The probability of randomly selecting a can that is not of grape is:
P = 0.67
How to find the probability?We know that on the ice chest there are:
6 cans of apple juice.9 cans of grape juice8 cans of orange juice.4 cans of pineapple juice.For a total of 6 + 9 + 8 + 4 = 27
The probability of selecting a can that is not of grape juice is equal to the quotient between the numbers of cans that are not of grape, and the total number of cans.
There are 27 cans in total and 18 of these aren't of grape, then the probability is:
P = 18/27 = 0.67
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Mariah made a cylinder out of clay that had a radius of 6 cm and a volume of 72π cm^3. She painted the entire surface of the cylinder purple. How many square centimeters of the cylinder did Mariah paint in terms of π?
Enter the correct answer in the box in terms of π.
The square centimeters of the cylinder did Mariah paint in terms of π will be 24π cm².
What is the surface area of a right circular cylinder?Let r be the radius and h be the height of the cylinder. Then the surface area of the cylinder will be given as,
SA = 2πrh square units
Mariah made a cylinder out of clay that had a radius of 6 cm and a volume of 72π cm³. The height of the cylinder is calculated as,
V = πr²h
72π = π x 6² x h
72 / 36 = h
h = 2 cm
Then the surface area of the cylinder is given as,
SA = 2 x π x 6 x 2
SA = 24π square cm
The square centimeters of the cylinder did Mariah paint in terms of π will be 24π cm².
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The distribution of age for players of a certain professional sport is strongly skewed to the right with mean 26.8 years and standard deviation 4.2 years. Consider a random sample of 4 players and a different random sample of 50 players from the population. Which of the following statements is true about the sampling distributions of the sample mean ages for samples of size 4 and samples of size 50 ? A Both will be skewed to the right, and the mean for size 50 will be closer to 26.8 than the mean for size 4. B Both will be skewed to the right, and the standard deviation for size 50 will be closer to 4.2 than the standard deviation for size C. Both will be approximately normal, and the mean for size 50 will be closer to 26.8 than the mean for size D. Only the sampling distribution for size 4 will be approximately normal, and the standard deviation for both will be 4.2. E Only the sampling distribution for size 50 will be approximately normal, and the mean for both will be 26.8.
The true statement about the sampling distributions is: Both will be approximately normal, and the mean for size 50 will be closer to 26.8 than the mean for size 4.
Thus, option (C) is correct.
According to the Central Limit Theorem, for a large enough sample size, the sampling distribution of the sample mean tends to follow a normal distribution, regardless of the shape of the population distribution.
Therefore, both sampling distributions of size 4 and size 50 will be normal.
Moreover, the mean of the sampling distribution of the sample mean will be equal to the population mean.
Since the population mean is given as 26.8 years, the mean for both size 4 and size 50 will be 26.8 years.
Thus, option (C) is correct.
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SOMEONE HELP ME QUICK I WILL GIVE BRAINLIEST
Answer: 1. 8, 2. 1680
Step-by-step explanation:
1. area is length times width, so 6 times 7 is 30. To find the missing one just do 40 divided by 5, which equals 8.
2. to find the total area do length x width x length x width. so 5 x 6 x 7 x 8 which equals 1680.
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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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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Cual es la propiedad de 6÷1=6
SEE IMAGE!! PLS HELP ASAPP
Find the quotient. If possible, rename the quotient as a mixed number. Select the answer that is in simplest form.
2 3
4 ÷ 3 = ?
A.
7
12
B.
11
12
C.
3
4
Answer:11/12
Step-by-step explanation:
Which sequence shows the numbers in order from least to greatest? a) 7.2/3 < √65 < |-7.82| b) 7.2/3 < |-7.82| < √65
c. |-7.82| < √65 < 7.2/3
d. √65 < |-7.82| < 7.2/3
The sequence from least to greatest is d) √65 < |-7.82| < 7.2/3.
To calculate this, we first need to find the square root of 65, or √65. This can be done using the formula √a = b, where a is the number to be square rooted and b is the result of the square root. In this case, a = 65, so we calculate √65 = 8.062257748. Next, we need to find the absolute value of -7.82, which is |-7.82| = 7.82. . To calculate this, the absolute value formula is used, which is |x| = |x|. Finally, 7.2/3 is the greatest of the three numbers. To calculate this, first convert 7.2/3 to a decimal by dividing it by 3. Then divide 7.2 by the result to get the final answer.Finally, we find 7.2/3, which is equal to 2.4. Therefore, the sequence from least to greatest is √65 < |-7.82| < 7.2/3, or 8.062257748 < 7.82 < 2.4.
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A company's cereal boxes advertise 9.65 ounces of cereal. In fact, the amount of cereal in a randomly selected box follows a Normal distribution with meanμ=9.70μ=9.70ounces and standard deviationσ=0.03σ=0.03ounces. What is the probability that a randomly selected box of the cereal contains less than 9.65 ounces of cereal? Show your work.
The probability that a randomly selected box of the cereal contains less than 9.65 ounces of cereal is 0.4608.
The probability that a randomly selected box of the cereal contains less than 9.65 ounces of cereal is 0.4608.
P(x < 9.65) = P(Z < (9.65 - 9.70) / 0.03)
= P(Z < -0.17)
= 0.4608
The first step is to calculate the Z-score for 9.65 ounces. To calculate the Z-score, you subtract the mean (9.70 ounces) from the value (9.65 ounces) and then divide by the standard deviation (0.03 ounces). The Z-score in this case is -0.17. The next step is to use a Z-table to find the probability of a Z-score less than -0.17. The probability is 0.4608. Therefore, the probability that a randomly selected box of the cereal contains less than 9.65 ounces of cereal is 0.4608.
The probability that a randomly selected box of the cereal contains less than 9.65 ounces of cereal is 0.4608.
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Given that alpha and beta are acute. Sin alpha = 1/√10 and Sin beta = 1/√5, prove without using calculator and table that Sin (alpha + beta) = 1/√2
If ∠A and ∠B are acute angles such that cos A = cos B we have proved that ∠A = ∠B since AC = BC and angles opposite to equal sides of a triangle are equal.
What are the definitions of two acute angles?Image outcome
An acute angle has a degree that is less than 90 degrees, i.e. less than a right angle. Acute angle degrees include 12°, 35°, 48°, 65°, 80°, and 89°. An acute angle triangle (also known as an acute-angled triangle) is a triangle with all of its internal angles being acute angles. Remember that an acute angle is one that is less than 90°. Because all three angles are less than 90°, ABC is an acute angle triangle or acute-angled triangle.
The sine function for the angle yields the ratio of the length of the opposing side to the length of the hypotenuse.
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math hw pls fast, What is the similarity ratio of the dilation?
Answer: the ratio is 1/2
The weight gains of beef steers were measured over a 140 -day test period. The average daily gains (lb/day) of 9 steers on the same diet were as follows: 23
3.89 3.51 3.97 3.31 3.21 3.36 3.67 3.24 3.27
Determine the mean and median
The weight gains of beef steers were measured over a 140 -day test period .then mean is 3.545 and median is 3.435
Given
3.89 3.51 3.97 3.31 3.21 3.36 3.67 3.24 3.27
mean = 3.89+3.51+3.97+3.31+3.21+3.36+3.67+3.24+3.27 /10
= 35.45/10
= 3.545
average is the sum of all terms present in the question divided by the number of terms that is the way of calculating average
Solving (a): The mean = 3.545
This is calculated as:
median = 10+1 /2 term
= 11/2
=5.5 term
So, we have:
median = 3.435
The weight gains of beef steers were measured over a 140 -day test period .then mean is 3.545 and median is 3.435
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What is the shape of the bases for the following polyhedron?
a. triangle
b. square
c.rectangle
d.circle
The shape of the bases for the following polyhedron depend on the type of polyhedron. For example, a triangular prism would have triangular bases, a square pyramid would have square bases, a rectangular prism would have rectangular bases, and a cone would have circular bases.
The shape of the bases for a polyhedron depend on the type of polyhedron. Polyhedra can have a variety of shapes, such as triangular, square, rectangular, and circular. For example, a triangular prism would have triangular bases, a square pyramid would have square bases, a rectangular prism would have rectangular bases, and a cone would have circular bases.
The shape of the bases for the following polyhedron depend on the type of polyhedron. For example, a triangular prism would have triangular bases, a square pyramid would have square bases, a rectangular prism would have rectangular bases, and a cone would have circular bases.
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You have 26 cards, labeled with the letters A-Z. The deck is shuffled randomly. You draw cards without replacement. (a) Let I be the number of letters we needed to draw to have seen both A and B. For example, if the 3rd letter is A and the 5th letter is B, then L = 5. What is the PMF of L? (b) Given that we have drawn k many cards already (for some k 0,..., 24) and neither A nor B have yet appeared, what's the probability that these two letters will be drawn consecutively (either AB or BA)? =
The PMF of the number of letters needed to draw to have seen both A and B is pmf(L=k) = $\frac{1}{26 \choose k-1}$ for k = 2, 3, ..., 26. Given that we have already drawn k many cards and neither A nor B have yet appeared, the probability that these two letters will be drawn consecutively is $\frac{2}{26-k}$.
(a) pmf(L=k) = $\frac{1}{26 \choose k-1}$ for k = 2, 3, ..., 26
(b) P(AB or BA | k cards already drawn) = $\frac{2}{26 - k}$
(a) Since the deck is shuffled randomly, we can assume that each of the 26 letters has an equal probability of being drawn. The probability of drawing A and B before any other letter is $\frac{1}{26 \choose k-1}$.
(b) The probability of drawing AB or BA is $\frac{2}{26-k}$, since there are two possible combinations and we have (26-k) cards left to draw.
The PMF of the number of letters needed to draw to have seen both A and B is pmf(L=k) = $\frac{1}{26 \choose k-1}$ for k = 2, 3, ..., 26. Given that we have already drawn k many cards and neither A nor B have yet appeared, the probability that these two letters will be drawn consecutively is $\frac{2}{26-k}$.
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(07.01 mc) a coolant is injected into engine fluid, causing the temperature of the engine fluid to decrease with respect to time at a rate that is proportional to the difference of the fluid's instantaneous temperature, t, and the fluid's original temperature, to. select the differential equation that represents the relationship.
The differential equation that represents the relationship is
dT/dt = - k * ( T - T0 ) from the condtion - dT/dt ∝ ( T - T0 ).
Given:
The engine fluid temperature rate of change, dT/dt
- dT/dt ∝ ( T - T0 ) [proportional to the difference of the fluid's instantaneous temperature, T, and the fluid's original temperature, T0.]
The Differential Equation that represents the relationship.
The framework is already set, there is only one thing that changes the proportionality to an equation. And that is a proportionality constant.
Let's call it 'k'
Since this is a directly proportional set up, the constant 'k' will be multiplied to the term of ( T - T0 )
- dT/dt = k * ( T - T0 )
And to make this look like a typical equation, multiply both sides by a -1 to get:
∴ dT/dt = - k * ( T - T0 )
As expected, this comes to represent the same thermodynamic effect you'd see in Newton's Law of Cooling; it is expected because most of all fluids exhibit this cooling effect when interacting with other fluids of differing temperatures.
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from the smith chart, find the normalized input admittances corresponding to the following normalized input impedances:
The normalized input admittances are the inverse of the normalized input impedances and can be found on the Smith Chart.
The Smith Chart is a graphical tool used to quickly calculate the reflection coefficient, impedance, admittance, and other characteristics of a transmission line. To find the normalized input admittances corresponding to the normalized input impedances, we first need to find the normalized reflection coefficient for the given normalized input impedances. This can be done by locating the given normalized impedance on the Smith Chart and then determining the normalized reflection coefficient from the radial and angular coordinates. Once the normalized reflection coefficient has been determined, the normalized input admittance can be found by using the formula Y = 1/Z. The normalized input admittance is then located on the Smith Chart in the same manner as the normalized input impedance, using the radial and angular coordinates.
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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
example 3
include critical numbers, sign chart , intervals of increasing / decreasing
show work and upload pics of work
include critical numbers, sign chart , intervals of increasing / decreasing
SHOW YOUR WORK
The critical points, sign chart and intervals of increase and decrease is plotted for the function f(x) = x³/4 - 3x.
Critical points - (−2,4) and (2,−4)
Interval of increase - (−∞,−2)∪(2,∞)
Interval of decrease - (-2,2)
What is a function?
In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.
A continuous function f with x in its domain has a critical point at that point xx if it satisfies either of the following conditions -
f'(x) = 0,
f'(x) is undefined
The given function is f(x) = x³/4 - 3x
So, f'(x) = 3x²/4 - 3
In the graph of the function, only two points are there on the graph where f'(x) = 0, which is -
(x,f(x))=(−2,4)
(x,f(x))=(2,−4)
A sign chart tells you when the value of a function f(x) is negative or positive, which is the same as when the graph of f(x) is below or above the x-axis, respectively.
Draw a solid line when the y-value of the function is greater than zero, which is when f(x)>0.
Draw a dashed line when the y-value of the function is less than zero, which is when f(x)<0.
In the graph it can be seen that the graph is below zero up to x = -3.464 and between x = 0 and x = 3.464. In these intervals the sign chart is dashed. The graph is above the x-axis between x = -3.464 and x = 0 and when x>3.464. In these areas the sign chart is solid.
In the graph it can be clearly seen that the graph is increasing from point -∞ up to -2 and from 2 to ∞.
Also in the graph it can be clearly seen that the graph is decreasing from point -2 up to 2 .
Therefore, the critical points, sign chart and intervals of increase and decrease is found.
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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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79=2x+3y solve for y
The results of the x-intercept and y-intercept results from above (39.5, 0) and (x1, y1) and (x2, y2) (0,26.333)
How to solve y?Here, we'll explain how to solve arithmetic issues with the formula 2x + 3y = 79 and offer answers.We'll start by figuring out the x-intercept and y-intercept of the equation 2x + 3y = 79 and then showing you the result.Then, we'll demonstrate how to obtain the coordinates for the graph plot for the equation 2x + 3y = 79 so that we may depict it on a graph.The slope of 2x + 3y = 79 will then be calculated and shown to you once we have solved 2x + 3y = 79 for both x and y.locate x-interceptWhere the graph veers off the x-axis is known as the x-intercept. We first set y1=0 and then solve for x to determine the x-intercept.2x + 3y = 79
2x + 3(0) = 79
x1 = 39.5 y1 = 0
the y-intercept
Where the graph crosses the y-axis is known as the y-intercept. We first solve for y before setting x2=0 to determine the y-intercept.
2x + 3y = 79
2(0) + 3y = 79
y2 = 26.333
x2 = 0
To draw a straight line on a graph, you must get two graph points. The format for the plot coordinates is (x1,y1) and (x2,y2).
Thus, we can obtain the graph plots for 2x + 3y = 79 as follows using the results of the x-intercept and y-intercept results from above:
(39.5, 0) and (x1, y1) and (x2, y2) (0,26.333)
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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]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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f(x) is a direct variation function f(3)= 12 f(x)= ? f(2)= ?
The function f(x) will be : f(x) = 4x and f(2) = 8.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is that f(x) is a direct variation function .
Direct variation of {y} with {x} means that as {x} increases, {y} increases uniformly with it. Mathematically -{K} = y/x = constant
Scale factor {K} is a dimensionless value that indicates the constant ratio value indicating direct variation.It is given that -
f(3) = 12
We can write the slope of f(x) as -
m = (12 - 0)/(3 - 0)
m = 12/3
m = 4
So, f(x) will be -
f(x) = 4x
and
f(2) will be -
f(2) = 8
Therefore, the function f(x) will be : f(x) = 4x and f(2) = 8.
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There are 31 players on the softball team. They are traveling to the District Championship in 8 cars. There are 4 players in each of the first 7 cars. How many players on the softball team are traveling in the eighth car?
Answer: 3 players
Step-by-step explanation: Hello! You know that there are 31 players on the softball team, there are 8 cars in total for everyone to drive in, and there are 4 players in each car for the first 7 cars. What you have to find out is how many players are traveling in the eighth car. The first thing that you want to know is how many players are already in the cars. You already know that 4 players are in each of the 7 cars. To know how many players are in the 7 cars, multiply the 4 players by the 7 cars. 4 x 7 = 28. 28 players are in the first 7 cars. You know that there are 31 players in total. To find the number of players in the last car, subtract the number of players in the first 7 cars from the total amount of players. 31 - 28 = 3. There are 3 players on the softball team in the eighth car. Hope this helps!
When an employer asks for "how many years of basic math experience do you have" on an application do you include school years? If so how many?
Answer:
Step-by-step explanation:
When an employer asks for "how many years of basic math experience do you have" on an application, it is generally understood that they are asking for the amount of time you have spent working with basic math concepts in a professional setting. This would typically include time spent working in roles that required the use of basic math skills, such as data entry, accounting, or similar positions.
However, if you have no professional experience in a math-related field but you have studied math in school, you can mention that you have a certain number of years of education in math and that you are familiar with basic math concepts.
It's up to you how many years of math classes you want to include, but it's not necessary to include all the years you've studied math since primary school, it's better to just mention that you have a certain level of education in math, or give an estimate of the number of years that you have studied it in a higher level education like college or university.
It's important to be honest and transparent in your application, but also to highlight the skills and experience that make you a good fit for the role.
find the slope of the line passing through points -1,5 and -5,7
Answer:
-1/2
Step-by-step explanation:
use slope formula
it'd be:
\frac{5-7}{-1+5} =\frac{-2}{4} =\frac{x}{y} -1/2\\
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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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)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.
To learn more about ordered arrays
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