x --> ∞ in the expression  [tex](1+1/x)^{x}[/tex], the output approaches е" is true.

Algebraic expressiοn can be defined as cοmbinatiοn οf variables and cοnstants.

As x apprοaches infinity, the term with the highest pοwer dοminates the expressiοn. In this case, the highest pοwer is x in the expοnent. Therefοre, we can write:

[tex]\lim_{x \to \infty}[/tex] [tex](1+1/x)^{x}[/tex]= [tex]e^ \lim_{x \to \infty} (1/x)(x)[/tex]

Since the (1/x) × x term simplifies to 1 as x approaches infinity, we have:

[tex]\lim_{x \to \infty}[/tex] [tex](1+1/x)^{x}[/tex] = [tex]e ^ \lim_{x \to \infty}1[/tex] = e

So, the limit of the expression as x approaches infinity is e.

Therefore, the statement "As x --> ∞ in the expression  [tex](1+1/x)^{x}[/tex], the output approaches е" is true.

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Since angles, DAB and MAX are congruent (statement 1) and angles ADB and AMX are congruent (statement 2), then the two triangles are similar.  Hence,  ∆DAB ≅ ∆MAX

Two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.

Here, we have

1. ∠DAB ≅ ∠MAX

2. ∠ADB ≅ ∠AMX (Vertical angles are congruent)

3. ∆DAB ~ ∆MAX (Angle-Angle Similarity Postulate)

Statement 1 is given in the problem. It states that angle DAB is congruent to angle MAX.

Statement 2 follows from the fact that angles ADB and AMX are vertical angles, which means they are congruent.

Statement 3 is the conclusion of the proof, which states that triangles DAB and MAX are similar by the Angle-Angle Similarity Postulate, which states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Since angles, DAB and MAX are congruent (statement 1) and angles ADB and AMX are congruent (statement 2), then the two triangles are similar.

Hence,  ∆DAB ~ ∆MAX.

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Answer:

new height of the figure will be 175% of the original height.

Step-by-step explanation:

To calculate the new height of the figure after using a vertical scale factor of 175%, we can use the following formula:

New height = Original height x Scale factor

Let's say the original height of the figure is h. Then, using a vertical scale factor of 175% means multiplying the original height by 1.75. Therefore:

New height = h x 1.75

Simplifying:

New height = 1.75h

So the new height of the figure will be 175% of the original height.

The correct answer to the given question is "it is inverse."

One of the most essential characteristics of the normal distribution is that it is bell-shaped.

This implies that it is symmetrical, with the highest point at the mean or center and the curve declining on either side of the mean.

Normal distribution is a crucial concept in statistics and is widely utilized in research and decision-making processes.

It is a probability distribution that is commonly used to describe the distribution of a set of data.

The normal distribution has several characteristics that distinguish it from other probability distributions.

The normal distribution is not inverse, meaning that the tails do not approach the x-axis as they extend.

Rather,

The normal distribution has a gradual decline, with the tails diminishing in size but never reaching zero.

It is also asymptotic, which means that the tails of the distribution continue indefinitely without ever touching the x-axis, although they become increasingly small as they move away from the mean.

Normal distribution is not skewed, which means that it is symmetrical around the mean.

A skewed distribution is one in which the mean, median, and mode are not equal, indicating that one side of the curve is longer than the other.

The normal distribution's symmetry is one of its most important features since it indicates that the mean, median, and mode are all equivalent, and the data is uniformly distributed on either side of the mean. Therefore, the correct answer to the given question is "it is inverse."

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In the absence of any specific story or context for "Secrets of Oak Park," However, based on the given prompt, here are five possible feelings that Zach might have if he believes his dad is behind the incidents.

Betrayed: Zach might feel a sense of betrayal if he believes that his dad, who he presumably trusts and loves, is responsible for the incidents in question.

Confused: Zach may feel confused about why his dad would do something like that or what his motives might be.

Angry: Zach could feel angry that his dad could cause harm or trouble, and be causing problems in their community or family.

Worried: Zach might feel worried about what will happen if his dad is found to be responsible for the incidents.

Determined: Zach might also feel determined to find out the truth, confront his father, and set things right.

Therefore, These are just a few possibilities, as the story's context would greatly influence Zach's feelings.

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The percentage error between the estimated age and the carbon-14 dating age of the fossil is 12.5% and an estimated age of approximately 2978 years would give the same percent error of 12.5%.

What is the definition of percentage?

Percentage is a way of expressing a fraction or proportion as a part of 100. It is denoted by the symbol "%". The term "percent" means "per hundred". For example, if 50% of a group of people are men, it means that 50 out of every 100 people in the group are men.

Now,

a). To find the percent error between the estimated age and the carbon-14 dating age of the fossil, we can use the formula:

percent error = |(estimated age - carbon-14 dating age) / estimated age| x 100%

Substituting the given values, we get:

percent error = |(3200 - 3600) / 3200| x 100%

= |-400 / 3200| x 100%

= 12.5%

Therefore, the percent error between the estimated age and the carbon-14 dating age of the fossil is 12.5%.

b). To find other estimates that give the same percent error, we can use the formula:

percent error = |(estimated age - carbon-14 dating age) / estimated age| x 100%

If we let x be the estimated age that gives the same percent error, then we can write:

12.5% = |(x - 3600) / x| x 100%

Simplifying, we get:

0.125 = |(x - 3600) / x|

Taking the positive square root of both sides (since percent error cannot be negative), we get:

√0.125 = (x - 3600) / x

Simplifying, we get:

x = 3600 / (1 - √0.125)

x ≈ 2978

Therefore, an estimated age of approximately 2978 years would give the same percent error of 12.5% as the original estimated age of 3200 years. This means that if the fossil were actually 2978 years old, the carbon-14 dating would have given an age estimate of approximately 3299 years, which is 12.5% higher than the true age.

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The probability of choosing a left-handed student was determined by observing a sample of students and calculating the proportion of left-handed students in that sample.

The given probability is an empirical probability. This is because it is based on actual data obtained by observing or measuring the occurrence of an event, in this case, the proportion of left-handed students in a certain population. Empirical probabilities rely on empirical evidence and can vary from one sample to another.

In contrast, classical probability is based on theoretical probabilities calculated by assuming that all outcomes are equally likely, while subjective probability is based on personal judgments or beliefs about the likelihood of an event.

In this case, the probability of randomly choosing a left-handed student is not based on theoretical assumptions or personal judgments but rather on actual data obtained from observation, making it an empirical probability.

The probability of choosing a left-handed student was determined by observing a sample of students and calculating the proportion of left-handed students in that sample.

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Answer:

24,968 revolutions

Step-by-step explanation:

We need to find the circumference of the wheel in order to determine how many revolutions it will take to cover a distance of 78.5 km. The formula for circumference is:

Circumference = π x diameter

Substituting the given values, we get:

Circumference = 3.14 x 1 = 3.14 meters

Now, we need to convert the distance of 78.5 km to meters, so that we can compare it with the circumference of the wheel.

1 km = 1000 meters

Therefore, 78.5 km = 78,500 meters

To find the number of revolutions taken by the wheel, we need to divide the distance traveled by the circumference of the wheel:

Number of revolutions = Distance ÷ Circumference

Number of revolutions = 78,500 ÷ 3.14

Number of revolutions = 24,968.15

Therefore, it will take approximately 24,968 revolutions for the wheel of the truck to cover a distance of 78.5 km.

Hopes this helps

Answer:

Jaime is correct, -1 x n is always equal to n ÷ (-1) for all values of n.

Thus, we can see that -1 x n and n ÷ (-1) both simplify to -n. Therefore, Jaime is correct, the value of -1 x n is always equal to the value of n ÷ (-1) for all values of n.

Answer:

The population of Tokyo is 50 times larger than that of Detroit.

Step-by-step explanation:

The population of Tokyo in 2011 was 3.5 x 10^7 (35,000,000) and the population of Detroit was 7 x 10^5 (700,000).

To calculate how many times larger the population of Tokyo is than that of Detroit, we can divide the population of Tokyo by the population of Detroit:

Population of Tokyo / Population of Detroit = (3.5 x 10^7) / (7 x 10^5)

Simplifying this expression, we get:

Population of Tokyo / Population of Detroit = 50

Therefore, the population of Tokyo is 50 times larger than that of Detroit.

To determine the number of triangles in any given level of the pyramid, we can use the formula:

f(l) = 3(l-1)^2 + 1

where l represents the level of the pyramid.

The formula can be derived by noticing that each level of the pyramid consists of a square with sides of length l-1, and four triangles attached to each side of the square. Each of these triangles has a base of length l-1 and a height of l-2. Therefore, the area of each triangle is (1/2)(l-1)(l-2), and the total area of the four triangles on each level is 2(l-1)(l-2). Adding this to the area of the square, which is (l-1)^2, gives the total number of toothpicks in the level: 3(l-1)^2. Finally, we add 1 to account for the top toothpick.

Thus, the function is:

f(l) = 3(l-1)^2 + 1

Answer:

7.3km

Step-by-step explanation:

Using Pythagoras Theorem

D² = 7² + 2²

D = 7.280

Rounding off D = 7.3 km

Answer:

  7.3 km

Step-by-step explanation:

You want to know the straight-line distance between Ryan's house and Emmy's house if Ryan's is 7 km north of Shannon's and Emmy's is 2 km east of Shannon's.

The problem can be modeled by a right triangle with legs equal to the distances from Shannon's house to Ryan's and Emmy's houses. The hypotenuse of the triangle is the straight line between Ryan's and Emmy's.

The length of the hypotenuse can be found using the Pythagorean theorem.

  c² = a² +b²

  c² = 7² +2² = 49 +4 = 53

  c = √53 ≈ 7.3

The straight-line distance between Ryan's and Emmy's houses is about 7.3 km.

Answer:

Below.

Step-by-step explanation:

Let the number of bananas I ate be x.

Number of apples [tex]= 2x[/tex].

Number of blueberries [tex]= 3x[/tex].

So we have the equation:

[tex]x + 2x +3x = 12[/tex]

[tex]6x = 12[/tex]

[tex]x = 2[/tex].

So I ate 2 bananas, 4 apples and 6 blueberries.

Step-by-step explanation:

2x + 2y = 108   and    x*y = 720   so  y = 720 /x    sub into first equation

2x + 2 * 720/x  = 108     multiply through by 'x'

2x^2 - 108x  + 1440 = 0   Use quadratic formula to find  x= 24,30

so the dimensions of the field are 24m  x 30m    

  to make a square by reducing a dimension means reducing 30 to 24 :

       a reduction of 6 out of 30

              6/30   * 100% = 20 %   reduction