The double integral of the given equation is 101.3π/4.
What is meant by integral?
In mathematics, an integral lends numerical values to functions to represent concepts like volume, area, and displacement that result from combining infinitesimally small amounts of data. Integration is the action of locating integrals.
The integrals of a function in two variables over a region in R², or the real number plane, are referred to as double integrals in mathematics.
The surface area of a 2D figure can be determined primarily using the double integral, which is indicated by the symbol "∫∫". By using double integration, we may quickly determine the area of a rectangular region. Double integration challenges will be easier to address if we are familiar with simple integration.
We are asked to find the double integral of
[tex]\int\int\limits_r \frac{4(1+x^2)}{1+y^2} dA,[/tex], where region r = {(x, y) | 0 ≤ x ≤ 4, 0 ≤ y ≤ 1}
dA = dx.dy
We can then rewrite the equation as
[tex]\int\limits^4_0\int\limits^1_0 \frac{4(1+x^2)}{1+y^2} dydx[/tex] = [tex]\int\limits^4_04(1+x^2)dx \ .\int\limits^1_0 \frac{1}{1+y^2} dy[/tex]
For the first part,
[tex]\int\limits^4_04(1+x^2)dx = 4\int\limits^4_0(1+x^2)dx = 4[ x + x^3/3]\limits^4_0[/tex]
= 4 [ 4 + 4³/3] - 4[ 0] = 101.3
For the second part,
[tex]\int\limits^1_0 \frac{1}{1+y^2} dy[/tex]
Take y = tan u
then dy = sec²(u) du
Then the above equation becomes,
[tex]\int\limits^1_0 \frac{1}{1+tan^2(u)}. sec^2(u) du = \int\limits^1_0 \frac{1}{sec^2(u)}. sec^2(u) du[/tex]
= [tex]\int\limits^1_0 (1.du) = [u]^1_0[/tex]
We know x = tan u
so u = tan⁻¹ x
Then the equation becomes
[tan⁻¹ x]¹₀ = [ tan⁻¹ 1 - tan⁻¹ 0] = π/4 - 0 = π/4
Therefore the double integral of the given equation is 101.3π/4.
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A pizza place changed the size of their large pizzas from a 28-inch diameter to a 24-inch diameter. What is the percent change?
some of our farm fields are being left unused. does this have any implications for the economy's ppf diagram (with agricultural products on one axis and all other products on the other axis)?
Yes, the implications of leaving farm fields unused for the economy's PPF diagram can be seen in the output of agricultural products.
The PPF diagram shows the maximum output of two products that an economy can produce at a given time, and it is shaped like a curve. When farm fields are left unused, the amount of agricultural products that can be produced is decreased, resulting in a shift in the PPF curve inwards. This means that the economy has to sacrifice other goods to maintain the same level of production of agricultural products, leading to an opportunity cost. Thus, leaving farm fields unused has implications for the economy's PPF diagram, as it reduces the output of agricultural products, resulting in a shift in the PPF curve inwards.
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what is a quadrilateral symmertrical across both diagonals
A rhombus has reflectional symmetry over either of its diagonals.
Define quadrilateral?In geometry, a quadrilateral is a four-sided polygon with four edges and four corners. The word is derived from the Latin words quadri (a variant of four) and latus (a side).Quadrilaterals are shapes with four sides and four corners, or vertices, and two of the sides are often parallel, which means they are always the same distance apart no matter how long they stretch. Squares and rectangles are the most common quadrilaterals, but there are also trapezoids, parallelograms, and kites.A quadrilateral is a closed shape and type of polygon with four sides, four vertices, and four angles. It is formed by connecting four non-collinear points.The sum of the interior angles of quadrilaterals is always 360 degrees.To learn more about quadrilateral refer to:
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How can you add and subtract positive and negative numbers in complicated expressions?
When dealing with complicated expressions that involve the addition and subtraction of positive and negative numbers, it is important to remember the following rules.
What is numbers?Nobody's a mathematical object used to count level and measure it is in the fundamental concept in mathematics and you used in many different way number can be used to represent anything from quantity of object to ideas in the concept additionally number are used to describe relationship between different object or concept.
First, when adding or subtracting two numbers with the same sign, the sum or difference will have the same sign as the numbers. For example, adding 5 + 3 gives 8 and subtracting 5 – 3 gives 2, both of which are positive numbers.
Second, when adding or subtracting two numbers with different signs, the sum or difference will have the sign of the number with the largest absolute value. For example, adding 5 + (-3) gives 2 and subtracting 5 – (-3) gives 8, both of which are positive numbers.
Third, when multiplying and dividing positive and negative numbers, the product or quotient will have the sign of the number with the largest absolute value. For example, multiplying 5 x (-3) gives -15 and dividing 5 / (-3) gives -1.67, both of which are negative numbers.
Finally, when dealing with expressions with multiple operations, it is important to use the correct order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This allows for the correct evaluation of an expression.
By remembering these rules, it is possible to add and subtract positive and negative numbers in complicated expressions.
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Which question can be represented by the equation 4 2/7 =
What is 4 groups of 2/7 s?
How many 2/7 s are in 4?
What is 2/7 of 4?
How many 4s are in 2/7?
The equation 4 2/7 = can be represented by the question: "What is 2/7 of 4?"
What is the arithmetic expressions?
An arithmetic expression is a mathematical phrase that can contain numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division.
The equation 4 2/7 = is asking for the value of 2/7 times 4. The right side of the equation is a mathematical expression that represents the product of 2/7 and 4. The value of this expression can be found by performing the calculation and replacing the expression with the resulting numerical value. In this case, 2/7 of 4 can be found by dividing 2 by 7 and then multiplying that result by 4. The answer to this question would give the value of 2/7 times 4.
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The variables x
and y
vary directly. Use the values to find the constant of proportionality, k
. Then write an equation that relates x
and y
.
y=72; x=3
Answer: k = 24, y = 24x
Step-by-step explanation:
y = kx (the variables x and y varies directly)
when x = 3, y = 72
k = y/x = 72/3 = 24
y = 24x
Find a power series representation for the function.
f(x) = x (1 + 8x)2
f(x) =[infinity] n = 0
Determine the radius of convergence, R.
R =
As a result, the radius of convergence, R, is equal to one. As a result, f(x) = x(1 + 8x)^2 = x(1 + 16x + 64x^2 +...) = x + 16x^2 + 64x^3 +... This is the function f's power series representation (x).
What is function?A function is an equation with just one solution for y for every x. A function produces exactly one output for each input of a certain type. Instead of y, it is usual to call a function f(x) or g(x). f(2) indicates that we should discover our function's value when x equals 2. Example.
Here,
The power series representation for f(x) = x(1 + 8x)^2 can be found by expanding the function (1 + 8x)^2 as a Taylor series centered at x = 0:
(1 + 8x)^2 = 1 + 16x + 64x^2 + ...
So, f(x) = x(1 + 8x)^2 = x(1 + 16x + 64x^2 + ...) = x + 16x^2 + 64x^3 + ...
This is the power series representation for the function f(x).
To determine the radius of convergence, R, we use the formula:
R = 1 / limsup |a_n|^(1/n)
where a_n is the nth coefficient in the power series and limsup refers to the limit superior of the sequence {|a_n|^(1/n)}.
In this case, a_n = 16n x^n, so |a_n|^(1/n) = |16n|^(1/n) * |x|^(1/n). Since |16n|^(1/n) is a constant, the limit superior is just equal to |x|^(1/n). Therefore,
R = 1 / limsup |a_n|^(1/n) = 1 / |x|^(1/n) = 1
So, the radius of convergence, R, is equal to 1. So, f(x) = x(1 + 8x)^2 = x(1 + 16x + 64x^2 + ...) = x + 16x^2 + 64x^3 + ... This is the power series representation for the function f(x).
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what is the resultant image of point B (9,-2) after a scale factor of 4, then a transition up 4 units?
Answer:
(36, -4).
Step-by-step explanation:
The scaled point B (9,-2) would be (36,-8) and after the translation up 4 units, the final image of B would be (36, -4).
A political scientist was interested in studying america's voting habits. So, he decided to make a least squares regression equation to predict the percentage of people in a state that would vote for obama in 2012 based on the percentage of people in a state that voted for obama in 2008. The least squares equation is y-hat = -4. 599 +1. 04x. In the state of wyoming, 32. 54% voted for obama in 2008; whereas, 27. 82% voted for him in 2012. What is the value of the residual?.
The residual in this case is 4.72%, which is calculated by taking the difference between the predicted value (32.54%) and the actual value (27.82%).
The political scientist's least squares regression equation predicted that 32.54% of people in Wyoming would vote for Obama in 2012. However, the actual percentage of people who voted for Obama in Wyoming in 2012 was 27.82%, which was lower than the predicted value. This difference between the predicted and actual value is the residual, which in this case is 4.72%. This suggests that the least squares equation was not very accurate in predicting the voting habits in Wyoming for the 2012 election.
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Consider the given matrix A and right-hand sides 6, and b2: [o.6 A = ~0.8 [ijp]. 6, = [o.8 [4 6= [1] 0.6 Find the pseudoinverse A+ Compute the projection matrices AtA and AA+ _ Find the ~best" solution z+ = A+b for each right-hand side. Which solutions fromn (c) arC exact? Which solutions are unique?
The "best" solution z+ = A^+b is the one that minimizes the 2-norm of the residual vector r = b - Az. In general, not all right-hand sides b will have a unique solution z+, and even if there is a solution, it may not be exact (i.e., Az ≠ b).
To find the pseudoinverse of a matrix A, the first step is to take the singular value decomposition (SVD) of A, which can be written as A = UΣV^T, where U and V are orthogonal matrices and Σ is a diagonal matrix with nonnegative singular values. The pseudoinverse of A is given by A^+ = VΣ^+U^T, where Σ^+ is the same as Σ except that the nonzero elements of Σ are replaced by their inverses.
Next, the projection matrices are given by AtA and AA+, which are used to project a vector onto the column space and the row space of A, respectively.
The "best" solution z+ = A^+b is the one that minimizes the 2-norm of the residual vector r = b - Az. In general, not all right-hand sides b will have a unique solution z+, and even if there is a solution, it may not be exact (i.e., Az ≠ b).
Whether a solution is exact or unique depends on the rank of A and the number of linearly independent columns and rows. If the rank of A is equal to the number of columns, then the solution is unique, otherwise, the solution may not be unique.
Therefore, The "best" solution z+ = A^+b is the one that minimizes the 2-norm of the residual vector r = b - Az. In general, not all right-hand sides b will have a unique solution z+, and even if there is a solution, it may not be exact (i.e., Az ≠ b).
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WILL GIVE BRAINLIEST TO FIRST ANSWER, EASY POINTS
Which of the following tables represents a linear relationship that is also proportional?
x −1 2 5
y −1 0 1
x −2 −1 0
y −1 0 1
x −2 −1 0
y −4 −2 0
x −4 −2 1
y −2 0 3
Answer:
x −2 −1 0
y −1 0 1
Step-by-step explanation:
Condition:
A linear relationship is proportional if the ratio of the change in the dependent variable to the change in the independent variable is constant. This constant is known as the proportionality constant and can be represented by "k" in the equation y = kx. When a linear relationship is proportional, there is a direct relationship between the variables and as one increases, the other increases and vice versa.
The second table (x = -2, -1, 0 and y = -1, 0, 1) represents a linear relationship that is also proportional. A proportional relationship means that the ratio of y to x remains constant, or in other words, y is a constant multiple of x. In this table, for every 1 unit increase in x, y increases by 1 unit, making it proportional.
The third table represents a linear relationship that is also proportional.
an event that has a probality of being 1 is said to be
Answer: Guaranteed
Step-by-step explanation:
Answer: Certain to happen
Hope this helps :)
Step-by-step explanation:
It's important to remember that when regarding probabilities with decimals, 0 is considered least likely, and 1 is most likely.
Need help ASAP PLEASE!
Using the fact that the sum of the interior angles must be 180°, we will get:
∠B = 101°
XYZ = 76°
How to find the measure of angle B?Let's solve the problem for the first triangle.
Ok, remember that the sum of the interior angles of any triangle is always equal to 180°, then we can write:
∠B + ∠C + ∠A = 180°
Replacing the values that we know we will get:
∠B + 44° + 35° = 180°
Now we can solve that equation to get:
∠B = 180° - 44° - 35° = 101°
So the answer is 101°.
For the second triangle, the interior angles are:
(5x + 1)
(3x + 11)
And, the one at the right is:
180° - 12x + 20
Then the sum gives:
(5x + 1) + (3x+ 11) + 180° - 12x + 20 = 180
5x + 1 + 3x + 11 - 12x + 20 = 0
-4x + 32 = 0
-4x = -32
x = 32/4 = 8
And XYZ = 12x - 20 = 12*8 - 20 = 76
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5 T-shirts cost R120. How much will 9 cost?
Riggoletto's Portraits specializes in making circular portraits of people. A large portrait is 18 in. In diameter. A medium portrait is 6 in. In diameter. Estimate the difference between the areas of the two sizes of portraits. Use 3. 14 for T. Round your answer to the nearest whole number.
The estimated difference between the areas of the two sizes of portraits is 226.
First, find the area of the large portrait:
A = πr^2 = π(9)^2 = 3.14 * 81 = 253.94
Next, find the area of the medium portrait:
A = πr^2 = π(3)^2 = 3.14 * 9 = 28.26
Finally, subtract the area of the medium portrait from the area of the large portrait to find the difference:
253.94 - 28.26 = 225.68
Rounding to the nearest whole number, the difference is 226.
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Your foot measures 10.8 inches. what proportion can you use to determine how many centimeters, x, it is? :A) 10.8in. 2.54in. xcm 1cm 2.54in. B) 10.8cm xin lcm C) 10.8in lin. хст 2.54cm D) 10.8in 2.54cm XCM lin
Your foot measure in centimeter is 4.25 centimeters and the proportion is 10.8 inches / 2.54 centimeters = x centimeters.
A proportion is an equation that states that two ratios are equal. It's often used to convert from one unit of measurement to another.
In your case, you want to convert 10.8 inches to centimeters, so you'll use a proportion to help you do that.
Here the correct proportion to use is 10.8 inches / 2.54 centimeters = x centimeters.
In this proportion, the first ratio on the left side of the equation represents 10.8 inches in terms of centimeters, and the second ratio on the right side of the equation represents x centimeters in terms of inches.
Since we know the conversion factor between inches and centimeters, 2.54 centimeters per inch, we can substitute that into the proportion.
=> 10.8 inches / 2.54 centimeters = x centimeters
=> 10.8 inches / 2.54 centimeters
=> 10.8 / 2.54 centimeters = 4.25 centimeters.
Therefore, 10.8 inches is equal to 4.25 centimeters.
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Find the distance between the points (–5,5) and (9.6,5).
The distance between the two points is 14.6 units. The distance is always positive.
What is a point?
A point is a basic concept in classical Euclidean geometry that represents an exact location in space and has no length, width, or thickness. In contemporary mathematics, a point more broadly refers to a component of a set known as a space.
Given points are (–5,5) and (9.6,5).
The y-coordinate of both points is the same. The distance between two points is the absolute value of the difference of x-coordinates.
The x-coordinate of (–5,5) is -5.
The x-coordinate of (9.6,5) is 9.6.
The distance between points is
|-5 - 9.6| units
=|-14.6| units
= 14.6 units
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Pls Help me on this problem it is number 19 I have no idea what to do
Answer:
$81.15
Step-by-step explanation:
So to find the total amount of money, you take the original price and add the sales tax.
We need to find the sales tax first, which is telling us to multiply the original price by 0.065.. so lets do that
(79.95)(0.065) = 5.19675
Since we got that answer there, we add that to the original price
$79.95 + $5.19675 = $81.14675
Which rounds to $81.15
For George’s party, his dad bought an ice cream cake. After the party,
1/3 of the cake was left. The next day, George ate half of what was left. Choose the model that shows what fraction of cake George ate.
Answer: 4 aka the one on the bottom
Step-by-step explanation: There is 3 sections, only one filled in to make 1/3, then you split that in half
Answer:
1/6
Step-by-step explanation:
If we know that only 1/3 of the ice cream cake was left, and George ate 1/2 of that, we can use the equation:
1/3 ÷ 2 = 1/6
If we were to divide a fraction, we would take the denominator, in this case its 3, and multiply that by the amount being divided, which is 2.
2 × 3 = 6
Now that we have 6, simply add it back to the fraction denominator with the same numerator which is 1. This would make it 1/6.
Therefore, the picture that best matches this is the last one.
find the z-score for which the area under the standard normal curve to its left is 0.96. round to two decimal places
The z-score for which the area under the standard normal curve to its left is 0.96 is approximately 1.75.
The z-score for which the area under the standard normal curve to its left is 0.96 can be found using a standard normal table or a calculator with a standard normal cumulative distribution function (CDF).
Using a standard normal table, we find that the z-score that corresponds to an area of 0.96 to its left is approximately 1.75.
Alternatively, using a calculator with a standard normal CDF, we can calculate the inverse CDF of 0.96:
z = invNorm(0.96)
This gives us a z-score of approximately 1.75.
Therefore, the z-score for which the area under the standard normal curve to its left is 0.96 is approximately 1.75, rounded to two decimal places.
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Please help i got stuck!!!!
Divide using polynomial long division.
[tex](6x^2-5x+9) / (2x-1)[/tex]
Please help!!!!!!
Answer: 36=2547^3
Step-by-step explanation: calulat0r your welcome
Simplify. Assume all variables are positive.
Answer:[tex]\sqrt[9^4]{10x}[/tex] there that should be ur answer ad ur hot
Step-by-step explanation:
-3 = -2(h-1) + 5
h =
Answer:
h = 5
Step-by-step explanation:
-3 = -2(h-1) + 5
-3 = -2h + 2 + 5
-3 = -2h + 7
-10 = -2h
h = 5
Let's check
-3 = -2(5-1) + 5
-3 = -2(4) + 5
-3 = -8 + 5
-3 = -3
So, h = 5 is the correct answer.
A personnel director interviewing 11 senior engineers forfour job openings has scheduled six interviews for the firstday and five for the second day of interviewing. Assumethat the candidates are interviewed in random order.
a. What is the probability that x of the top four candidatesare interviewed on the first day?
b. How many of the top four candidates can be expected tobe interviewed on the first day?
All parts of the probelm given regarding probability have been solved and provided below.
We have been told that a personnel director interviewed 11 senior engineers for four job openings and had scheduled six interviews for the first day and five for the second day of interviewing. Now let us assume that the candidates are interviewed in a random order.
(a) The probability that any quantity, lets say x, of the top four candidates are interviewed on the first day can be calculated using -
= h(x; 6,4,11) = (4;x) (7;6-x) / (11;6)
(b) The expected number of candidates, if top four candidates to be interviewed on the first day can be found out using -
= E[X]=nM/N
= 6*4 / 11
= 2.1812
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A company plans to build houses. The company will use 23
acres of land to build the houses. Each home will be built on acres. How many houses will the company build?
The number of houses that can be built are 94.
What is division?A division is a process of splitting a specific amount into equal parts.
Given that, a construction company plans to build houses. The company will use [tex]23\frac{1}{2}[/tex] acres of land to build houses. Each house will be built on a 1/4 acre lot.
We need to find the number of houses that can be built,
To find the number of houses, we will divide total area by area of one house.
Therefore,
Number of house = [tex]23\frac{1}{2}[/tex] ÷ 1/4
= 47/2 ÷ 1/4
= 47 x 2
= 94
Hence, the number of houses that can be built are 94.
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Monthly maintenance costs are distributednormally with a μ=$250 and σ=$50. What percent of months have maintenancecosts in the range of $200 to $300?
68.26% percent of months have maintenance costs in the range of $200 to $300
How do you calculate maintenance costs?The monthly maintenance cost per square foot is calculated by dividing the total cost by the property's total area. It is multiplied by the unit area of each home to determine the individual maintenance contributions.
Your home's square footage can also be used to estimate your yearly maintenance expenses. If you follow this guideline, you should begin by figuring annual maintenance expenditures at $1 per square foot. Your estimated annual maintenance cost for a 2,000 square foot home is $2,000.
Costs incurred during normal measures to maintain an asset's original state are referred to as maintenance costs. Simple electrical repairs, bulb replacement, paint touch-ups, pool cleaning, etc. are a few examples of maintenance charges.
Remember that, [tex]P(X < A) = P(Z < (A - \mu )/\sigma)[/tex]
and given that \mu = [tex]$250\ and \ \sigma = \$50[/tex]
So, P(maintenance cost is in the range $200 to $300) = P(200 < X < 300)
= P(X < 300) - P(X < 200)
= P(Z < (300 - 250)/50) - P(Z < (200 - 250)/50)
= P(Z < 1) - P(Z < -1)
= 0.8413 - 0.1587
= 0.6826
= 68.26%
Thus, 68.26% percent of months have maintenance costs in the range of $200 to $300
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Determine the intervals on which the following function is continuous. f(x) = x + 6/x^2 - 36 The function is continuous on the interval(s) (Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed.)
Function is continuous everywhere expect x = 6 and x = -6
What does a math interval mean?
In mathematics, an interval is quantified in terms of numbers. The range of numbers between any two specific numbers is known as the interval.
Every real number between those two values is included in this range. All kinds of numbers, including imaginary ones, are real numbers.
The function will not be continuous if the denominator becomes zero .
f(x) = x+6/(x² - 36 )
to become denominator zero -
x² - 36 = 0
(x + 6) ( x - 6 ) =0
x = 6 or x = -6
so at x = 6 and x = -6 the function is not continuous .
the interval for continuity is -
= ( -∞ , - 6), ( -6 , 6) , ( 6, ∞ )
In other words function is continuous everywhere expect x = 6 and
x = -6
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how to determine how many rectangles we should use in order for our estimate to be accurate within some range.
To determine the number of rectangles needed to achieve a desired accuracy, one could start with a small number of rectangles, calculate the approximation error, and then increase the number of rectangles until the error is within the desired range.
How determine the number of rectangles?The number of rectangles to use in a rectangle approximation of a definite integral depends on several factors:
The desired accuracy: The smaller the desired accuracy, the more rectangles will be required.
The shape of the function: If the function is highly curved, more rectangles will be required to capture the shape of the function than if the function is more linear.
The range of integration: The larger the range of integration, the more rectangles will be required.
In general, to determine the number of rectangles needed to achieve a desired accuracy, one could start with a small number of rectangles, calculate the approximation error, and then increase the number of rectangles until the error is within the desired range. Alternatively, one can use mathematical techniques such as adaptive quadrature methods that automatically adjust the number of rectangles to achieve a desired accuracy.
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a student in a statistics class decided to analyze the number of candies per package for a project. a dot plot of the counts of candies in one package is shown below. what was the most common number of candies in a package?
To determine the most common number of candies in a package, the student would need to examine the dot plot and identify the value(s) with the highest frequency.
A dot plot is a type of graph that can be used to display the distribution of data by plotting individual observations as dots on a number line. To determine the most common number of candies in a package, the student would need to look at the dot plot and identify which value appears most frequently.
Typically, the most common number of candies in a package, also known as the mode, would be the value that has the highest number of dots plotted on the number line. If there are multiple values that have the highest frequency, then the data set is considered to have multiple modes. If all values in the data set appear with equal frequency, then the data set is considered to have no mode.
Thus, In conclusion, to determine the most common number of candies in a package, the student would need to examine the dot plot and identify the value(s) with the highest frequency.
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The most common number of candies in a package is 60
To analyze the number of candies per package for the project
60 candidates make up the majority of packages.
For a project in statistics class, a student chose to examine how many candies are contained in each container.
The maximum number of dots in a dot plot of the candy counts in one package is 60.
60 are the maximum limit of candies that can be placed in each and every package so these are the maximum number of candies that can be placed
The average candidate package has 60
So the most number of candies in each package are 60
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Please help im begging!
Answer:(3*7)
Step-by-step explanation: