Answer:
The inequality can be solved by isolating the variable x on one side of the inequality. To do this, we can first add 4x to both sides of the inequality:
-2 ≤ 4x + 3 ≤ 4
Next, we can subtract 3 from both sides of the inequality:
-5 ≤ 4x ≤ 1
Finally, we can divide both sides of the inequality by 4 to solve for x:
x ≥ -5/4 and x ≤ 1/4
So the solution of the compound inequality is x ≥ -5/4 and x ≤ 1/4
We can also express this solution in interval notation: [-5/4, 1/4]
Note that the compound inequality has two separate inequality signs, indicating that the solution is the set of all x-values that make both inequalities true.
Step-by-step explanation:
Find the area of all shaded regions. Give your answer as a completely simplified exact value in terms of pi. (no approximations) WHO EVER ANSWERS FIRST GETS 100 BRAINLY POINTS
Answer:
18 π cm^2.Step-by-step explanation:
The area of the whole circle = π r^2 = 81π cm^2.Since there are 360 degrees in a circle the area of the shaded region = 80/360 * area of the circle.
This = 80/360 * 81 π
= 2/9 * 81 π
= 18 π
hope it helps<3Answer:
Area = 18π cm²
Step-by-step explanation:
The shaded region is a sector of a circle with radius 9 cm.
[tex]\boxed{\begin{minipage}{6.4 cm}\underline{Area of a sector}\\\\$A=\left(\dfrac{\theta}{360^{\circ}}\right) \pi r^2$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in degrees.\\\end{minipage}}[/tex]
From inspection of the give diagram:
θ = 80°r = 9 cmSubstitute the values into the formula for area of a sector:
[tex]\implies A=\left(\dfrac{80^{\circ}}{360^{\circ}}\right) \pi \cdot 9^2[/tex]
[tex]\implies A=\dfrac{2}{9} \pi \cdot 81[/tex]
[tex]\implies A=\dfrac{162}{9} \pi[/tex]
[tex]\implies A=18 \pi\;\; \f cm^2[/tex]
Therefore, the area of the shaded region is 18π cm².
if two differnt people are randomly selected, without replacement, from the 884 subjects, find the probability that they are both women. Round to four decimal places.O 0.00003906O 3274O 2500O 3276
The probability of randomly selecting two women from 884 subjects is 1.
The probability of randomly selecting two women from 884 subjects can be calculated using the following formula: P(A and B) = P(A) x P(B). In this case, A is the event of randomly selecting a woman from 884 subjects and B is the event of randomly selecting a second woman from the remaining 883 subjects.
Therefore, P(A) = 884/884 = 1 and P(B) = 883/883 = 1. Thus, P(A and B) = 1 x 1 = 1. Then, the probability of randomly selecting two women from 884 subjects is 1. To round to four decimal places, the probability is 0.0000.
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Anything help’s please!!
The value of x is 6, which can be calculated using triangle proportionality theorem.
What is the Triangle proportionality theorem?A triangle's other two sides are divided in the same ratio if a line drawn parallel to either one of its sides intersects the other two sides at two different spots.The Triangle Proportionality Theorem, often known as the Thales Theorem, was developed by the well-known Greek mathematician Thales.Triangle Proportionality Theorem asserted that the ratio of any two related sides is always the same for any two equiangular triangles.Thales provided the Triangle Proportionality Theorem based on this idea (BPT). Similar triangles have introduced this idea before. The corresponding angles of two triangles are equal if they are identical.From the figure, we have both triangles' corresponding sides proportionate to one another.
so, PR/ PR' = RQ/ RQ'
so, x/10 = 9/15
x = 6
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Rewrite these numbers in an ordered array: 312 158 73 43 305 226 107
Do not use the factorial key on your calculator. 103! / 101! = 122!/(119! 3!)
An ordered array is an arrangement of numbers in a specific order, usually in increasing or decreasing order. 43, 73, 107, 158, 226, 305, 312
To rewrite the numbers in an ordered array: you need to arrange the numbers in either ascending or descending order. For example, the ordered array of the numbers in ascending order: 43 73 107 158 226 305 312
Note that it's important to read the instruction carefully when you have to order the numbers without using the factorial key on your calculator.
The factorial key on a calculator is typically represented by the symbol "!", and it is used to calculate the factorial of a number. The factorial of a number is the product of all the positive integers less than or equal to that number.
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A bakery wants to bag at most 100 cookies into cookie packs. Each cookie pack has 4 cookies. They have already bagged 20 cookies. How many more bags of cookies could they make?
Write an inequality describing the situation and solve. Which inequality below is correct?
They may bake 20 additional bags of cookies. The problem is represented by the inequality x < 80.
What is inequality?In mathematics, inequalities specify the connection between two non-equal numbers. Equal does not imply inequality. Typically, we use the "not equal sign ()" to indicate that two values are not equal. But several inequalities are utilized to compare the numbers, whether it is less than or higher than.
Given, A bakery wants to bag at most 100 cookies into cookie packs. Each cookie pack has 4 cookies. They have already bagged 20 cookies.
Since each cookie pack can hold 4 cookies
Thus, 100 cookies will require = 100/4 packs
100 cookies will required = 25 packs
Since They have already bagged 20 cookies.
Thus,
The total pack they used = 20/4
The total pack they used = 5
Inequality:
let "x" be the number of cookies they can make
80 > x
Therefore, 20 more bags of cookies they could make. the inequality that represents the situation is x< 80.
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the expression 5t+16 can be used to approximate the hieght,in inches of a tree t yesrs after it was planted. what is the hieght of the tree 7 years after it was planted?
Using the expression 5t + 16, the height of the tree after 7 years it was planted is: 51 inches.
How to Evaluate an Expression?To evaluate implies that we find the value of the expression by plugging in the value of the variable and simplify accordingly.
Given that t represents the number of years after a tree is planted, and the expression, 5t + 16 represents the approximate height of the tree, to find its height after 7 years, substitute t for 7 into the expression and evaluate:
5(7) + 16
35 + 16
= 51
Thus, the height of the tree after 7 years it was planted is: 51 inches.
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This is due in 20 minutes PLEASE HELP
The slope of the line passing through the point (-6, 0) and (0, 3) is 0.5
What is an equation?An equation is an expression showing the relationship between numbers and variables.
The slope intercept form of a straight line is:
y = mx + b
Where m is the slope and b is the y intercept.
The standard form of a straight line is:
Ax + By = C
Where A, B and C are constants
The line shown in the graph passes through the point (-6, 0) and (0, 3). Hence:
Slope = (3 - 0)/(0 - (-6)) = 3/6 = 0.5
The slope is 0.5
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A fair coin is flipped 30 times. Let X denote the number of heads among the first 20 coin flips and Y denote the number of heads among the last 20 coin flips: Compute the correlation coefficient of X and Y.
the correlation coefficient of X and Y is Cov ( X, Y ) = 1/2
X = Number of heads in the first 20 flips
Y = Number of heads in the last 20 flips
Given that X and Y are binomial variables hence
P( probability ) = 1/2
Find Cov( X; Y )
xi = result of the ith flip ∴ X = x1 + x2 + x3 + x4+....x20
yj = result of the jth flip ∴ Y = y3 + y4 + y5 + y6+ ....x20
covariance of xi and yi = 1/2 * 1/2 = 1/4 when i = j and it is = 0 when i ≠ j
hence Cov( X; Y ) can be expressed as
Cov( X; Y ) = ∑^4 ∑^6 ∴ Cov( Xi , Yj ) = 2/4 = 1/2 ( given that i = j )
Hence he correlation coefficient of X and Y is Cov ( X, Y ) = 1/2
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got ya nose
354 times 23
Answer:
8142
Step-by-step explanation:
Answer: 354x23=8142
because if you brake 300+50+4 and multiply 300x23, 50x23, and 4x23, then you get 8142
The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds.
ft (smaller value)
ft (larger value)
The lower limit that she traveled in three seconds is 33.45 feet and the upper limit that she traveled in three seconds is 43.45 feet
The first three seconds of a race saw a runner's pace rapidly grow.
The time is therefore 3 seconds.
The table provides her speed at half-second intervals.
We are aware,
Distance x Time Speed
Similarly
The distance equals speed x time.
t equals 0.5 seconds
Then, the minimum distance traveled overall is equal to 0.5(0 + 6.7 + 9.2 + 14.1 + 17.5 + 19.4).
Terms are multiplied and added.
= 0.5 × 66.9
= 33.45 feet
The maximum distance travelled is equal to 0.5 (6.7 + 9.2 + 14.1 + 17.5 + 19.4 + 20)
= 0.5 × 86.9
= 43.45 feet
Hence, the lower limit that she traveled in three seconds is 33.45 feet and the upper limit that she traveled in three seconds is 43.45 feet
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The graph shows the cube root parent function.
-5
5-
-5
5
K
Which statement best describes the function?
OA. The function is positive when x < 0 and when x > 0.
B. The function is always positive.
C. The function is positive when x > 0.
OD. The function is positive when x<0
The correct option, C. The function is positive when x > 0, for the graph of cube root parent function.
Define the term cube root parent function?The opposite of such cubic function is the cube root function.
We are aware that its parent cubic function is ascending, one-one, and onto, and that its form is f(x) = x³. It results in a bijection as a result. As a result, its cube root inverse function, of the type f(x) = ∛x, also forms a bijection.The definition of the cube root for all numbers may be found in the introduction, as we have already seen (positive, real, and 0). As a result, there is no x for which any cube root function, f(x), is not defined. Therefore, the set among all real numbers is its domain (R).Similar to the square root function, the range of a cube root function is the set among all real numbers because it produces all values (positive, real, and 0). Consequently, a cube root function is f(x): R → R.Thus, using the domain and range we can see from the graph that, for the positive values of the function, the graph is increasing for positive values.
Therefore, The function is positive when x > 0.
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 solve the following equation for g:
m+n^2 = g5M
Answer:
To solve for g in the equation m + n^2 = g5M, we first need to isolate g on one side of the equation. We can do this by subtracting m and n^2 from both sides:
m + n^2 = g5M
m + n^2 - m - n^2 = g5M - m - n^2
0 = g5M - m - n^2
Now we can divide both sides by 5M:
0 = (g5M - m - n^2) / 5M
And we get the equation for g:
g = (m + n^2) / 5M
if i helped you, can you mark my answer as best?
the number of guests at a theme park can be modeled by function p(t) where t is measured in hours. p is a solution to the logistic differential equation dp over dt equals p over 2 minus p squared over 60000 comma where p(0)
Answer: 10,000 guests per hour
the following graphs show the sampling distributions for two different point estimators, r and w, of the same population parameter. the figure presents 2 histograms, each containing 7 bars. the histogram on the left is labeled point estimator r and the histogram on the right is labeled point estimator w. the heights of the bars in both histograms appear to be the same and appear symmetric about the fourth bar. the bars gradually increase in height from left to right, reach a peak at the fourth bar, and then gradually decrease in the same manner. in the point estimator r histogram, the third bar is labeled population parameter. in the point estimator w histogram, the fourth bar is labeled population parameter. which of the following statements is true?'
Histogram representing point estimator w labelled fourth bar as a population parameter is a true statement.
As given in the question,
The given graph represents the sampling distributions of two different point estimators,
Two point estimators are r and w represents same population parameters.
Two histogram each one with 7 bars:
Left histogram represents point estimator r and right histogram represents point estimator w.
Increase of the heights of the bars in both r and w histogram left to right.
Peak point is at fourth bar.
In point estimator r histogram : Third bar represents population parameter.
In point estimator w histogram : Fourth bar represents population parameter.
Here fourth bar represents the peak point.
Therefore, point estimator w histogram represents the population parameter is a true statement.
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Solve for the indicated variable using the given value of the other variable.
Solve for y when x=0 using the equation y=2x+5
Answer: y is equal to 5
Solve for y :
x=0
y=2(0)+5
y=0+5
y=5
Hope this helped =)
-YinAhara
Find the minimum value of the function z = 4x+3y subject to the following constraints.
x≤ 15
y≤ 13
3x+y≥ 16
2x+11y252
Note that the ALEKS graphing calculator can be used to make computations easier.
Z=
The minimum value for given linear equation 4x+3y will be 24.
What are linear equations?
A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant. The standard form of a linear equation in two variables is of the form Ax + By = C. Here, x and y are variables, A and B are coefficients and C is a constant.
Now,
The minimum value for z=4x+3y will be at (6,0) i.e. 24
because this point follows the given linear inequalities too
as given in the graph provided in image.
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when representing a function as a power series do you have to ratio test to find the radius of convergence if geometric test does nto work
No, you do not have to ratio test to find the radius of convergence if geometric test does into work,
when representing a function as a power series, it is not necessary to ratio test to find the radius of convergence if geometric test does into work. A geometric test does not always converge to the right value if it is used to find the radius of convergence of a function at a given point.
The radius of convergence is the distance between two points within a specified radius. If the geometry test fails, the convergence radius may not be found. To find the radius of convergence, use another test like a Venn diagram to see if there is a compatibility pattern.
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An indoor staircase has a 14 inch vertical rise per step to get to the second floor of the house. If the angle of elevation is 47 degrees and there are 13 feet of horizontal space total for each of the landings
combined, answer the following questions showing each step of your work:
a) What is the landing space per step (in inches, rounded to the nearest inch)?b) Find the vertical rise from the 1st floor to the 2nd floor (in feet, rounded to the nearest foot).
The landing space per step for the indoor staircase is solved to be
13 inchesThe vertical rise from first floor to second floor is 14 feet
How to find the landing spaceThe figure defined to be a right triangle and the dimensions are worked using SOH CAH TOA
Sin = opposite / hypotenuse - SOH
Cos = adjacent / hypotenuse - CAH
Tan = opposite / adjacent - TOA
The right angle triangle comprises of
opposite = total vertical height
adjacent = total horizontal height = 13 feet
The total vertical height is calculated using tan, TOA
let the angle be x
tan x = opposite / adjacent
tan 47 = opposite / 13
opposite = (13 * tan 47) feet
opposite = 12 * (13 * tan 47) inch
opposite = 167.2895 inch
For a 14 inch vertical rise per step say number of step is y
14 y = total vertical height
14 y = 167.2895
y = 11.9493 steps ≈ 12 steps
The landing space for 12 steps is solved by
= horizontal space total in inch / number of steps
= 13 * 12 / 12
= 13 inches
vertical rise from the 1st floor to the 2nd floor
opposite = (13 * tan 47) feet
opposite = 13.94 feet
opposite = 14 feet (to the nearest foot)
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In preparation for catering a picnic event, a restaurant bought potatoes that weighed 5 pounds less than the ones bought for catering an office party. The potatoes for the office party weighed 16 pounds. How much did the potatos for the picnic event weigh?
Answer:
11 pounds
Step-by-step explanation:
16 lbs - 5 lbs = 11 pounds
The two-way table given shows the results from the lunch orders from an upperclassman barbecue.
11th Graders 12th Graders
Hamburger 72 39
Hotdog 65 24
Chicken Wings 23 27
A segmented bar chart was created showing the lunch preference by grade. What percent would the bar for 12th graders show for preferring chicken wings?
The percentage of 12th graders preferring chicken wings is 30%.
What is the percentage?A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.
From the given information total number of 12th graders are,
(39 + 24 + 27)
= 90.
The number of 12th graders preferring chicken wings is 27.
Therefore, The percentage bar for 12th graders shows for preferring chicken wings is,
= (27/90)×100%.
= 30%.
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what is the slope through the line of (-5, 7) and (1, 1)
Answer:
m = -1
Step-by-step explanation:
Slope = rise/run = [tex]\frac{-5-1}{7-1} = \frac{-6}{6} = -1[/tex]
Which of the following expense items should be included when determining whether a taxpayer who wishes to file as head of household paid more than half the cost of keeping up their home? Education. Medical expenses. Real estate taxes. Rental value of the home owned by the taxpayer.
The item would be the rental value of the home owned by the taxpayer. The correct option is D.
What is a taxpayer?A taxpayer is a person or organization that is required to pay taxes. Modern taxpayers may have an identification number, which is a reference number issued by the government to individuals or businesses. The term "taxpayer" generally refers to someone who pays taxes.
A recurring payment made by a tenant to a landlord in exchange for the use of land, a building, an apartment, an office, or other property. a payment made by a lessee to an owner in exchange for the use of machinery, equipment, etc.
A taxpayer who wishes to file as head of household paid more than half the cost of keeping up their home will be the rental value of the home owned by the taxpayer.
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711
Put the measures of the angles in order from least to greatest
Answer: [tex]m\angle YOZ, m\angle VOS, m\angle UOZ, m\angle XOS, m\angle TOZ[/tex]
Step-by-step explanation:
[tex]m\angle VOS=90^{\circ}\\\\m\angle UOZ=180^{\circ}-56^{\circ}=124^{\circ}\\\\m\angle YOZ=180^{\circ}-143^{\circ}=37^{\circ}\\\\m\angle XOS=126^{\circ}\\\\m\angle TOZ=180^{\circ}-33^{\circ}=147^{\circ}[/tex]
Use the confidence level and sample data to find a confidence interval for estimating the population μ. Find the margin of error E and round your answer to the nearest tenth.
Test scores: n = 82, = 89.1, σ = 6.3; 99% confidence
The margin of error round to the nearest tenth will be 1.8.
What is the margin of error?The probability or the chances of error while choosing or calculating a sample in a survey is called the margin of error.
The margin of error is given as
[tex]\rm ME = z\times \sqrt{\dfrac{\sigma ^2}{n}}[/tex]
The z-value for the 99% confidence will be 2.576. Then the margin of error is calculated as,
ME = 2.576 × √[(6.3)² / 82]
ME = 2.576 × √0.4840
ME = 2.576 × 0.6957
ME = 1.8
The margin of error round to the nearest tenth will be 1.8.
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Is this right? The equivalent of 8.55 in fraction form?
Answer:
[tex]\frac{171}{20}[/tex] is equal to 8.55
Step-by-step explanation:
[tex]\frac{171}{20}[/tex] in decimal form is 8.55
[tex]\frac{17}{2}[/tex] in decimal form is 8.5
[tex]\frac{55}{8}[/tex] in decimal form is 6.875
[tex]\huge\text{Hey there!}[/tex]
[tex]\textbf{Let us say:}[/tex]
[tex]\mathsf{8.55 \rightarrow \dfrac{8.55}{1}}[/tex]
[tex]\textbf{This method should make this question easier to solve (in some cases}\\\textbf{of course)}[/tex]
[tex]\mathsf{8.55 \rightarrow \dfrac{8.55}{1}}[/tex]
[tex]\mathsf{= \dfrac{8.55}{1}\times\dfrac{100}{100}}[/tex]
[tex]\mathsf{= \dfrac{8.55\times100}{1\times100}}[/tex]
[tex]\mathsf{= \dfrac{855}{100}}[/tex]
[tex]\mathsf{= \dfrac{855\div5}{100\div5}}[/tex]
[tex]\mathsf{= \dfrac{151}{21}\rightarrow 8 \dfrac{11}{20}}[/tex]
[tex]\huge\text{Therefore, your answer should be:}[/tex]
[tex]\huge\boxed{\mathsf{Option\ A. \ \dfrac{171}{20}}}\huge\checkmark[/tex][tex]\textsf{What you have chosen as the answer to this question is correct}\ \heartsuit[/tex]
[tex]\huge\text{Good luck on your assignment \& enjoy your day!}[/tex]
~[tex]\frak{Amphitrite1040:)}[/tex]Which of the following is the dependent variable in an experiment? Select the correct answer below: O lurking variable O explanatory variable O response variable O treatment
The dependent variable is the factor that is affected by the changes in the independent variable and is measured to observe the effects.
The dependent variable is the factor that is affected by the changes in the independent variable and is measured to observe the effects. This is the variable that is expected to change in response to changes in the independent variable. The dependent variable is the factor that is affected by the changes in the independent variable and it is measured to observe the effects of the changes in the independent variable. For example, in an experiment about plant growth, the dependent variable would be the height of the plant and the independent variable would be the amount of water added to the soil. By changing the amount of water, the height of the plant can be observed and measured to determine the effect of the water on the growth of the plant.
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HELP ASAP!!!! NO LINKS OR SPAM
Malek owns a home, has a bi-weekly gross income of $3,115.00, and has total minimum monthly debt payments of $3,165.00. Choose the inequality that shows the amount Malek needs to reduce his monthly debt payments by to have a debt-to-income ratio of 35%. (2 points)
x ≤ $802.79
x ≥ $802.79
x ≤ $984.50
x ≥ $984.50
The inequality that shows the amount Malek needs to reduce his monthly debt payments by to have a debt-to-income ratio of 35% is x ≤ $984.50.
What is Debt to Income Ratio?Debt to income ratio is usually calculated as a percentage of the gross monthly debt payments divided by the gross monthly income.
Given that Malek has a biweekly gross income of $3115.00.
Monthly gross income of Malek = 2 × $3115.00 = $6230
Monthly debt payments is minimum of $3165.00
We need the debt to income ratio as 35%.
Let x be the monthly debt payment to get the ratio 35%.
x / 6230 = 0.35
x = 2180.5
$3165.00 - $2180.5 = 984.5
So the inequality is x ≤ $984.50
Hence the inequality showing the amount to be reduced to get the debt top income ratio as 35 % is x ≤ $984.50.
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A materials engineer wishes to compare the durability of two different types of paving material. She has 40 different one-mile stretches of interstate highway that she’s been authorized to repave for this study. She decides to carry out a matched pairs experiment. Which of the following is the best way for her to carry out the randomization for this study?
answer choices
O Use a table of random digits to divide the 40 roadways into 20 pairs and then, for each pair, flip a coin to decide which pavement to use on which member of the pair.
O Subjectively divide the 40 roadways into 20 pairs (making the roadways within each pair as different as possible) and then, for each pair, flip a coin to decide which pavement to use on which member of the pair.
O Use a table of random digits to divide the 40 roadways into two groups of twenty, and then use the table of random digits a second time to decide which pavement to use on which group.
O Let each of the 40 roadways act as its own pair, dividing each roadway into the first half-mile and the second half-mile. Flip a coin for each of the 40 roadways to decide which half-mile gets which pavement.
O Let each of the 40 roadways act as its own pair, dividing each roadway into the first half-mile and the second half-mile. Flip a coin once to decide which pavement is put on the first half-mile of all the roadways.
The best way for the materials engineer to carry out the randomization for this study is by using a table of random digits to divide the 40 roadways into 20 pairs and then, for each pair, flipping a coin to decide which pavement to use on which member of the pair.
This can be represented by the formula P(A|B) = P(A) * P(B|A), where P(A|B) is the probability of pavement type A being used on a given roadway, P(A) is the probability of pavement type A being used on any given roadway, and P(B|A) is the probability of a given roadway being used for pavement type A.
The probability of pavement type A being chosen for any given roadway is 1/2 (since there are two pavement types). The probability of a given roadway being used for pavement type A is also 1/2 (since there are two members of each pair). This means that the probability of pavement type A being chosen for a given roadway is equal to P(A|B) = 1/2 * 1/2 = 1/4.
By randomly assigning pavement types to each pair of roadways, the materials engineer is ensuring that her experiment is as fair and unbiased as possible. This will allow her to accurately compare the durability of the two different types of paving material.
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How do you graph the ellipse? (x - 6)²/36 + (y + 3)²/100 = 1
Drag choices into the boxes to correctly complete the statements.
Answer:
center: [tex](6, -3)[/tex]
distance from center:
major axis - 10 unitsminor axis - 6 unitsconnect: [tex](16, -3), (6, -9), (-4, -3), (6, 3)[/tex]
Step-by-step explanation:
Ellipse Equation:An ellipse has two forms, although they're essentially the same:
[tex]\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1[/tex]
and
[tex]\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1[/tex]
Where a > b, and if the "a^2" is under the [tex](x-h)^2[/tex] then that means the ellipse has a horizontal major axis, but in general the value below this numerator relates to the length of the horizontal axis. Now if "a^2" is under the [tex](y-k)^2[/tex] then that means the ellipse has a vertical major axis, but in general the value below this numerator relates to the length of the vertical axis.
The length of the major axis is defined as 2a and the minor axis as 2b, but the distance from the center to the end points will be half of this, so either "a" or "b" depending on which axis the end point is on.
The other thing to note is that (h, k) is the center of the ellipse.
Analyzing the Equation:We're given the equation: [tex]\frac{(x-6)^2}{36}+\frac{(y+3)^2}{100}=1[/tex], in this case the bigger value is under the (y+3)^2, so we know that we will have a horizontal major axis. Since the denominator's represent the square of "a" and "b" we first have to take the square root of them. So the following is true:
[tex]a=\sqrt{100}=10\\b=\sqrt{36}=6[/tex]
The "a" gives us the distance from the center to the end point on the major axis while the "b" gives us the distance from the center to the end point on the minor axis, so the distance from the center to the end point on the minor axis is 6 and for the major axis it's 10.
We also know the center is at (6, -3) by looking at the signs and what is being added/subtracted from the x and y values in the equation. We can use this to calculate the end points since the end points on the horizontal major axis can be calculated as: [tex](6\pm10, -3)[/tex] and the end points on the vertical major axis can be calculated as: [tex](6, -3\pm 6)[/tex], which gives us four points: [tex](16, -3), (6, -9), (-4, -3), (6, 3)[/tex] which we can connect to draw a graph.
Answer:
[tex]\textsf{The center of the ellipse is $\boxed{(6,-3)}$}\;.[/tex]
[tex]\textsf{The endpoints of the major axis are $\boxed{10}$ units from the center}\;.[/tex]
[tex]\textsf{The endpoints of the minor axis are $\boxed{6}$ units from the center}\;.[/tex]
[tex]\textsf{To graph the ellipse, connect $\boxed{(12,-3),(6,-13),(0,-3), \;\textsf{and}\; (6,7)}$ with a smooth curve}\;.[/tex]
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{7.2 cm}\underline{General equation of an ellipse}\\\\$\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$\\\\where:\\\phantom{ww}$\bullet$ $(h,k)$ is the center\\ \phantom{ww}$\bullet$ $a$ and $b$ are the radii.\\ \phantom{ww}$\bullet$ $(h\pm a,k)$ and $(h,k\pm b)$ are the vertices.\\ \end{minipage}}[/tex]
Given equation:
[tex]\dfrac{(x-6)^2}{36}+\dfrac{(y+3)^2}{100}=1[/tex]
As b > a, the given ellipse is vertical.
The major axis is the longest diameter and the minor axis is the shortest diameter, therefore:
Major axis = 2bMinor axis = 2aVertices = (h, k±b)Co-vertices = (h±a, k)Determine the values of h and k:
[tex](x-h)=(x-6) \implies h=6[/tex]
[tex](y-k)=(y+3) \implies k=-3[/tex]
Therefore, the center of the ellipse is:
[tex](h,k)=(6, -3)[/tex]Calculate the values of a and b:
[tex]a^2=36 \implies a=6[/tex]
[tex]b^2=100 \implies b=10[/tex]
As the major axis is 2b, then the major radius is b.
Therefore, the endpoints of the major axis are "b" units from the center, so they are 10 units from the center.
As the minor axis is 2a, then the minor radius is a.
Therefore, the endpoints of the minor axis are "a" units from the center, so they are 6 units from the center.
To graph the ellipse, connect the vertices and co-vertices with a smooth curve.
Vertices = (h, k±b) = (6, -3±10) = (6, -13) and (6, 7)Co-vertices = (h±a, k) = (6±6, -3) = (0, -3) and (12, -3)math hw pls fast, What is the similarity ratio of the dilation?
Answer: the ratio is 1/2
