Answers
Answer:
x = 14√2 or 19.8 units---------------------------------
Given is the special right triangle.
It is a 45° isosceles right triangle. It has a property that the hypotenuse is √2 times the leg.
We have a hypotenuse of 28 units, so the leg is:
x = 28/√2 =28√2/2 = 14√2 or 19.8 unitsAnswer:
14√2
Step-by-step explanation:
Here we need to find out the value of "x" in the given right angled triangle, here we can make use of trigonometric ratios to find out the value of x .
In a right angled triangle, sine is defined as the ratio of perpendicular and hypotenuse . With respect to angle 45° , x is the perpendicular and 28 is hypotenuse (side opposite to 90° ) .
So we have;
sin45° = x/28
value of sin45° is 1/√21/√2 = x/28
x = 28*1/√2
x = 14 * (√2)² / √2
x = 14√2
Hence the value of x is 14√2 .
Related Questions
Ribbon wands are made from strips of ribbon tied to sticks. Connie has 84 feet of red ribbon, 48 feet of blue ribbon, and 72 feet of white ribbon. She wants to cut the ribbons into equal lengths that are as long as possible so that no ribbon is wasted
How many pieces of each color will she have?
Answers
Answer
Connie can cut the ribbon into 12 foot pieces.
Step-by-step explanation:
For these problems we should find the GCF, which is 12.
Match the conic equations to the descriptions. A. StartFraction (x + 5) squared Over 100 EndFraction + StartFraction (y minus 4) squared Over 225 EndFraction = 1 B. StartFraction (x minus 4) squared Over 16 EndFraction minus StartFraction (y + 5) squared Over 9 EndFraction = 1 C. StartFraction (y + 5) squared Over 64 EndFraction + StartFraction (x minus 4) squared Over 81 EndFraction = 1 D. StartFraction (y minus 4) squared Over 16 EndFraction minus StartFraction (x + 5) squared Over 9 EndFraction = 1 Choose the letter of the equation from the drop down menu. Ellipse with center at (4, –5): Ellipse with center at (–5, 4): Hyperbola with center at (–5, 4): Hyperbola with center at (4, –5): ‘
Answers
Correct option is A - Ellipse with center at (-5,4) ; B - Ellipse with center at (4,-5) ; C - Hyperbola with center at (4,-5) ; D - Hyperbola with center at (-5,4).
What is conic section ?
Conic sections are curves that are formed by the intersection of a plane and a double cone. The conic sections include circles, ellipses, parabolas, and hyperbolas. Each of these curves has a unique set of characteristics that can be described by mathematical equations.
Explanation of the correct matching :
A - The equation represents an ellipse with center at (-5,4). The values 100 and 225 in the equation represent the squared lengths of the major and minor axes, respectively. The center of the ellipse is (h,k), which is (-5,4) in this case.
B - The equation represents an ellipse with center at (4,-5). The values 16 and 9 in the equation represent the squared lengths of the major and minor axes, respectively. The center of the ellipse is (h,k), which is (4,-5) in this case.
C - The equation represents a hyperbola with center at (4,-5). The values 64 and 81 in the equation represent the squared distances between the center and the vertices on the y-axis and x-axis, respectively. The center of the hyperbola is (h,k), which is (4,-5) in this case.
D - The equation represents a hyperbola with center at (-5,4). The values 16 and 9 in the equation represent the squared distances between the center and the vertices on the x-axis and y-axis, respectively. The center of the hyperbola is (h,k), which is (-5,4) in this case.
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Answer:
C A D B
Step-by-step explanation:
on edge
Calculate the 90% confidence interval for the proportion of voters who cast their ballot for the candidate.
Answers
We can say with 90% confidence that the true proportion of voters who cast their ballot for the candidate lies between 0.564 and 0.636. We can calculate it in the following manner.
To calculate the 90% confidence interval for the proportion of voters who cast their ballot for the candidate, we need to use the following formula:
CI = p ± z√(p(1-p)/n)
where:
CI is the confidence interval
p is the sample proportion
z is the z-score corresponding to the desired confidence level (90% in this case)
n is the sample size
Assuming we have a sample of size n and a sample proportion of p who voted for the candidate, we need to find the value of z for the 90% confidence level. The z-score can be found using a z-table or a calculator, and for a 90% confidence level, the z-score is 1.645.
Substituting the values into the formula, we get:
CI = p ± 1.645√(p(1-p)/n)
For example, if the sample size is 1000 and the sample proportion is 0.6 (60% of voters voted for the candidate), then the 90% confidence interval would be:
CI = 0.6 ± 1.645√(0.6(1-0.6)/1000) = (0.564, 0.636)
Therefore, we can say with 90% confidence that the true proportion of voters who cast their ballot for the candidate lies between 0.564 and 0.636.
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Full question here:
Calculate the 90% confidence interval for the proportion of voters who cast their ballot for the candidate. Number of votes: 125
Voter Response Dummy Variable
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Solve for X, please help
Answers
Answer:
x = 6
Step-by-step explanation:
If 3 or more parallel lines are intersected by two or more transversals , the parallel lines divide the transversals proportionally.
here 3 parallel lines are intersected by two transversals , then
[tex]\frac{1+4x}{20}[/tex] = [tex]\frac{15}{27-15}[/tex]
[tex]\frac{1+4x}{20}[/tex] = [tex]\frac{15}{12}[/tex] ( cross- multiply )
12(1 + 4x) = 20 × 15 = 300 ( divide both sides by 12 )
1 + 4x = 25 ( subtract 1 from both sides )
4x = 24 ( divide both sides by 4 )
x = 6
The surface area of a globe in Mr.Patton’s classroom is about 452.39 square inches. Find its volume in cubic inches . Use 3.14 for pi. Round to the nearest whole number
Answers
The volume of the globe is 905 cubic inches, for the given surface area.
What is surface area?The area is the area occupied by a two-dimensional flat surface. It has a square unit of measurement. The surface area of a three-dimensional object is the space taken up by its outer surface. The region that includes the base(s) and the curved portion is referred to as the total surface area. It is the overall area that the object's surface occupies. The total area of a form with a curved base and surface is equal to the sum of the two areas.
The volume of the globe is given as:
V = 4/3πr³
The surface area if given as:
SA = 4πr²
Substituting the values of the given SA we have:
452.39 = 4 * 3.14(r²)
r = 6.01
Now, substitute the value of r in the equation of volume we have:
V = 4/3(3.14)(6.01)³
V = 904.78.
Hence, the volume of the globe is 905 cubic inches, for the given surface area.
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State whether the triangles could be proven congruent as SSS or SAS Theorem.
Answers
Using SSS theorem of congruency in triangles, we can prove that in all the cases, each triangle is congruent to the other.
What do you mean by congruent triangles?Whether two or more triangles are congruent depends on the size of the sides and angles. A triangle's size and shape are consequently determined by its three sides and three angles. If the pairings of the respective sides and accompanying angles are equal, two triangles are said to be congruent. Both of these are the exact same size and shape. Triangles may satisfy a number of distinct congruence requirements.
The SSS criterion is also known as the Side-Side-Side criterion. This standard states that two triangles are congruent if the sum of the three sides of each triangle is the same.
Here in the question,
It is given that the two sides of each triangle are equal to the corresponding sides of the other triangle.
Now as two sides of a triangle is equal to the two sides of another triangle, it is obvious that he third side will be equal to the corresponding sides of the other triangle.
Now as per the SSS criteria, as all the sides are equal to the corresponding sides of the other triangle, the triangle are congruent.
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A group of 15 athletes participated in a golf competition. Their scores are below:
Would a dot plot or a histogram best represent the data presented here? Why?
A) Histogram, because a large number of scores are reported as ranges
B) Histogram, because a small number of scores are reported individually
C) Dot plot. because a large number of scores are reported as ranges
D) Dot plot, because a small number of scores are reported individually
Answers
Hello, I think the answer is D. Since the scores arent that huge of a gap, dot plot because small number scores are reported individually
A group of 8 friends went to lunch and spent a total of $76, which included the food bill and a tip of $16. They decided to split the bill and tip evenly among themselves. Which equations and solutions describe the situation? Select two options. The equation StartFraction 1 over 8 EndFraction (x + 16) = StartFraction 76 over 8 EndFraction represents the situation, where x is the food bill. The equation StartFraction 1 over 8 EndFraction (x + 16) = 76 represents the situation, where x is the food bill. The solution x = 60 represents the total food bill. The solution x = 60 represents each friend’s share of the food bill and tip. The equation 8 (x + 16) = 76 represents the situation, where x is the food bill.
YALL PLEASE HELP ASAP
Answers
Answer:
The solution x = 5 represents each friend's share of the food bill and tip.
RATE 5 stars and mark brainliest
Part a: The number of transistors per IC in 1972 seems to be about 4,000 (a rough estimate by eye).
Using this estimate and Moore's Law, what would you predict the number of transistors per IC to be 20
years later, in 1992?
Prediction = ?
Part b: From the chart, estimate (roughly) the number of transistors per IC in 2016. Using your estimate
and Moore's Law, what would you predict the number of transistors per IC to be in 2040?
Part c: Do you think that your prediction in Part b is believable? Why or why not?
Answers
Moores law that number of transistors per IC Prediction = 4,096,000 and Prediction = 25.6 trillion.
What is probability ?
Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, with 0 indicating that an event is impossible and 1 indicating that an event is certain. For example, the probability of flipping a fair coin and getting heads is 0.5, while the probability of rolling a six on a fair six-sided die is 1/6 or approximately 0.1667.
In mathematical terms, probability is calculated by dividing the number of ways an event can occur by the total number of possible outcomes. Probability theory is widely used in many fields, including statistics, finance, science, engineering, and economics, to help predict and analyze the likelihood of various outcomes and make informed decisions based on the probabilities involved.
According to the question:
Part a: Moore's Law predicts that the number of transistors per IC doubles every 18-24 months. Since 20 years is approximately 10 doublings (20/2), we can estimate the number of transistors in 1992 to be [tex]4,000 * 2^{10} = 4,096,000.[/tex]
Prediction = 4,096,000
Part b: Based on the chart, the number of transistors per IC in 2016 appears to be around 10 billion [tex](1 * 10^{10})[/tex]
Using Moore's Law, we can estimate the number of transistors per IC in 2040 to be [tex]1 * 10^{10} * 2^{24/18} = 25.6 * 10^{12} (or 25.6 trillion)[/tex].
Prediction = 25.6 trillion
Part c: The prediction in Part b may not be entirely believable, as there are physical limits to the number of transistors that can be placed on a single chip. Moore's Law has been slowing down in recent years, with transistor density growth rates dropping below historic trends. Additionally, new technologies beyond traditional silicon-based chips may become necessary to continue improving transistor density at the same pace as in the past. Therefore, while the prediction is technically possible, it may not be achievable without significant breakthroughs in semiconductor technology.
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X man can complete a work in 40 days.If there were 8 man more the work should be finished in 10 days less the original number of the man
Answers
In linear equation, 24 is the original number of the man .
What in mathematics is a linear equation?
A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.
Equations with power 1 variables are known as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.
Original job = x men * 40 days = 40x man days to complete
now add 8 men = x+8 men
man days now is (x+8) (30) to complete job
so 40x = (x+8)(30)
40x = 30x + 240
10 x = 240
x = 24 men originally.
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If cos is the third quadrant find sin
Answers
The value of sinθ = -[tex]\frac{7}{8}[/tex] using trigonometry.
Trigonometry: What Is It?One of the most significant areas of mathematics, trigonometry has a wide range of applications. Trigonometry is a field of mathematics that primarily focuses on the analysis of how a right-angle triangle's sides and angles relate to one another. Therefore, using trigonometric formulas, functions, or trigonometric identities can be helpful in determining the absent or unknown angles or sides of a right triangle. Angles in geometry can be expressed as either degrees or radians. 0°, 30°, 45°, 60°, and 90° are some of the trigonometric angles that are most frequently used in computations.
In this question,
sin²θ + cos²θ= 1
sin²θ + (-1/4)² = 1
sin²θ = 1- (1/8)
sinθ = ± √(7/8)
since, it is in the third quadrant,
sinθ= -[tex]\frac{7}{8}[/tex]
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g find the mean and variance for x suppose that a random variable x has a continuous uniform distribution 1/2 2
Answers
The mean of x is 1.25 and the variance of x is approximately 0.1458.
The continuous uniform distribution has a constant probability density function (PDF) between its minimum and maximum values. In this case, the minimum value is 1/2 and the maximum value is 2.
The mean of a continuous uniform distribution is the average of the minimum and maximum values:
mean = (minimum + maximum) / 2
= (1/2 + 2) / 2
= 1.25
The variance of a continuous uniform distribution is defined as:
[tex]variance = (maximum - minimum)^2 / 12[/tex]
[tex]= (2 - 1/2)^2 / 12[/tex]
= 7/48
≈ 0.1458
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through: (2,5), slope = 3
Answers
The equation of the line passing through (2,5) with a slope of 3 is y = 3x - 1.
This question is incomplete, the complete question is:
What is the equation of line passing through: (2,5), and with a slope = 3?
What is the equation of the line with the given point and slope?The equation of a line in slope-intercept form is expressed as:
y = mx + b
Where m is the slope and b is the y-intercept.
Given that, the point (2, 5) and the slope of the line is 3.
We can use the point-slope form of the equation of a line to find the equation in slope-intercept form:
y - y1 = m(x - x1)
Where x1 and y1 are the coordinates of the given point ( 2,5 ) and m is slope 3.
Substituting the given values, we get:
y - y1 = m(x - x1)
y - 5 = 3(x - 2)
Expanding and rearranging, we get:
y - 5 = 3x - 6
y = 3x - 1
Therefore, the equation of the line is y = 3x - 1.
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Find the standard normal area for each of the following (round your answer to 4 decimal places)
Answers
b.0.0218
c.0.9321
d.0.3096
URGENT PLEASE HELP add the polynomials
Answers
Answer:
Step-by-step explanation:
Combine like terms: [tex]8x^{2} -5x-8x+3x^3+3[/tex] = [tex]3x^3+8x^2-13x+3[/tex]
can you help me to solve these two questions?
Answers
Case 1: The constant c of the piecewise function is equal to 1 / 7.
Case 2: The value of the constant b of the piecewise function with the greater absolute value is equal to 20.
How to determine the value of a variable such that a piecewise function is continuous
A piecewise function is function formed by two or more functions relative to intervals. A piecewise function is continuous if they do not have any jump on graph. For two functions, we must solve the following equation for the case of a piecewise function formed by two functions:
g(a) = h(a)
Case 1 - g(y) = c · y + 3, h(y) = c · y² - 3, a = 7
c · a + 3 = c · a² - 3
c · (a² - a) = 6
c = 6 / (a² - a)
c = 6 / (7² - 7)
c = 6 / 42
c = 1 / 7
The value of the constant c is equal to 1 / 7.
Case 2 - g(x) = b - 2 · x, h(x) = - 150 / (x - b), a = 5
b - 2 · a = - 150 / (a - b)
(b - 2 · a) · (a - b) = - 150
a · b - b² - 2 · a² + 2 · a · b = - 150
- b² + 3 · a · b - 2 · a² = - 150
b² - 3 · a · b + 2 · a² - 150 = 0
b² - 15 · b - 100 = 0
(b - 20) · (b + 5) = 0
b₁ = 20 or b₂ = - 5
The solution with the greater absolute value is b = 20.
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PLS HELP FAST 20 POINTS + BRAINLIEST!!
Answers
360 - 95 - 44 - 42 - 78 = 101
And it is on a straight line so p and a add up to 180
If p is 101
Then q is 179
Answers of angle degrees of p and q
q = 79
p = 101
Explanation
Each angle given has an adjacent angle creating a straight angle of 180 degrees.
First I found the missing adjacent angle degree - see attachment.
We know four of the five interior angle degrees are 85, 136, 138, 102.
Since we know the sum of angles in a pentagon = 540°, we can subtract the known angles from 540 to find “q”
540 - 85 - 136 - 138 - 102 = 79 angle q
To find “p” we know p and q create a straight angle of 180 degrees. We can subtract to find p.
180 - 79 = 101 angle p.
I attached a picture to help
she works a 35
-hour week earning $17.10
an hour.
How much does she earn in one year? (Use 52
weeks in one year.)
Answers
Answer:
$31122.00
Step-by-step explanation:
We know
She works 35 hours a week, earning $17.10 an hour.
17.10 x 35 = $598.50 a week
How much does she earn in one year?
We Take
598.50 x 52 = $31122.00
So, she earns $31122.00 one year.
the possible degree of a polynomial function is at least one more, or 3 more,or 5 more,... than the total number of local maximas and minimas
Answers
This is not generally accurate. A polynomial function's degree can vary between one, three, or five more or less than the sum of its local peaks and minima.
Consider the formula f(x) = x3 - 3x as an illustration. Two local extrema exist for this function: a local maximum at x = -1 and a local minimum at x = 1. The polynomial's degree, however, is only 3, not one, three, or five more than the sum of the local peaks and minima.
Consider the formula g(x) = x5 - 5x3 + 4x as an alternative. Moreover there are two local extrema for this function: a local maximum at x = -1 and a local minimum at x = 1. The polynomial's degree, however, is 5, which is three more than the sum of its local peaks and minima.
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what are the transformations of the following 1) f(x)=3x2^x+4-1
2) f(x)=-1/2x5^x-2+6
3) g(x)=1/5log(x+5)+3
4) g(x)=-4log(x)-2
Answers
1. The functiοn [tex]f(x) = 3x2^x+4-1[/tex]undergοes the fοllοwing transfοrmatiοns
A vertical translatiοn dοwnward by 1 unit (the [tex]"-1[/tex]" at the end)
An upward vertical stretch by a factοr οf 3 (the "3" cοefficient in frοnt)
An expοnential grοwth with base 2 (the expοnent "x" in the term [tex]"2^x"[/tex])
A hοrizοntal shift tο the left by 4 units (the "-4" in the expοnent οf [tex]"2^x"[/tex])
2. The functiοn [tex]f(x) = -1/2x5^x-2+6[/tex] undergοes the fοllοwing transfοrmatiοns:
A vertical translatiοn upward by 6 units (the "+6" at the end)An upward vertical cοmpressiοn by a factοr οf 1/2 (the [tex]"-1/2"[/tex]cοefficient in frοnt)An expοnential grοwth with base 5 (the expοnent "x" in the term [tex]"5^x[/tex]")A hοrizοntal shift tο the left by 2 units (the[tex]"-2"[/tex] in the expοnent οf [tex]"5^x[/tex]")3. The functiοn [tex]g(x) = 1/5log(x+5)+3[/tex] undergοes the fοllοwing transfοrmatiοns:
A vertical translatiοn upward by 3 units (the "+3" at the end)A hοrizοntal shift tο the left by 5 units (the "+5" inside the lοgarithm)A vertical stretch by a factοr οf 1/5 (the [tex]"1/5"[/tex] cοefficient in frοnt)4. The functiοn [tex]g(x) = -4log(x)-2[/tex] undergοes the fοllοwing transfοrmatiοns:
A vertical translatiοn dοwnward by 2 units (the [tex]"-2"[/tex] at the end)A vertical cοmpressiοn by a factοr οf 4 (the[tex]"-4"[/tex] cοefficient in frοnt)A hοrizοntal shift tο the right (there is nο explicit shift, but the dοmain οf the functiοn is restricted tο[tex]x > 0[/tex], which means the graph is shifted tο the right οf the y-axis)
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What is the constant of proportionality between the corresponding areas from Rectangle A to Rectangle B?
Rectangle A: area = 5 in²
Rectangle B: area = 125 in²
Responses
5
10
15
25
Answers
Answer:
its 5
Step-by-step explanation:
I did this question
light of 650 nm wavelength illuminates a single slit of width 0.20 mm . (figure 1) shows the intensity pattern seen on a screen behind the slit.
Answers
the intensity pattern visible on a screen at 1.168 metres behind the slit.
That is the answer to the question "650 nm light shines on two slits that are separated by 0.20 mm. The image depicts the intensity pattern visible on a screen hidden behind the slits (Figure 1).
How far away from you is the screen?"
It is possible to specify that the distance to the screen is d=1.168m.
The answer to the question is that 650 nm light illuminates two slits that are 0.20 mm apart. The image depicts the intensity pattern visible on a screen hidden behind the slits (Figure 1).
How far is the screen from you?
The equation for the distance is typically presented mathematically as
d=1.168m
As a result,
d=1.168m is the distance to the screen.
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What are the values of the interior angles?
Round each angle to the nearest degree.
A) m∠X = 131º, m∠Y = 16º, m∠Z = 33º
B) m∠X = 120º, m∠Y = 15º, m∠Z = 30º
C) m∠X = 145º, m∠Y = 18º, m∠Z = 36º
Answers
We can see here the values of the interior angles will be: A) m∠X = 131º, m∠Y = 16º, m∠Z = 33º.
What is interior angle?An interior angle is an angle created between two adjacent sides of a polygon. To put it another way, it is the angle created by two polygonal sides that have a shared vertex.
Sum of interior angles of a triangle = 180°
[tex]2p + \frac{1}{4} p + \frac{1}{2} p = 180[/tex]
11p/4 = 180°
p = 720°/11
m∠X = 2p = 2 × 720°/11 = 130.9 ≈ 131°
m∠Y = [tex]\frac{1}{4} p[/tex] = 1/4 × 720°/11 = 16.3 ≈ 16°
m∠Z = [tex]\frac{1}{2} p[/tex] = 1/2 × 720°/11 = 32.7 ≈ 33°
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What is the correct numerical expression for "one-half the difference of 6 and 8 hundredths and 2?"
one half x (8 − 6) + 2
one half x (6 + 8 + 2)
one half x (6.08 − 2)
one half − (6.08 ÷ 2)
Answers
Answer: c
Step-by-step explanation: i dont have one
1/2 x (6.08 - 2) (6.08 - 2) is the correct numerical expression for "one-half the difference of 6 and 8 hundredths and 2 " .
what is expression ?An expression, as used in computer programming, is a grouping of values, variables, operators, and/or function calls that the computer evaluates to produce a final value. For instance, the equation 2 + 3 combines the numbers 2 and 3 using the + operator to produce the number 5. Similar to this, the equation x * (y + z) produces a value based on the current values of the variables x, y, and z by combining the variables x, y, and z with the * and + operators.
given
In terms of numbers, the phrase "one-half the difference of 6 and 8 hundredths and 2" is expressed as follows:
1/2 x (6.08 - 2) (6.08 - 2)
1/2 x (6.08 - 2) (6.08 - 2) is the correct numerical expression for "one-half the difference of 6 and 8 hundredths and 2 " .
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Find the slope of the line passing through the points (-9, 2) and (-9, -6)
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Answer:
Step-by-step explanation:
use the formula of gradient:
slope=change in y/ change in x
= [tex]\frac{Y2-Y1}{X2-X1}[/tex]
= [tex]\frac{-6-2}{-9--9}[/tex]
=[tex]\frac{-8}{0}[/tex]
the answer is definite because we cannot divide by 0.
Suppose that A is the set of sophomores at your schooland B is the set of students in discrete math at your school.Express each of the following sets in terms of A and B.a. The set of sophomores taking discrete math at yourschool.That’s the intersection A ∩ B.b. The set of sophomores at your school who are nottaking discrete math.This is the difference A − B. It can also be expressed byintersection and complement A ∩ B.c. The set of students at your school who either are sophomores or are taking discrete math.The union A ∪ B.d. The set of students at your school who either are notsophomores or are not taking discrete math.Literally, it’s A ∪ B. That’s the same as A ∩ B.
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Set of sophomores taking discrete math = A ∩ B. Set of sophomores not taking discrete math = A - B or A ∩ B^c. Set of students who are sophomores or in discrete math = A ∪ B. Set of students who are not sophomores or not in discrete math = (A ∩ B)^c or A ∪ B^c.
The set of sophomores taking discrete math at your school is the intersection of the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A ∩ B.
The set of sophomores at your school who are not taking discrete math is the difference between the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A - B or A ∩ B^c, where B^c is the complement of B (i.e., the set of students who are not in discrete math).
The set of students at your school who either are sophomores or are taking discrete math is the union of the set of sophomores A and the set of students in discrete math B. So, it can be expressed as A ∪ B.
The set of students at your school who either are not sophomores or are not taking discrete math is the complement of the intersection of the set of sophomores A and the set of students in discrete math B.
This can be expressed as (A ∩ B)^c or as A ∪ B^c, where B^c is the complement of B (i.e., the set of students who are not in discrete math). Note that this set includes all students who are either juniors, seniors, or not enrolled in discrete math.
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A store is selling 5 types of hard candies: cherry, strawberry, orange, lemon and pineapple. How many ways are there to choose: (a) 24 candies? (b) 24 candies with at least a piece of each flavor? (b) 24 candies with at least 2 cherry and at least 2 lemon?
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The solution for question a), b) and c) are as follows. The number of ways to choose 24 candies is 20475. And the number of of ways to choose 24 candies with at least a piece of each flavor is 7700. Similarly, the total number of ways to choose 24 candies with at least 2 cherry and at least 2 lemon is 272.
(a) To choose 24 candies from 5 types of hard candies, we can use the stars and bars method. We need to distribute 24 candies among 5 types, where each type can have 0 or more candies.
We can represent this by 24 stars (*) and 4 bars (|) to separate the candies of different types. The number of ways to arrange these stars and bars is the same as the number of ways to choose 4 positions out of 28 to place the bars. Therefore, the number of ways to choose 24 candies is:
(28 choose 4) = 20475
(b) To choose 24 candies with at least a piece of each flavor, we can first choose 5 candies, one of each flavor, and then choose 19 candies from the remaining candies. The number of ways to choose 19 candies from 20 candies (excluding one candy of each flavor) is:
(19 + 4 - 1) choose (4 - 1) = 22 choose 3 = 1540
Therefore, the total number of ways to choose 24 candies with at least a piece of each flavor is:
5 * 1540 = 7700
(c) To choose 24 candies with at least 2 cherry and at least 2 lemon, we can use the inclusion-exclusion principle. Let A be the event that we choose at least 2 cherry, and let B be the event that we choose at least 2 lemon. Then the number of ways to choose 24 candies with at least 2 cherry and at least 2 lemon is:
P(A union B) = P(A) + P(B) - P(A intersect B)
To calculate P(A), we can choose 2 cherry and then choose 20 candies from the remaining 3 types, which is:
(2 choose 2) * (20 + 3 - 1) choose (3 - 1) = 22 choose 2 = 231
Similarly, P(B) is also 231.
To calculate P(A intersect B), we can choose 2 cherry, 2 lemon, and then choose 18 candies from the remaining 3 types, which is:
(2 choose 2) * (2 choose 2) * (18 + 3 - 1) choose (3 - 1) = 20 choose 2 = 190
Therefore, the total number of ways to choose 24 candies with at least 2 cherry and at least 2 lemon is:
2 * 231 - 190 = 272
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Jina rolled a number cube 40 times and got the following results.
Outcome Rolled
1
Number of Rolls 7
2
6
3
9
4
6
5
3
Answer the following. Round your answers to the nearest thousandths.
6
9
(a) From Jina's results, compute the experimental probability of rolling a 3 or 6.
0.45
(b) Assuming that the cube is fair, compute the theoretical probability of rolling a 3 or 6.
0
(c) Assuming that the cube is fair, choose the statement below that is true.
With a small number of rolls, it is surprising when the experimental probability is much
greater than the theoretical probability.
With a small number of rolls, it is not surprising when the experimental probability is much
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When there are few rolls, it is expected that the experimental probability will be significantly higher than the theoretical chance.
what is probability ?The study of random occurrences or phenomena falls under the category of probability, which is a branch of mathematics. It is used to determine how likely or unlikely an occurrence is to occur. An event's likelihood is expressed as a number between 0 and 1, with 0 denoting impossibility and 1 denoting certainty of occurrence. The symbol P stands for the probability of an occurrence A. (A). It is determined by dividing the number of positive results of event A by all the potential outcomes.
given
(a) The result of rolling 3 or 6 times is 6 + 9 = 15.
Experimental chance = (Total number of rolls) / (Number of times 3 or 6 were rolled) = 15/40 = 0.375
(b) The theoretical likelihood of rolling either a 3 or a 6 on a fair number cube is equal to the total of those odds, which is 1/6 + 1/6 = 1/3 = 0.333. (rounded to three decimal places).
(c) When there are few rolls, it is expected that the experimental probability will be significantly higher than the theoretical chance.
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write a quadratic function in standard form that passes through the points (-8,0) ,(-5, -3) , and (-2,0) .
F(x)=
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A quadratic function in standard form that passes through the points [tex](-8,0), (-5,-3), and (-2,0)[/tex] is equals to the [tex]f(x) = (1/3)( x^{2} + 10x + 16)[/tex].
What are some examples of quadratic functions?f(x) = ax2 + bx + c, in which a, b, and c are integers and an is not equal to zero, is a quadratic function. A parabola is the shape of a quadratic function's graph.
How do you determine whether an equation is quadratic?In other terms, you have a quadratic equation if a times the squares of the expression after b plus b twice that same equation not square plus c equals 0.
[tex]f(x) = ax^{2} + bx + c ----(1)[/tex]
is determined by three points and must be [tex]a[/tex] not equal [tex]0[/tex]. That is for determining the f(x) we have to determine value of three values a, b, and c. Now, we have three ordered pairs [tex](-8,0), (-5,-3)[/tex], and [tex](-2,0)[/tex] and we have to determine quadratic function passing through these points. So, firstly, plug the coordinates of point[tex]( -8,0), x = -8, y = f(x) = 0[/tex] in equation [tex](1)[/tex],
[tex]= > 0 = a(-8)^{2} + b(-8) + c[/tex]
[tex]= > 64a - 8b + c = 0 -------(2)[/tex]
Similarly, for second point [tex]( -5,-3) , f(x) = -3, x = -5[/tex]
[tex]= > - 3 = a(-5)^{2} + (-5)b + c[/tex]
[tex]= > 25a - 5b + c = -3 --(3)[/tex]
Continue for third point [tex](-2,0)[/tex]
[tex]= > 0 = a(-2)^{2} + b(-2) + c[/tex]
[tex]= > 4a -2b + c = 0 --(4)[/tex]
So, we have three equations and three values to determine.
Subtract equation [tex](4)[/tex] from [tex](2)[/tex]
[tex]= > 64 a - 8b + c - 4a + 2b -c = 0[/tex]
[tex]= > 60a - 6b = 0[/tex]
[tex]= > 10a - b = 0 --(5)[/tex]
subtract equation [tex](4)[/tex] from [tex](3)[/tex]
[tex]= > 21a - 3b = -3 --(6)[/tex]
from equation (4) and (5),
[tex]= > 3( 10a - b) - 21a + 3b = -(- 3)[/tex]
[tex]= > 30a - 3b - 21a + 3b = 3[/tex]
[tex]= > 9a = 3[/tex]
[tex]= > a = 1/3[/tex]
from [tex](5)[/tex] , [tex]b = 10a = 10/3[/tex]
from [tex](4)[/tex], [tex]c = 2b - 4a = 20/3 - 4/3 = 16/3[/tex]
So,[tex]f(x)= (1/3)( x^{2} + 10x + 16)[/tex]
Hence, required values are [tex]1/3, 10/3,[/tex] and [tex]16/3[/tex].
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Answer:
f(x) = (1/3)x² + (10/3)x + 16/3-------------------------------------
Given 3 points of a quadratic function and two of them lie on the x-axis:
(-8, 0) and (-2, 0)These two points are representing the roots of the function. With known roots we can show the function in the factor form:
f(x) = a(x - x₁)(x - x₂), where a - coefficient, x₁ and x₂ are rootsSubstitute the roots into the equation and use the third point with coordinates x = - 5, f(x) = - 3, find the value of a:
-3 = a(- 5 + 8)((-5 + 2)- 3 = a(3)(-3)3a = 1a = 1/3This gives us the function in the factor form:
f(x) = (1/3)(x + 8)(x + 2)Convert this into standard form of f(x) = ax² + bx + c:
f(x) = (1/3)(x + 8)(x + 2)f(x) = (1/3)(x² + 10x + 16)f(x) = (1/3)x² + (10/3)x + 16/3Pick out greatest and smallest numbers from 9929 , 9829 , 9289 , 9982.
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Answer:
smallest no-- 9829, 9289.
greatest no. 9982,9929.