Write the equation for a parabola with a focus at (2,2) and a directrix at x=8
Answers
Answer:
(y - 2)² = -12(x - 5)
Step-by-step explanation:
A parabola is a locus of points, which are equidistant from the focus and directrix;
Generic cartesian equation of a parabola:
y² = 4ax, where the:
Focus, S, is: (a, 0)
Directrix, d, is: x = -a
a > 0
Put simply, a is the horinzontal difference between the directrix and the vertex or between the vertex and focus;
Always a good idea to do a quick drawing of the graph;
We are the told the focus, F, is: (2, 2) and directrix, d, is: x = 8;
First thing to note, the vertex, or turning point will be in line with the focus vertically, i.e. they will share the same y-coordinate;
Horizonatally, it will be halfway between the focus and the directrix, i.e. halfway between 8 and 2;
Therefore, the vertex will be will be (5, 2);
We can also work out a:
a = 8 - 5 = 5 - 2
a = 3
Substituting this value of a into the generic cartesian equation:
y² = 4(3)x
y² = 12x
The focus and directrix will be:
S: (3, 0)
d: x = -3
Next thing to note, a parabola curves away from the directrix;
In this case, the directrix is x = 8, so the vertex will be the right-most point on the parabola, it will curve off to the left and the focus will also be to the left;
What we want to do is compare with y² = 12x;
This parabola, has a vertex (0, 0), which is the left-most point that curves off to the right and a focus also to the right;
Since we know the formula of this parabola, if we figure out how to transform it into the one in the question, we can find out it's equation;
What we should recognise first is that the parabola in the question is reflected in the y-axis, compared to y² = 12x;
So we apply the transformation that corresponds to this, i.e. use the f(-x) rule:
y² = 12(-x)
y² = -12x
Now the two graphs will have the same shape and orientation;
The focus and directrix will also be affected:
S: (-3, 0)
d: x = 3
Now, the only remaining difference would be the coordinates of the focus and directrix of the two graphs;
The focus of the graph in the question is 5 units to the right and 2 units upwards compared to the focus of y² = -12x;
The directrix is 5 units to the right of that of y² = -12x;
So we apply a translation transformation of 5 units right and 2 units up, like so:
(y - 2)² = -12(x - 5)
Replace y with (y - 2) to translate up 2 units;
Replace x with (x - 5) to translate 5 units right.
We know have a parabola with focus, (2, 2), directrix, x = 8 and vertex, (5, 2), i.e. the parabola in the question;
Hence, the equation of the parabola in the question is:
(y - 2)² = -12(x - 5)
It might seem a bit long and complicated to begin with, but can be done very quickly if you can get used to it.
Related Questions
Por favor, necesito ayuda con esto es de estadística. Muchas gracias
Las calificaciones de 20 alumnos que presentaron exámen de admisión a cierta facultad, utilizando la escala de 0 a 100, fueron:
83 64 51 46 82 91 73 82 65 61 74 64 75 81 94 65 42 81 56 61 72 65 54 39 70 93 42 46 54 72
•Elaborar: diagrama de tallo y hoja
•Calcular: coeficiente de variación
•Realizar un diagrama de caja
•Percentil 85, decil 2
Answers
Therefore, the coefficient of variation for the given data is approximately 24.71%.
What is box plot?A box plot, also known as a box-and-whisker plot, is a graphical representation of a data set that shows the distribution of the data using quartiles. It is a standardized way of displaying the distribution of data based on the five-number summary: minimum, first quartile, median, third quartile, and maximum. The box represents the middle 50% of the data, with the median marked by a line inside the box. The whiskers extend from the box to the minimum and maximum values, or to a certain range if there are outliers. Box plots are useful for comparing the distributions of different data sets and identifying potential outliers.
Here,
1. Stem-and-Leaf Plot:
A stem-and-leaf plot is a way to display data that separates the tens digit of each number from the ones digit. Here is the stem-and-leaf plot for the given data:
3 | 9 9
4 | 2 2 6 6
5 | 1 4 4 6
6 | 1 4 5 5 5 5 5 5
7 | 0 2 3 4
8 | 1 2 2 3 5
9 | 3 4
In this plot, the stem represents the tens digit and the leaves represent the ones digit.
2. Coefficient of Variation:
The coefficient of variation is a measure of the relative variability of a data set. It is calculated as the ratio of the standard deviation to the mean, expressed as a percentage. Here is how to calculate the coefficient of variation for the given data:
Calculate the mean of the data:
Mean = (83+64+51+46+82+91+73+82+65+61+74+64+75+81+94+65+42+81+56+61+72+65+54+39+70+93+42+46+54+72)/20 = 68.25
Calculate the standard deviation of the data:
Standard deviation = sqrt((1/20) * ((83-68.25)^2 + (64-68.25)^2 + ... + (72-68.25)^2))
Standard deviation ≈ 16.88
Calculate the coefficient of variation:
Coefficient of variation = (Standard deviation / Mean) * 100
Coefficient of variation ≈ 24.71%
3. Box Plot:
A box plot is a way to visualize the distribution of data. It displays the minimum value, first quartile (Q1), median, third quartile (Q3), and maximum value of the data. Here is the box plot for the given data:
| +----------+
94 | |
| +----------+
93 | |
| +-----+-----+
82 | | |
| | |
81 | | |
| +-----+ |
80 | | |
| | |
79 | | |
| | |
78 | | |
| | |
77 | | |
| | |
76 | | |
| | |
75 | | |
| | |
74 | | |
| | |
73 | | |
| | |
72 | +-----------------+
|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
In this plot, the horizontal line inside the box represents the median, the bottom and top edges of the box represent the first and third quartiles (Q1 and Q3), respectively, and the vertical lines extending from the box represent the minimum and maximum values, excluding outliers.
4. To find the 85th percentile, we need to arrange the data in order from smallest to largest:
39, 42, 42, 46, 46, 51, 54, 56, 61, 61, 64, 64, 65, 65, 65, 70, 72, 72, 73, 74, 75, 81, 81, 82, 82, 83, 91, 93, 94
There are a total of 20 scores, so the 85th percentile would be the score at the 0.85(20) = 17th position:
85th percentile = 72
To find the 2nd decile, we first need to calculate the number of scores in each decile. Since there are 20 scores, each decile would have 2 scores. The 2nd decile would be the score at the 0.2(20) = 4th position:
2nd decile = 46
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Complete question:
Please, I need help with this, it's about statistics. Thank you very much. The grades of 20 students who took an admission exam to a certain faculty, using a scale of 0 to 100, were: 83 64 51 46 82 91 73 82 65 61 74 64 75 81 94 65 42 81 56 61 72 65 54 39 70 93 42 46 54 72
• Make: stem-and-leaf plot
• Calculate: coefficient of variation
• Create a box plot
• 85th percentile, 2nd decile.
In the diagram below, IJK~LJK Find g. 5
Answers
In the diagram given IJK≅LJK ,the length cannot be negative, the only valid solution is g = 6. Therefore, the length of the segment KJ is 6 meters.
What is length?Length is a physical dimension that measures the distance between two points or the size of an object along one dimension.
It is commonly measured using standard units of length such as meters, feet, inches, and centimeters.
Given that KM is the bisector line of line IM and J is the bisector point on line IL. Two triangles are formed, △IJK and △LJK, on line IL in the upward and downward direction, respectively. We are given the lengths of IK, JN, LM, and KJ, and we need to find the value of g such that IJK≅LJK.
Since the two triangles are similar, their corresponding sides are proportional. Using the side proportionality theorem, we have:
KJ/LM = IJ/JL
Substituting the given values, we get:
g/10 = 5/(4+g)
Cross-multiplying, we get:
g(4+g) = 50
Expanding and simplifying, we get:
g²+ 4g - 50 = 0
Using the quadratic formula, we get:
g = (-4 ± √(4² + 4(50)))/2
g = (-4 ± √(256))/2
g = (-4 ± 16)/2
g = -10 or g = 6
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what is 7 in x 3 in x 6 in x 4 in x 15 in=
Answers
Answer:
7,560 inches.
Step-by-step explanation:
Given: 7 in x 3 in x 6 in x 4 in x 15 in = ?
First, multiply 7 and 3:
21 in x 6 in x 4 in x 15 in
Then multiply 21 and 6:
126 in x 4 in x 15 in
Then multiply 4 and 15:
126 in x 60 in
Finally, multiply 126 and 60:
= 7,560 inches.
6x³+4xy²+3y³+13xy+8÷x+2 Divide polynomials
Answers
The quοtient is [tex]6x^2 - 8xy + 17y^2 - 34y + 70[/tex], and the remainder is [tex]8y^3 - 40.[/tex]
What is polynomial?A polynomial is a mathematical expression made up of variables (like x, y, and z) and coefficients (constants multiplied by the variables), as well as addition, subtraction, and multiplication operators. The variables and coefficients are raised to n-negative integer powers, also known as degrees.
We can use long division to divide fractions. Divide 6x³ + 4xy² + 3y³ + 13xy + 8 by x + 2 as shown:
[tex]6x^2 - 8xy + 17y^2 - 34y + 70[/tex]
-------------------------------------
[tex]x + 2 | 6x^3 + 4xy^2 + 3y^3 + 13xy + 8[/tex]
[tex]- 6x^3 - 12x^2[/tex]
---------------
[tex]12x^2 + 4xy^2[/tex]
[tex]- 12x^2 - 24x[/tex]
--------------
[tex]4xy^2 + 24x[/tex]
[tex]- 4xy^2 - 8y^3[/tex]
--------------
[tex]24x - 8y^3 + 8[/tex]
[tex]- 24x - 48[/tex]
------------
[tex]8y^3 - 40[/tex]
Therefοre, the quοtient is [tex]6x^2 - 8xy + 17y^2 - 34y + 70[/tex], and the remainder is [tex]8y^3 - 40[/tex].
Thus, we can express the οriginal pοlynοmial as:
[tex]6x^3+4xy^2+3y^3+13xy+8[/tex]
[tex]= (x + 2)(6x^2 - 8xy + 17y^2 - 34y + 70) + (8y^3 - 40) \div (x + 2)[/tex]
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Second chance! Review your workings and see if you can correct your mistake. The table below shows how much a healthy kitten is expected to weigh. Betsy's kitten is 2 weeks old and its mass is 113% of its expected mass. What is the mass of Betsy's kitten? Give your answer in grams (g) and give any decimal answers to 1 d.p. Age (weeks) 1 2 3 4 5 6 7 8 Mass (g) 210 310 390 460 600 680 790 910 Watch video Answer >
Answers
Answer:
350.3 g
Step-by-step explanation:
310 * 113% = 310 * 1.13 = 350.3 g
Hope this helps!
do the following statements describe red pulp or white pulp? drag and drop the labels into the corresponding boxes.
Answers
After labeling with the corresponding boxes , The statements which describes Red pulp and white pulp are as follow:
1. Red pulp 2. Red Pulp 3. White Pulp 4. Red pulp
5. White Pulp 6. White Pulp 7. Red Pulp.
Red Pulp:
The red pulp of the spleen is made up of connective tissue (also known as Bill Roth's cord) and numerous sinusoids of the spleen which appear red due to excess blood. Its main function is to filter antigens, microorganisms and defective or worn out red blood cells from the blood.
The spleen consists of red and white pulp, separated by the marginal zone; 76-79% of a normal spleen is made up of red pulp. Unlike white pulp, which contains mostly lymphocytes (such as T cells), red pulp is made up of several types of blood cells, including platelets, granulocytes, red blood cells, and plasma.
White Pulp:
White pulp is the histological name for the areas of the spleen (so called because they appear whiter in rough sections than the surrounding red pulp), which make up approximately 25% of the spleen. The white pulp is entirely composed of lymphoid tissue.
Specifically, the white pulp contains several regions with distinct functions:
The periarteriolar lymphatic sheaths (PALS) are usually associated with the arteriolar supply to the spleen; they contain T lymphocytes.
Lymphoid follicles with dividing B lymphocytes located between the PALS and the marginal zone bordering the red pulp. This region produces IgM and IgG2.
These molecules play a role in the opsonization of extracellular organisms, in particular encapsulated bacteria.
The marginal zone exists between the white pulp and the red pulp. It is located away from the central arterioles, close to the red pulp. It contains antigen-presenting cells (APCs), such as dendritic cells and macrophages. Some white pulp macrophages are of a special type called metallolipid macrophages.
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The cost C (in dollars) of supplying recycling bins to p% of the population of a rural township is given by C=25000p/100-p, 0 ≤ p ≤ 100, Use a graphing utility to graph the cost function.
Answers
Here's a graph of the cost function C = 25000p / (100 - p), using graphing utility. The graph shows that the cost function is continuous and defined for all values in its domain.
The domain of the function is restricted to 0 ≤ p ≤ 100, since it doesn't make sense to supply recycling bins to more than 100% of the population, or to a negative percentage of the population.
As p approaches 100%, the cost of supplying recycling bins becomes very large, since it becomes increasingly expensive to provide bins to a large portion of the population. As p approaches 0%, the cost also approaches 0, since there are fewer bins needed for a smaller population.
The function has a vertical asymptote at p = 100, since the denominator of the fraction approaches 0 as p approaches 100 from the left. This means that the cost becomes infinitely large as the percentage of the population approaches 100%.
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c. The expression ax² + 11x + 3 can be factorised. List the possible values of a given that a is a positive integer.
Answers
Step-by-step explanation:
If the polynomial 3x^3+ax^2-11x+3 is exactly divisible by (x – 1), then find the value of a. Hence, factorize the polynomial. can anyone help
What is the average rate of change between
the points (17, 5) and (19, --1)?
Answers
The average rate of change between the points (17, 5) and (19, -1) is -3.
What is the average rate of change?If we have a given function y = f(x) with two known points (a, f(a)) and (b, f(b)), then the average rate of change in that interval [a, b] is:
R = ( f(b) - f(a))/(b - a)
Here we have the two points (17, 5) and (19, -1)
So we have:
a = 17 and f(a) = 5
b = 19 and f(b) = -1
Replacing that in the formula for the average rate of change we will get:
R = (-1 - 5)/(19 - 17)
R = -6/2
R = -3
The average rate of change is -3
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Let A and B be events with P(A) = 0.3, P(B) = 0.6, and P(A and B) = 0.03. Are A and B mutually exclusive? Explain why or why not.
Answers
Answer:
A and B are not mutually exclusive
Step-by-step explanation:
A and B are not mutually exclusive because P(A and B) > 0. If A and B were mutually exclusive, then they would have no outcomes in common and the probability of their intersection would be zero. However, in this case, they do share some outcomes, since P(A and B) is greater than zero.
Question 8 Suppose that W is random variable. Given that P(W ≤6)=0.935 find the probability of it complement, P(W>6)
Answers
The required probability of W being greater than 6 is 0.065 or 6.5%.
What is Probability?A probability value represents the likelihood that an occurrence will take place. In terms of percentage notation, it is stated as a number between 0 and 1, or between 0% and 100%. The higher the probability, the more probable it is that the event will take place.
According to question:The complement of an event A is the event that A does not occur. In this case, the event A is "W ≤ 6", and its complement is "W > 6".
We are given that P(W ≤ 6) = 0.935. Using the complement rule of probability, we can find the probability of W > 6 as follows:
[tex]$$P(W > 6) = 1 - P(W \leq 6)$$[/tex]
Substituting the given value, we have:
P(W > 6) = 1 - 0.935 = 0.065
Therefore, the probability of W being greater than 6 is 0.065 or 6.5%.
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Work out the value of the missing angle
x
.
The diagram is not drawn to scale.
Answers
Answer:
No diagram provided here
The rectangle can be made to have rotation symmetry of order 2 by colouring one of the squares blue. Put a cross in the middle of the square which would have to be made blue.
Answers
To make a rectangle with a rotation symmetry of order 2, first draw the rectangle on a piece of paper. Then draw a cross in the middle of one of the squares in the rectangle. This is the square that needs to be coloured blue in order to create the rotation symmetry. Finally, colour the square with the cross blue to create the rotation symmetry.
Consider relation Ron set A, where R = {(a,b) a divides b), and A is the set of positive Integers. Which of the following properties does the relation have? Check all that apply. Antisymmetric Transitive Reflexive Symmetric
Answers
Consider relation Ron set A, where R = {(a, b)| a divides b), and A is the set of positive Integers. The relation have antisymmetric. So the option A is correct.
A relation is antisymmetric if and only if it is not symmetric and for all elements a and b in the set, if (a,b) is in the relation, then (b,a) is not in the relation. In other words, if a is related to b and b is related to a, then a must be equal to b.
A relation is antisymmetric if for all elements (a,b) in the relation, if a divides b and b divides a then a=b. In this case, if a divides b and b divides a, then a and b must both be the same number since a positive integer cannot divide itself. Therefore, Ron set A is antisymmetric. So the option A is correct.
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The complete question is:
Consider relation Ron set A, where R = {(a, b)| a divides b), and A is the set of positive Integers. Which of the following properties does the relation have? Check all that apply.
A. Antisymmetric
B. Transitive
C. Reflexive
D. Symmetric
construct shear and bending diagrams for the following beams. show your equations used to create the plots. p p p p l/2 l/4 l/4 p p p l/3 l/3 l/3
Answers
The shear force and bending moment diagrams for the given beam will have multiple segments of different shapes and slopes, reflecting the variation of loads along the length of the beam.
To construct the shear and bending diagrams for the given beam, we need to analyze the beam for the different sections where the load is applied. We can break down the beam into five sections:
Leftmost section (0 ≤ x ≤ L/4)
Second section (L/4 < x ≤ L/2)
Third section (L/2 < x ≤ 5L/12)
Fourth section (5L/12 < x ≤ 7L/12)
Rightmost section (7L/12 < x ≤ L)
We can use the equations for shear and bending moments to create the plots:
For section 1: 0 ≤ x ≤ L/4
The shear force diagram will be constant since there is no load applied in this section. The bending moment diagram will be a sloping line, which will be zero at x = 0 and will increase linearly with x as we move toward the right end of the section.
For section 2: L/4 < x ≤ L/2
The shear force diagram will start from the value of P at x = L/4 and remain constant up to x = L/2. The bending moment diagram will be a parabolic curve, which will be zero at x = L/4 and L/2 and will reach a maximum value at the midpoint of the section.
For section 3: L/2 < x ≤ 5L/12
The shear force diagram will start from the value of P at x = L/4 and remain constant up to x = L/2. At x = 5L/12, a load of P/3 is added, causing the shear force to increase suddenly. The bending moment diagram will be a cubic curve, which will be zero at x = L/4 and L/2, and will have a local minimum at x = 5L/12.
For section 4: 5L/12 < x ≤ 7L/12
The shear force diagram will start from the value of P + P/3 at x = 5L/12 and remain constant up to x = 7L/12. The bending moment diagram will be a cubic curve, which will be zero at x = L/4 and L/2, and will have a local maximum at x = 7L/12.
For section 5: 7L/12 < x ≤ L
The shear force diagram will start from the value of P + P/3 at x = 5L/12 and remain constant up to x = 7L/12. At x = L/3, a load of P/3 is added, causing the shear force to decrease suddenly. The bending moment diagram will be a parabolic curve, which will be zero at x = L/4 and L/2 and will reach a minimum value at the midpoint of the section.
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Both descriptive statistics (mean, median, mode, and range) and probability (the likelihood that something will happen) can be useful in our academic, professional, and personal lives. • Determine which of the two (descriptive statistics or probability) you find to be the most useful in your life and explain why using two (2) specific examples.
Answers
Descriptive statistics are the most useful in life. Descriptive statistics provide information about a data set and can help to summarize and interpret data. Specifically, I find the mean and median to be the most useful.
What does Descriptive statistics mean?Descriptive statistics involves the use of measures such as the mean, median, mode, and range, as well as graphical representations of the data, such as histograms, box plots, and scatter plots.
The mean is the average of a set of data and is useful for summarizing and interpreting data. For example, when I am studying for a test, I often use the mean of my practice test scores to understand my overall performance.
The median is the middle value of a set of data and is useful for understanding the spread of the data. For example, when I am tracking my monthly expenses, I often use the median to understand how much I am spending each month. By taking the median of my monthly expenses, I can get an idea of which expenses are taking up the most of my budget.
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Solve the equation. (Find all solutions of the equation in the interval [0, 2pi). Enter your answers as a comma-separated list.)
Answers
The solutions to the given equation in the interval $[0,2\pi)$ are:
[tex]$x=\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$$[/tex]
How to deal with trigonometric equation?The given equation is:
[tex]$$\cos(2x) + \sin(x) = 0$$[/tex]
We can write [tex]$\cos(2x)$[/tex] as [tex]$1-2\sin^2(x)$[/tex]using double angle formula for cosine. Substituting this in the given equation, we get:
[tex]$$1-2\sin^2(x) + \sin(x) = 0$$[/tex]
Rearranging and factoring, we get:
[tex]$$(2\sin(x) - 1)(\sin(x) + 1) = 0$$[/tex]
So, either [tex]$2\sin(x) - 1 = 0$[/tex] or [tex]$\sin(x) + 1 = 0$[/tex].
Solving [tex]$2\sin(x) - 1 = 0$[/tex], we get:
[tex]$$\sin(x) = \frac{1}{2}$$[/tex]
which has solutions [tex]x=\frac{\pi}{6}$[/tex] and $[tex]x=\frac{\pi}{6}$[/tex] in the interval [tex]$[0,2\pi)$[/tex].
Solving [tex]$\sin(x) + 1 = 0$[/tex], we get:
[tex]$$\sin(x) = -1$$[/tex]
which has solution [tex]x=\frac{3\pi}{2}$[/tex]in the interval [tex]$[0,2\pi)$[/tex]
Therefore, the solutions to the given equation in the interval $[0,2\pi)$ are:
[tex]$x=\frac{\pi}{6}, \frac{5\pi}{6}, \frac{3\pi}{2}$$[/tex]
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What is the difference between the longest and
shortest pieces of scrap wood
Answers
The variation in length between the longest and shortest scraps of wood is 0.875.
Explain about the mixed fractions?The sum of both an entire number and a legal fraction is contained in a mixed fraction or number.
We translate an improper fraction into such a mixed fraction and use the basic arithmetical operations. An incorrect fraction is one in which the numerator exceeds the denominator.
Longest value ; 5
Shortest value; 4 1/8 (mixed fraction)
Convert mixed fraction in to proper fraction.
4 1/8 = ( 4*8 + 1 )/8
4 1/8 = 33/8
difference = longest pieces of scrap wood - shortest pieces of scrap wood
difference = 5 - 33/8
difference = 0.875
Thus, the difference in length between the longest and shortest scraps of wood is 0.875.
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Driving at a constant speed, Sharon usually
takes 180 minutes to drive from her house to
her mother's house. One day Sharon begins
the drive at her usual speed, but after driving
1 of the way, she hits a bad snowstorm and
reduces her speed by 20 miles per hour. This
time the trip takes her a total of 276 minutes.
How many miles is the drive from Sharon's
house to her mother's house?
Answers
The total number of miles from Sharon's house to her mother's house would be = 60 miles.
How to calculate the total number of miles from Sharon's house?The total amount of time it takes Sharon to reach the mother's house at constant speed = 180 minutes
The speed she used after the storm = 20 mile/hr
Therefore the miles she needs to cover before she reaches the mother's house = ?
That is;
20 miles = 1hr or 60 mins
X miles = 180 minutes
make X the subject of formula;
X miles = 20×180/60
= 3600/60
= 60 miles.
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Conduct a survey with a minimum of 20 people.Complete the designed questionnaire in 1.2.Remind participants why your doing the survey and that their information will kept confidential
Answers
Some of the tips which is used to conduct a survey are:
Determine the purpose of the survey and define the population of interest.Design the questionnaire and ensure that the questions are clear and concise.Select a sample from the population and distribute the questionnaire to the participants.Remind participants why you are conducting the survey and assure them that their responses will be kept confidential.What is the need to conduct a survey (questionnaire)?Conducting a survey is an important tool for gathering information from a large and diverse group of people. Its allows allow researchers to obtain data from a representative sample of the population, which can then be used to make informed decisions, identify trends, and measure changes over time.
Also important, its can provide insight into people's attitudes, beliefs, and behaviors, which can be valuable in developing marketing strategies, designing programs and policies, and making important decisions.
A paragraph in a designed questionnaire which can remind participants why your doing the survey and that their information will kept confidential is as follows: "Thank you for taking the time to participate in this survey. The purpose of this questionnaire is to gather information about your experiences and opinions on a particular topic. Your participation is crucial to help us gain insights and make informed decisions. All of the information provided in this survey will be kept strictly confidential and will only be used for research purposes".
Full question "In a designed questionnaire, remind participants why your doing the survey and that their information will kept confidential".
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Jane has been practicing sewing, and she wants to make a rectangular blanket to give as a gift to her best friend. So that the blanket is not too small, Jane decides the blanket will have an area of approximately 40 square feet, or 5,760 square inches. She also wants the blanket to be 18 inches longer than it is wide to have room to embroider her friend's name along one edge.
To the nearest tenth of an inch, what is the width of the blanket?
Answers
The width of the blanket to the nearest tenth of an inch is approximately 67.4 inches.
What is width in rectangle ?
In a rectangle, the width refers to the measurement of the shorter side of the rectangle, which is perpendicular to its length.
Let x be the width of the blanket in inches. Then the length of the blanket is x + 18 inches.
The area of the blanket is given by:
Area = Length x Width
5760 sq inches = (x + 18) inches * x inches
Expanding the right-hand side and solving for x, we get:
[tex]x^2 + 18x - 5760 = 0[/tex]
We can solve this quadratic equation using the quadratic formula:
[tex]x = (-b \pm \sqrt{(b^2 - 4ac)}) / 2a[/tex]
where a = 1, b = 18, and c = -5760. Plugging in these values, we get:
[tex]x = (-18 \pm \sqrt{(18^2 - 4(1)(-5760))}) / 2(1)[/tex]
[tex]x = (-18 \pm \sqrt{(18^2 + 23040)}) / 2[/tex]
x ≈ 67.42 or x ≈ -85.42
Since the width cannot be negative, the width of the blanket to the nearest tenth of an inch is approximately 67.4 inches.
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Given a doubling time of 934 minutes, how long will it take a culture in the exponential growth phase to grow from 15 cells per mL, to 86,059 cells per mL? Explain.
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It will take approximately 7.12 days for a culture with a doubling time of 934 minutes to grow from 15 cells/mL to 86,059 cells/mL. This is calculated using the exponential growth equation and logarithms.
If the doubling time is 934 minutes, it means that the number of cells in the culture doubles every 934 minutes. Let t be the time in minutes required to grow from 15 cells/mL to 86,059 cells/mL. We can set up the following equation:
86,059/15 = 2^(t/934)
Taking the logarithm of both sides, we get:
log(86,059/15) = (t/934) log(2)
Solving for t, we get:
t = (log(86,059/15)/log(2)) * 934 ≈ 10,288 minutes ≈ 7.12 days
Therefore, it will take approximately 7.12 days for the culture to grow from 15 cells/mL to 86,059 cells/mL under the given conditions.
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Fill in this table as you work through the lesson. You may also use the glossary to help you.
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Answer:
1. cosine
2. function
3. tangent
4. complementary
5. sine
6. cofunctions
the graph shows the preimage shaded in grey and the image outlined in black. what is the scale factor of the dilation?
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The scale factor of dilation of the shaded in gray to the shaded in black is 3
Calculating the scale factor dilationGiven that
The preimage = shaded in gray
The image = shaded in black
From the graph, we have the following values on the image and the preimage
The preimage = shaded in gray = 4
The image = shaded in black = 12
The scale factor of dilation is then calculated as
Scale factor = shaded in black/shaded in gray
So, we have
Scale factor = 12/4
Evaluate
Scale factor = 3
Hence, the scale factor dilation is 3
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show that the subspace (a, b) of r is homeomorphic with (0, 1) and the subspace [a, b] of r is homeomorphic with [0, 1].
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The process to show that subspace (a, b) of r is homeomorphic with (0, 1) and the subspace [a, b] of r is homeomorphic with [0, 1] is explained below.
To prove that the subspace (a, b) of the real numbers is homeomorphic to the interval (0, 1), we can define a function f(x) = (x-a)/(b-a).
This function maps the interval (a, b) onto (0, 1) by scaling and shifting the values of x so that "a" maps to "0" and "b" maps to "1", while preserving the order and continuity of the values.
The inverse function is g(y) = ay + (1-y)b, which maps (0, 1) onto (a, b) by reversing the scaling and shifting of the values of y so that "0" maps to "a" and "1" maps to "b", while preserving the order and continuity of the values.
So, f(x) and g(y) are both continuous and bijective, and their inverses are also continuous, so (a, b) and (0, 1) are homeomorphic.
Similarly, to show that the subspace [a, b] of the real numbers is homeomorphic to the interval [0, 1], we can define a function h(x) = (x-a)/(b-a).
This function maps the interval [a, b] onto [0, 1] by scaling and shifting the values of x so that "a" maps to "0" and "b" maps to "1", while preserving the order and continuity of the values.
The inverse function is k(y) = ay + (1-y)b, which maps [0, 1] onto [a, b] by reversing the scaling and shifting of the values of y so that "0" maps to "a" and "1" maps to "b", while preserving the order and continuity of the values.
So, h(x) and k(y) are both continuous and bijective, and their inverses are also continuous, so [a, b] and [0, 1] are homeomorphic.
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Slope of the tangent line=?
m=?
b=?
Answers
The function for this problem is given as follows:
[tex]f(x) = \sqrt{57 - x}[/tex]
The derivative of the function is given as follows:
[tex]f^{\prime}(x) = -\frac{1}{2\sqrt{57 - x}}[/tex]
The slope of the tangent line at point (8,7) is given by the numeric value of the derivative of f(x) at x = 8, hence:
[tex]f^{\prime}(8) = -\frac{1}{2\sqrt{57 - 8}}[/tex]
m = f'(8) = -1/14.
Hence the equation of the tangent line, which is a linear function, is given as follows:
y = -x/14 + b.
When x = 8, y = 7, hence the intercept b is obtained as follows:
7 = -8/14 + b
b = 7 + 8/14
b = 49/7 + 4/7
b = 53/7.
Hence the equation is given as follows:
y = -x/14 + 53/7.
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Using the set of data below, which statements are true?
13, 11, 16, 12, 42, 8
The mean of this data set is 17.
The median is 12.5.
The median is 14.5.
The mean is more affected by the outlier.
The median is more affected by the outlier.
The mean of this data set is 12.
Answers
Statements 1,2 and five are correct.
Explanation:
Statement one:
To calculate the mean we must add all the numbers first
13+11+16+12+42+8= 102
Now we must divide the total by the amount of numbers, which in this case will be 6
102/6 = 17
Mean= 17
Statement 2:
First we should organize the numbers from smallest to biggest
8,11,12,13,16,42
Now since there is no middle number, we must find the average
12+13=25
25/2=12.5
Statement 5:
We know the median is an outlier as it’s the only number with a decimal
Write v as a linear combination of u1, u2, and u3, if possible. (Enter your answer in terms of u1, u2, and u3. If not possible, enter IMPOSSIBLE.)v = (-1, 7, 2), u1 = (3, 2, 8), u2 = (-1, -3, -1), u3 = (-2, 1 , -3)
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v = u1 - 2u2 + u3 This means that v can be expressed as a linear combination of u1, u2, and u3. Specifically, we can take one copy of u1, subtract two copies of u2, and add one copy of u3 to obtain v.
To write v as a linear combination of u1, u2, and u3, we need to find constants a, b, and c such that v = au1 + bu2 + c*u3.
We can set up a system of equations to solve for these constants:
3a - b - 2c = -1
2a - 3b + c = 7
8a - b - 3c = 2
We can solve this system using row reduction, which yields:
a = 1
b = -2
c = 1
Therefore, we have:
v = u1 - 2u2 + u3
This means that v can be expressed as a linear combination of u1, u2, and u3. Specifically, we can take one copy of u1, subtract two copies of u2, and add one copy of u3 to obtain v.
It's worth noting that not every vector can be expressed as a linear combination of a given set of vectors. However, in this case, we were able to find constants that satisfy the equation v = au1 + bu2 + c*u3, so it is possible to write v as a linear combination of u1, u2, and u3.
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please help !! Given l, m, n, find the value of x.
(7x-2)° (5x+14)°
Answers
Answer:
Step-by-step explanation:
7x - 2 = 5x + 14
2x - 2 = 14
2x = 16
x = 8
I need help with this
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In the figure below, YE bisects ZFYD. Use the figure to answer are mZEYF.
What is figure?Figure is a page in a book, magazine, website, or other printed material that contains a graphical representation of data, such as a chart, diagram, drawing, or photograph. Figures are often used to help explain complex concepts or to present data in an easily digestible way. Figures can also be used to highlight important information or to make an argument more visually appealing.
The angle YEZ is an inscribed angle, and it bisects the central angle ZFYD. Therefore, the measure of angle YEZ is equal to one-half the measure of angle ZFYD, which is mZYEF. Thus, mZEYF = mZYEF.
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Intersect Combining these two facts, we get mZCFR + mZRFY = 2mZEYF, which simplifies to m2FYD.
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We can use the fact that when two lines intersect, the opposite angles formed are equal. Therefore, we can say that mZCFR = mZRFY.
In addition, since YE bisects ZFYD, we can say that mZFYD = 2mZEYF.
Combining these two facts, we get mZCFR + mZRFY = 2mZEYF, which simplifies to m2FYD.
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