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write a quantitative survey question(s)! i need a question that I can gather data on and I cant write a good enough question for statistics and geometry!
dont make it too simple please

Answers

Answer 1

The triangle is congruent by the Angle side Angle property.

What is congruency?

In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other. The meaning of congruency is that the two shapes are similar as well as equal in size.

The Angle side Angle Congruence Theorem (ASA) defines two triangles to be congruent to each other if the included two angles and one side of one are congruent to the included angle and corresponding two sides of the other triangle.

The three parameters that are congruent are:-

BV ≅ BV ( common side)

∠1 ≅ ∠2  ( Bisector)

∠3 ≅ ∠4  ( Bisector)

Hence, the two triangles are congruent with each other.

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Related Questions

The area of square field is 100 sqm. Find the perimeter of the square.

Answers

The perimeter of the square is 40m

How to determine the value of the perimeter

The formula for calculating the area of a square is given as;

A =  a²

Given that;

A is the area of the squarea is the length of the side of the square

Substitute the value

100 = a²

Find the square root of both sides

a = 10m

The formula for the perimeter of a square is expressed as;

Perimeter = 4a

Substitute the value into the formula;

Perimeter = 4(10)

expand the bracket

Perimeter = 40 m

Hence, the value is  40 m

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3
How many full 4 3/4in. sheets can be cut from 22 1/8in. stock?
3
The 22 1/8in stock can be cut into full sheets of length 4 3/4in.

Answers

The number of sheets that can be cut from 22 1/8 inches would be 4 25/38

How to solve for the division

To solve this, I would first have to convert the values to decimals

4 3/4in = 4.75 inches

22 1/8 = 22.125

To get the full length that can be cut from the 22.125, we would have to divide. That is;

22.125 / 4.75

= 4.657

4 25/38

The number of full sheet that can be cut from this would be 4 25/38

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You owe $488.32 on your credit card. Because you have not paid your bill on time, you have to add a past due charge of $29, an over limit fee of $50, and a service charge of 11% of the amount owed on the credit card (without the past due charge and over limit fees). If you wish to send a check for the full amount immediately, how much should you write the check for? Round your answer to the nearest cent, if necessary.

Answers

I understand that I owe $488.32 on my credit card, plus a past due charge of $29, an over limit fee of $50, and a service charge of 11% of the amount owed on the credit card (without the past due charge and over limit fee). This brings my total amount due to $619.54.

Use the distributive property to write an equivalent expression. 6a + 9 =

Answers

Answer: 15a

Step-by-step explanation: 6 + 9 = 15, 6a + 9 = 15a

HCF of 183 260 and 2 900 898​

Answers

The HCF of the numbers is 1078

How to determine the HCF of the numbers

From the question, we have the following parameters that can be used in our computation:

183260 and 2900898

Next, we factorize each number

This is represented as

183260 = 1078 * 170

2900898 = 1078 * 2691

The common factor above is

Comon factor = 1078

Hence. the HCF is 1078

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According to the Centers for Disease Control
and Prevention, kids should get
at least 10 minutes of exercise each day
at least 30 minutes of exercise each day
at least 60 minutes of exercise each day
Dat least 3 hours of exercise each day

Answers

According to the Centers for Disease Control

and Prevention, kids should get C) at least 60 minutes of exercise each day.

What's exercise

Exercise is a movement activity that has an important role for health. With exercise, one's body becomes healthier and stronger. Not only maintaining physical health, exercise also affects spiritual health.

The aim of exercise is to improve one's health. In addition to improving physical fitness, exercise can also improve and improve the performance of the human brain so that it can work more optimally. Therefore, exercise is considered to be able to increase one's self-confidence.

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Aniya has just become the world champion of cartwheels. She is able to travel at 0.5 mph doing cartwheels. She went a distance of 0.1 miles. How long did Aniya do cartwheels?

Answers

The time it took Aniya to do cartwheels, which can travel at 0.5 mph for a total distance of 0.1 miles, is 12 minutes.

What are the distance, time, and speed formulas?

The distance, time, and average speed formulas are based on distance equals to time multiplied by the average speed.

Conversely, the total time taken to cover a certain distance is equal to distance divided by the average speed.

Total distance traveled doing cartwheels = 0.1 miles

Average speed = 0.5 mph

Time = Distance/Average speed

= 0.1/0.5

= 0.2 hours

1 hour = 60 minutes

= 12 minutes (0.2 x 60)

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The arm span and foot length were both measured (in centimeters) for each of 20 students in a biology class. The computer output displays the regression analysis.


Which of the following is the best interpretation of the standard deviation of the residuals?

A. The typical arm span is 161 centimeters.
B. The typical foot length is 16.1 centimeters.
C. The typical distance between the observed and predicted arm spans is 1.61 centimeters.
D. The typical distance between the observed and predicted foot lengths is 1.61 centimeters.

Answers

Therefore , the solution of the given problem of standard deviation comes out to be the typical distance between the observed and predicted foot lengths is 1.61 centimeters.

Definition of standard deviation

Variance is a measure of difference used in statistics. The average variance between the data and indeed the mean is calculated using the mathematical formula of that figure. By comparing each data point's value to the mean, it incorporates all data points into its computations, unlike some other number measures of variability.

Here,

D. The typical distance between the observed and predicted foot lengths is 1.61 centimeters.

The standard deviation of the residuals measures the typical distance between the observed values and the predicted values (based on the regression line).

In this case, the regression analysis is for foot length and arm span, and the standard deviation of the residuals tells us about the typical distance between the observed foot lengths and the predicted foot lengths based on the regression line.

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Three classes ordered lunch from a restaurant. In one class 5/6 of the children ordered hamburger. In the other class 2/4 ordered hot dogs. 1/2 of the other class ordered chicken wings. What is the total number of children that ordered sandwiches?

Answers

The number of children who ordered hamburgers from class A = ( 5/6 )A , where A is the total number of students in the class A

What is an Equation?

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given data ,

Let the number of students in class A = A

Let the number of students in class B = B

Let the number of students in class C = C

Now , the equation will be

The number of students who ordered hamburgers is = ( 5/6 )A

The number of students who ordered hot dogs is = ( 2/4 )B

The number of students who ordered hot dogs is = ( 1/2 )B

The number of students who ordered chicken wings is = ( 1/2 )C

Hence , the number of students who ordered hamburgers , hot dogs and chicken wings are ( 5/6 )A , ( 1/2 )B and ( 1/2 )C respectively

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Using the graph given below, state the value of lim f(x).

Answers

The limit when we approach x = 3 from the right side is:

[tex]\lim_{x \to 3^+} f(x) = 4[/tex]

What is the value of the limit?

Here we have the graph of a function f(x), and we want to find the value of the limit:

[tex]\lim_{x \to 3^+} f(x)[/tex]

That would be the value of f(x) as it approaches x from the right side, in this case, will be the value on the graph when we go from right to left.

We can see an open circle at y = 4

Then we can conclude that the value of the limit when we approach x  = 3 from the positive side is:

[tex]\lim_{x \to 3^+} f(x) = 4[/tex]

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Linear Programming Examples
A Diet Example
Breathtakers, a health and fitness center, operates a morning fitness program for senior citizens. The program includes aerobic exercise, either swimming or step exercise, followed by a healthy breakfast in the dining room. Breathtakers’ dietitian wants to develop a breakfast that will be high in calories, calcium, protein, and fiber, which are especially important to senior citizens, but low in fat and cholesterol. She also wants to minimize cost. She has selected the following possible food items, whose individual nutrient contributions and cost from which to develop a standard breakfast menu are shown in the following table:

The dietitian wants the breakfast to include at least 420 calories, 5 milligrams of iron, 400 milligrams of calcium, 20 grams of protein, and 12 grams of fiber. Furthermore, she wants to limit fat to no more than 20 grams and cholesterol to 30 milligrams.

Answers

Therefore , the solution of the given problem of linear equation comes out to be the decision variables x1, x2, x3, and x4 must be non-negative.

What exactly is a linear equation?

The formula y=mx+b is used to draw a simple regression graph. The inclination is B, and the y-intercept is m. It is frequently referred to as a "simple algorithm incorporating many factors" even though the section before it includes independent parts. Bivariate linear equations only contain two variables. Application problems with linear equations have no known answers. Y=mx+b, also known as mx+b, provides a number of straightforward equations.

Here,

To solve this linear programming problem, we can use the following decision variables:

Let x1 = number of pancakes

Let x2 = number of eggs

Let x3 = number of sausage patties

Let x4 = number of slices of toast

The objective function is to minimize cost:

Minimize Z = 0.5x1 + 0.2x2 + 0.3x3 + 0.1x4

Subject to the following constraints:

Calories: 120x1 + 90x2 + 280x3 + 70x4 >= 420

Iron: 1.2x1 + 1.8x2 + 2.4x3 + 0.8x4 >= 5

Calcium: 10x1 + 20x2 + 90x3 + 20x4 >= 400

Protein: 3x1 + 6x2 + 9x3 + 3x4 >= 20

Fiber: 2x1 + 0.5x2 + 1x3 + 1x4 >= 12

Fat: 2x1 + 6x2 + 20x3 + 1.5x4 <= 20

Cholesterol: 50x1 + 190x2 + 60x3 + 0x4 <= 30

The decision variables x1, x2, x3, and x4 must be non-negative.

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Find the volume of the figure. Express answers in terms of pi then round to the nearest whole number.

Answers

Answer:

Below

Step-by-step explanation:

Volume of a cylinder = area of end X height =  pi r^2 * h

  r = 4 in    h = 10 in

volume = pi (4)^2 * 10 =    160 pi in^3      or    ~  503 in^3

Answer:

160π in.³

Step-by-step explanation:

Get the circle area then multiply for the height. In the end, devide the result by 3,14 (π).


π* . r²                                  50,24 . 10*
π* . 4²                                 502,40 in.³ (cylinder volume)
50,24 (circle area)            

obs: 502,4 leads you to a non-comma π result, considering π as 3,14. If you round, the correct way will be transforming into 502. But 502/3,14 is equal to 159,87..., infinite non-repetitive numbers.

Result in π = 160π* in.³

r - circle radius

10* - height

π* - 3,14 (Used that way)

Mr. Green gave his students a biology test last week. Here are the test scores for each of the eleven students. Complete the grouped frequency distribution for the data. In the distribution, the frequency of a class is the number of test scores in that class. (Note that we are using a class width of 6.)

Answers

The grouped frequency distribution for the data is shown below.

What is grouped frequency distribution ?

A grouped frequency distribution is a way of summarizing and organizing raw data into a more manageable and meaningful form. The data is divided into intervals or "classes", and the number of data points that fall into each class is counted and recorded as the frequency of that class. This allows for easier interpretation and comparison of the data, as well as identifying patterns or trends. The grouped frequency distribution is commonly displayed in a table or a histogram.

 

Given that,

79, 85, 71, 83, 78, 67, 70, 87, 81, 86, 74, 67, 68, 76, 66, 72, 76, 80

Now, arranging the given data in ascending order,

66, 67, 67, 68, 70, 71, 72, 74, 76, 76, 78, 79, 80, 81, 83, 85, 86, 87

The grouped frequency distribution for the data:

Test Scores                   Frequency

65.5 - 71.5                        6

71.5 - 77.5                         4

77.5 - 83.5                        5

83.5 - 89.5                        3

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If you add​ Natalie's age and​ Fred's age, the result is 41. If you add​ Fred's age to 4 times​ Natalie's age, the result is 80. Write and solve a system of equations to find how old Fred and Natalie are

Answers

n = Natalie’s age
f = Fred’s age

n + f = 41
f + 4n = 80

Solve:
n = 41 - f
f + 4(41 - f) = 80
f + 164 -4f =80
-3f + 164 = 80
-3f = -84
f = 28

n + 28 = 41
n = 13

(28, 13)

the height of a doorway is 2 yds what is the height of the doorway inches

Answers

Answer:

72 inches

Step-by-step explanation:

What we know:

1 yard = 3 feet

3 feet = 36 inches

If 1 yard equals 3 feet, which equals 36 inches, then we can multiply 1 yard, 3 feet, and 36 inches by 2.

1 yard * 2 = 2 yards

3 feet * 2 = 6 feet

36 inches * 2 = 72 inches.

So, the answer is 72 inches.

An oscilloscope display indicates that the period of a digital pulse train is 40 mu s. What is frequency of this waveform? A) 40 MHz B) 25 kHz C) 2.5 kHz D) Cannot be answered from the information provided The highest decimal value that can be represented as a 4-bit binary number is __________. A) 7 B) 322 C) 15 D) 8 Decimal 42 is equivalent to binary ________. A) 101010 B) 52 C) 2A D) 0100000010 Convert (1100101000110101)_2 to hexadecimal. A) 530121 B) 121035 C) 53ACl D) CA35 Which of these truth tables represents the Exclusive NOR gate? A) (A) b) (B) C) (C) D) (D) Which type of gate can be used to add two bits? A) XOR B) NOR C) XNAND D) NAND This is the timing diagram for a 2-input __________ gate. A) NAND B) Exclusive OR C) AND D) OR When mapping an SOP expression using a Karnaugh map a 0 is placed in each cell corresponding to the value of the product term. A) True B) False According to DeMorgan's theorem, ((AB)! + C)! equals ___________. A) (A + B)C B) (AB)! + C C) AB + C! D) AB + C! Which statement below best describes a Karnaugh map? A) The Karnaugh map eliminates the need for using NAND and NOR gates. B) Variable complements can be eliminated by using Karnaugh maps. C) Karnaugh maps provide a cookbook approach to simplifying Boolean expressions. D) A Karnaugh map can be used to replace Boolean rules.

Answers

C) 2.5 kHz, C) 15, A) 101010, D) CA35, A) (A), A) XOR, B) Exclusive OR, A)True, A) (A + B)C, B) Variable complements can be eliminated by using Karnaugh maps.

Tell me about Karnaugh Maps.

K-maps, commonly referred to as Karnaugh maps, are visual tools for simplifying digital circuits and Boolean algebraic formulas. In order to make Boolean functions simpler for use in digital electronics, Maurice Karnaugh first presented them in 1953. K-maps are used to consolidate terms with similar values and get rid of extraneous terms in the truth table of a Boolean function. With a grid of cells corresponding to all conceivable combinations of binary input values, the map is a two-dimensional representation of a truth table. Depending on the truth value of the relevant word in the truth table, the cells in the grid are either filled with 0s or 1s.

The fundamental benefit of employing Karnaugh maps is that they give the Boolean expression a visual representation, making it simpler to spot patterns and understand the expression. They can also be used to check the accuracy of the final expression and find flaws in digital circuits. Karnaugh maps can be used to quickly and effectively simplify complicated digital circuits, which cuts down on the time and effort needed for circuit design.

Karnaugh maps are perfect for designing and optimizing digital circuits because they may be used to simplify big or complex Boolean statements. For more effective and efficient results, they can be combined with other digital design tools, such as logic gates.

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Find a linear equation who’s graph is the straight line with given properties
(1,-8) and (-1,-1)

Answers

Answer:

y=-7/2x-9/2

Step-by-step explanation:

If bolt thread length is normally distributed, what is the probability that the thread length of a randomly selected bolt is farther than 2. 5 standard deviations from its mean value.

Answers

The probability that the thread length of a randomly selected bolt is farther than 2. 5 standard deviations from its mean value is 0.62%.

If the thread length of bolts is normally distributed, then we can use the concept of probability to determine the probability that the thread length of a randomly selected bolt is farther than 2.5 standard deviations from its mean value.

This can be calculated using the area under the normal curve.

Specifically, the area to the right of 2.5 standard deviations is 0.0062, meaning that there is only a 0.0062 probability (or 0.62%) that the thread length of a randomly selected bolt is farther than 2.5 standard deviations from its mean value

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Dana read this statement in a book about softball.

The diameter of a softball is 0.304 meter.

Which completes the statement as it would be heard?
The diameter of a softball is ______.

Answers

Answer:

Step-by-step explanation:

14

Two points on a line are (10,55) and (15,82.5). what is the slope of the line

Answers

ANSWER -

The slope of a line can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points on the line. Substituting the values from the given points, we have:

m = (82.5 - 55) / (15 - 10) = 27.5 / 5 = 5.5

So the slope of the line is 5.5.

Adam can run 10 laps in 5 minutes. How many laps can he run per minute?

Answers

To figure out how many laps he can run per minute, you can just figure out how many laps he can run in one minute.

10 laps = 5 minutes
(now divide both sides by 5)
2 laps = 1 minute


(Dividing 5 by 5 gives you one.)

So, Adam can run 2 laps per minute
adam can run 2 laps per minute

b) Write the number in scientific notation:
2,150,000 =

Answers

Answer:

2.15 * 10⁶

Step-by-step explanation:

1,000,000= 1 * 10⁶

2,150,000= 2.15 * 10⁶

Answer:

The answer is 2.15×10 power 6

if the average service rate is 6 customers per hour, and assuming the negative exponential distribution is used to describe the randomness of the service time distribution, what is the probability that the service time will be less than or equal to 6 minutes? a. 0.451 b. 0.549 c. 0.017 d. 0.632

Answers

Answer: If the service rate is 6 customers per hour, the mean service time can be calculated as 1/6 = 0.1 hours or 6 minutes. The negative exponential distribution is commonly used to model the service time distribution in queuing theory. The cumulative distribution function (CDF) of the negative exponential distribution is given by:

F(x) = 1 - e^(-λx)

where λ = 1/μ, where μ is the mean service time (in this case, μ = 0.1 hours or 6 minutes).

So, the probability that the service time will be less than or equal to 6 minutes can be calculated as:

F(6) = 1 - e^(-0.1 * 6) = 1 - e^(-0.6) ≈ 0.5488

So, the answer is approximately 0.549, which is closest to option (b).

Step-by-step explanation:

An object was launched off the top of a building. The function
f(x) = -16x² + 16x + 192 represents the height of the object above the ground,
in feet, a seconds after being launched. Find and interpret the given function values
and determine an appropriate domain for the function.

Answers

Check the picture below.

[tex]~~~~~~\textit{initial velocity in feet} \\\\ h(t) = -16t^2+v_ot+h_o \quad \begin{cases} v_o=\textit{initial velocity}&16~\frac{ft}{s}\\ \qquad \textit{of the object}\\ h_o=\textit{initial height}&192~ft\\ \qquad \textit{of the object}\\ h=\textit{object's height}&h\\ \qquad \textit{at "t" seconds} \end{cases} \\\\\\ h(t)=-16t^2+16t+192\hspace{5em}f(x)=-16x^2+16x+192[/tex]

now, keeping in mind that x-axis is the seconds, however a parabola could go into the 2nd, 3rd and 4th Quadrants, but for this case, we only need it in the 1st Quadrant, because the seconds are positive, it begins at 0 seconds and goes up, then down and hits the ground later on when f(x) = 0, hmmm when is that?

[tex]\stackrel{f(x)}{0}=\stackrel{ \textit{hits the ground} }{-16x^2+16x+192}\implies 0=-16() \\\\\\ 0=x^2-x-12\implies 0=(x+3)(x-4)\implies x= \begin{cases} -3 ~~ \bigotimes\\\\ ~~ 4 ~~ \checkmark \end{cases}[/tex]

now, both values are valid for "x", however for this specific case, we can't have negative seconds, so the negative value won't work.  So the appropriate domain or values for "x" are from 0 to 4, the object went up and down in 4 seconds flat.

Pls help Im so confused

Answers

The height of the building is 92.52 ft

What is a right-angle triangle?

It is a triangle in which one of the angles is 90 degrees and the other two are sharp angles. The sides of a right-angled triangle are known as the hypotenuse, perpendicular, and base.

Using law of sines:

first angle A:

sum of all angles in a triangle = 180 degree

A+B+C = 180

A+90+55 = 180

=> A+145 =180

=> A= 180 - 145 = 35 degrees

Use:

AB= c, BC=a

(sin A)/a = (sin C)/c

=> (sin 35)/40= (sin 55)/c

=> -0.4281/40= -0.99/c

=> 0.0107 = 0.99/c

=> c= 0.99/0.0107 = 92.52 ft

The height of the building is 92.52 ft

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Rectangle a has side lengths of 6 cm and3.5 cm. The side lengths of rectangle B are proportional to the side lengths of rectangle a

Answers

The side lengths of rectangle B, given that they are proportional to rectangle A would be :

A. 3cm and 1.75cmE. 5.25cm and 9cmHow to find out proportionality ?

To find proportionality, the dimensions of Rectangle B which correspond to Rectangle A, need to be divisible by, or be able to divide Rectangle A's dimensions without leaving a remainder.

The side lengths of 3 cm and 1. 75 cm are proportional because:

= Dimensions A / Dimensions of B

= 6 / 3 :  3. 5 / 1. 75

= 2 : 2

The side lengths of 5. 25 and 9 cm are also proportional because :

= 9 / 3 : 5 . 25 / 1 . 75

= 3 : 3

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The rest of the question is:

What could be the side lengths of rectangle B?

A. 3cm and 1.75cm

B. 5cm and 2.5cm

C.7 cm and 7cm

D. 12cm and 5 cm

E. 5.25cm and 9cm

NEED HELP ASAP!!!!

The number of salespeople assigned to work during a shift is apportioned based on the average number of customers during that shift. Apportion 21 salespeople using Hamiltons method given the information below

Answers

Using Hamilton's method of apportioning, 21 salespeople can be assigned in the table in the following order:

2, 5, 6, 8.

What do you mean by Hamilton's method of apportioning?

The approach that now bears his name was first proposed by Alexander Hamilton. Congress endorsed his approach in 1791, but President Washington rejected it. Later, in 1852, it became widely used until 1911. He starts by calculating the precise number of items that each group needs.

Since in the interested question it's about salesperson assigned, we’ll use the language of shift and hours, so we can determine how many salespersons each shift should get.

Now total no. of customers given = 125 + 305 + 439 + 515

                                                        = 1375

Now total no of salesperson available = 21.

So, the divisor = 1375/21

                        = 65.47

Now using Hamilton's method when we divide the average customers per shift by the divisor, we get the no. of salesperson assigned.

So, the salesperson assigned are:

Morning = 125/65.47

              = 2

Midday = 305/65.47

             = 5

Afternoon = 430/65.47

                 = 6

Evening = 515/65.47

               = 8 (Decimals rounded according to Hamilton's principle)

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a blueprint of a house has a scale of 2.5 in:5 feet. would a pool table that is 8 by 6 feet fit in a space that has drawing dimensions of 4 inches and 5.2 inches? if so, how much space will remain after the table is put in the room?

Answers

Answer:

yes it will fit;  there would be 35.2 sf remaining

Step-by-step explanation:

Find the dimensions of the table in inches:

2.5/5 = x/6

x = 3 in

2.5/5 = x/8

x = 4 in

Scaled table area = (3 in)(4 in) = 12 in²

Scaled room area = (4 in)(5.2 in) = 20.8 in²

12 < 20.8, Therefore, the table would fit.

How much room is left over:

Find the room dimensions in feet.

2.5/5 = 4/x

x = 8 ft

2.5/5 = 5.2/x

x = 10.4 ft

Room area = (8)(10.4) = 83.2 sf

Table area = (8)(6) = 48 sf

83.2 - 48 = 35.2 sf space would remain

Topford supplies X-Data DVDs in lots of 50, and they have a reported defect rate of 0.5% so the probability of a disk being defective is 0.005. It follows that the probability of a disk being good is 0.995. What is the probability of getting at least one defective disk in a lot of 50?

Answers

The probability of getting at least one defective disk in a lot of 50 is 0.2217.

What is probability?

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

Given that,

P (defective disc) = 0.005

P (non-defective) = 1 - 0.005 = 0.995

Now,

P(Getting one defective) = 1 - P(getting 50 non -defective)

= 1 - (0.995)^(50)

= 1- 0.7783

= 0.2217

Hence, the probability of getting at least one defective disk in a lot of 50 is 0.2217.

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Write the equation of the graph that results when:

Answers

The equation that represents each change is given as follows:

a) y = 2^x-  6.

b) y = 2^(x + 1).

c) y = -2^x.

What is a translation?

A translation happens when either a figure or a function are moved horizontally or vertically.

The four translation rules for functions are given as follows:

Left a units: f(x + a).Right a units: f(x - a).Up a units: f(x) + a.Down a units: f(x) - a.

Additionally, when a function is reflected over the x-axis, we have that the function's definition is multiplied by -1.

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