if cot(θ)<0 and cos(θ)<0, then we can conclude that θ lies in the second quadrant.
What is trigonometry functions ?
Trigonometry functions are a set of mathematical functions that relate the angles of a triangle to the lengths of its sides. The three main trigonometric functions are sine, cosine, and tangent, which are commonly abbreviated as sin, cos, and tan. These functions can be defined as ratios of the sides of a right triangle.
In a right triangle, the side opposite an acute angle is called the "opposite" side, the side adjacent to the angle is called the "adjacent" side, and the hypotenuse is the side opposite the right angle. With this terminology, we can define the following trigonometric functions:
Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In other words, sin(θ) = opposite/hypotenuse.
Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, cos(θ) = adjacent/hypotenuse.
Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In other words, tan(θ) = opposite/adjacent.
There are also three reciprocal trigonometric functions: cosecant (csc), secant (sec), and cotangent (cot). These functions are the reciprocals of sine, cosine, and tangent, respectively.
Trigonometric functions have many practical applications in fields such as engineering, physics, and astronomy, where they are used to solve problems involving angles and distances. They also have many applications in pure mathematics, such as in the study of periodic functions and Fourier analysis.
According to given information :
To understand why θ must be in the second quadrant, we need to recall the signs of the trigonometric functions in each quadrant.
In the first quadrant, all trigonometric functions are positive.
In the second quadrant, sine is positive and cosine is negative.
In the third quadrant, both sine and cosine are negative.
In the fourth quadrant, sine is negative and cosine is positive.
Since we know that cos(θ) is negative, we can conclude that θ lies in either the second or third quadrant. However, since cot(θ) = cos(θ)/sin(θ) and cot(θ) is negative, we know that cos(θ) and sin(θ) must have opposite signs. In the third quadrant, both cos(θ) and sin(θ) are negative, so their quotient cot(θ) would be positive, which is not the case.
Therefore, we can conclude that θ lies in the second quadrant.
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Answer:
If cot(θ)<0 and cos(θ)<0, then we can conclude that θ lies in the second quadrant.
What are trigonometry functions ?
Trigonometry functions are a set of mathematical functions that relate the angles of a triangle to the lengths of its sides. The three main trigonometric functions are sine, cosine, and tangent, which are commonly abbreviated as sin, cos, and tan. These functions can be defined as ratios of the sides of a right triangle.
In a right triangle, the side opposite an acute angle is called the "opposite" side, the side adjacent to the angle is called the "adjacent" side, and the hypotenuse is the side opposite the right angle. With this terminology, we can define the following trigonometric functions:
Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In other words, sin(θ) = opposite/hypotenuse.
Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, cos(θ) = adjacent/hypotenuse.
Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In other words, tan(θ) = opposite/adjacent.
There are also three reciprocal trigonometric functions: cosecant (csc), secant (sec), and cotangent (cot). These functions are the reciprocals of sine, cosine, and tangent, respectively.
Trigonometric functions have many practical applications in fields such as engineering, physics, and astronomy, where they are used to solve problems involving angles and distances. They also have many applications in pure mathematics, such as in the study of periodic functions and Fourier analysis.
According to given information :
To understand why θ must be in the second quadrant, we need to recall the signs of the trigonometric functions in each quadrant.
In the first quadrant, all trigonometric functions are positive.
In the second quadrant, sine is positive and cosine is negative.
In the third quadrant, both sine and cosine are negative.
In the fourth quadrant, sine is negative and cosine is positive.
Since we know that cos(θ) is negative, we can conclude that θ lies in either the second or third quadrant. However, since cot(θ) = cos(θ)/sin(θ) and cot(θ) is negative, we know that cos(θ) and sin(θ) must have opposite signs. In the third quadrant, both cos(θ) and sin(θ) are negative, so their quotient cot(θ) would be positive, which is not the case.
Therefore, we can conclude that θ lies in the second quadrant.
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Please help meeee.
i need step by step please
here is the picture of the problem.
The required interval of domain for the composite function is [-2, 3].
What is Domain?
The range of values that we are permitted to enter into our function is known as the domain of a function. The x values for a function like f make up this set (x). A function's range is the collection of values it can take as input. After we enter an x value, the function outputs this sequence of values.
According to question:
The domain of fog(x) is the set of all values of x for which the composition function fog(x) is defined.
fog(x) means that we plug g(x) into f(x), so we have:
fog(x) = f(g(x)) = f(x^2 - x) = √(6 - (x^2 - x))
= √(6 - x^2 + x)6
For the expression under the square root to be defined, we must have 6 - x^2 + x ≥ 0. This is a quadratic inequality that can be factored as:
-(x - 3)(x + 2) ≤ 0
The solutions of this inequality are -2 ≤ x ≤ 3.
However, we also need to check that the expression under the square root is non-negative, so we need to exclude the values of x that make 6 - x^2 + x < 0. This inequality holds for x < -2 or x > 3.
Therefore, the domain of fog(x) is [-2, 3].
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The perimeter of a pentagon is 41.5 cm. write and solve and equation to detrmine the length of each side of the pentagon?
A pentagon has a 41.5 cm circumference, and its sides are each 8.3 cm long.
When you already know the value of one side, calculating the perimeter of a pentagon is made much simpler. The five sides of each pentagon are equal. If the length of one side is given to you.
Since a pentagon has five sides, we will abbreviate each side's length as "s". The lengths of all five sides add up to the perimeter of the pentagon, allowing us to formulate the following equation:
5s = 41.5
We must put "s" on one side of the equation alone in order to solve for it. Divide both sides by 5 to do this: s = 8.3
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The double dot plot below shows the number of hours Kayla and Carmen studied during a two week period in college. Determine the most appropriate measure of variation for each data set. What is the difference between the centers?
The measure of variation include range, variance, iqr etc
What are the measure of variationIn statistics, a measure of variation is a numerical value that describes how spread out or dispersed a set of data is. The most common measures of variation are:
Range: The range is the difference between the maximum and minimum values in a data set. It gives an idea of the spread of the data, but is sensitive to outliers.
Interquartile range (IQR): The IQR is the range of the middle 50% of the data. It is less sensitive to outliers than the range.
Variance: The variance is the average of the squared differences of each data point from the mean. It measures how much the data is spread out from the mean.
Standard deviation: The standard deviation is the square root of the variance. It is a common measure of variation and indicates the typical amount that each data point deviates from the mean.
Coefficient of variation: The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage. It is used to compare the variation of data sets with different means.
An overview was given due to incomplete information.
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Find a vector equation with parameter t for the line through the point (4,0,-3) and parallel to the vector 2i - 4j + 6k
Answer r(t)=
The vectorial equation of the line that passes through (4, 0, - 3) and is parallel to the vector <2, - 4, 6> is equal to (x, y, z) = (4, 0, - 3) + t · (2, - 4, 6).
How to derive the vectorial equation of a line
According to linear algebra, three-dimension lines can be derived from the knowledge of a point and a vector slope, whose formula is introduced below:
(x, y, z) = P(x, y, z) + t · M(x, y, z)
Where:
P(x, y, z) - Point on the linet - ParameterM(x, y, z) - Vector slopeTwo lines are parallel when they have the same vector slope. If we know that P(x, y, z) = (4, 0, - 3) and M(x, y, z) = (2, - 4, 6), then the vectorial equation of the line is:
(x, y, z) = (4, 0, - 3) + t · (2, - 4, 6)
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How many solutions does this equation have? -16 + n + 14n = -16 + 15n
Answer:
Infinitely many solutions
Step-by-step explanation:
Given the equation:
[tex]\displaystyle{-16+n+14n=-16+15n}[/tex]
Which can be simplified (for left side) to:
[tex]\displaystyle{-16+15n=-16+15n}[/tex]
Both sides are same, meaning that there are infinitely many solutions since no matter which n-values you substitute in, you'll end up getting true equation.
Can you find the area of these shapes
Answer:
1. 40
2. 351
3. 49
Step-by-step explanation:
just multiply the 2 numbers!
5 times 8 is 40
13 times 27 is 351
7 times 7 is 49
(square is the same on every side so every side is 7)
A scientist places a cell in a Petri dish. At the end of 1 hour, the cell divides so
that there are 2 cells in the Petri dish. At the end of 2 hours, those cells divide so
there are 4 cells in the Petri dish. The cells continue to divide this way every hour.
Use the expression 2 to the 10th power to find the number of cells in the Petri dish after 10 hours.
Show your work.
The number of cells in the Petri dish after 10 hours is 2¹⁰ = 1024 cells in the dish.
What is simplification?The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.
Here,
The number of cells in the Petri dish after each hour can be represented by the expression 2ⁿ, where n is the number of hours that have passed.
So, after 1 hour, there are 2¹ = 2 cells in the dish.
After 2 hours, there are 2² = 4 cells in the dish.
After 3 hours, there are 2³ = 8 cells in the dish.
And so on.
Therefore, after 10 hours, there are 2¹⁰ = 1024 cells in the dish.
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Graph the function f(x) = 3√x -2.
the graph of the function is attached below.
What is the function?A relationship between a group of inputs and one output is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function.
Given a function, f(x) = 3√x -2
To graph the function f(x) = 3√x -2, we can follow these steps:
Choose some x values to evaluate the function. It's a good idea to pick values that will give us an idea of the general shape of the graph. Let's choose x = 0, 1, 8, and 27.Plug in these values of x into the function to find the corresponding y values:f(0) = 3√0 - 2 = -2
f(1) = 3√1 - 2 = 1
f(8) = 3√8 - 2 ≈ 4.9
f(27) = 3√27 - 2 ≈ 7.2
Plot the points (0, -2), (1, 1), (8, 4.9), and (27, 7.2) on a coordinate plane.Draw a smooth curve through the points to show the shape of the graph.Here, the graph of f(x) = 3√x -2 is attached below.
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2
Let the width and the length be real numbers. The perimeter
remains a constant 24cm. Draw the graph for 1 ≤ w ≤11
Label the axes appropriately.
Please find attached the graph of the function of the length with respect to the width of the rectangle, created with MS Excel, where, 1 ≤ W ≤ 11, and the perimeter of the rectangle is 24 cm.
What is a graph of a function?A graph of a function, is a representation of the ordered pairs of the points of the function, (x, f(x)), on the coordinate plane.
Whereby the figure is a rectangle, we get;
Perimeter = 2 × Length of the rectangle + 2 × Width of the rectangle
Let P represent the perimeter of the rectangle and let L represent the length of the rectangle and let W represent the width of the rectangle, we get;
P = 2·L + 2·W
The perimeter of the rectangle. P = 24
Therefore;
24 = 2·L + 2·W
2·L = 24 - 2·W
L = (24 - 2·W)/2 = 12 - W
L = 12 - W
The graph of the above linear equation representing the length of the rectangle, L, where the width, W is; 1 ≤ W ≤ 11.
Please find attached the required graph of length, L to the width W of the rectangle created with MS Excel
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Determine the coordinates of the point on the unit circle corresponding to the given central angle. If
necessary, round your results to the nearest hundredth.
202⁰
a. (-0.93,-0.37)
b. (-0.37, -0.93)
56:2
c.
(1, -0.37)
d. (-0.93, 0)
Answer:
a. (-0.93,-0.37)
Step-by-step explanation:
The unit circle is a circle with a radius of 1 that is centered at the origin (0, 0) of a coordinate plane. To find the coordinates of a point on the unit circle corresponding to a given central angle, we can use the relationship between the angle and the coordinates of the point.
For a central angle of 202 degrees, the point on the unit circle would be found by using the formula:
x = cos(202°)
y = sin(202°)
Rounding the results to the nearest hundredth, we have:
x = -0.927
y = -0.3746
So the coordinates of the point on the unit circle corresponding to the central angle of 202 degrees are (-0.93, -0.37).
Answer:
a. (-0.93, -0.37)
Step-by-step explanation:
A unit circle has its center at (0, 0) and a radius of 1.
The coordinates on the unit circle (x, y) are equivalent to (cos θ, sin θ), where θ is the angle (measured anticlockwise from the positive x-axis).
Therefore, given θ = 202°, the coordinates of the corresponding point on the unit circle are:
x = cos 202° = -0.93 (nearest hundredth)y = sin 202° = -0.37 (nearest hundredth)Therefore, the point on the unit circle is (-0.93, -0.37).
caitlin took both the sat and the act college entrance exams. her scores on both exams are shown in the table, as well as the national mean scores and standard deviations. exam caitlin's exam score national mean exam score national standard deviation sat 1850 1500 250 act 28 20.8 4.8 she wants to know on which test she performed better. find the z-scores for her result on each exam. provide your answer below: sat z-score: ____
act z-score: ____
On the SAT her z-score was 1.4 and on the ACT her z-score was 1.5 Thus, due to the higher z-score, she performed better on the ACT.
The amount of standard deviations by which the value of a raw score differs from or differs above the mean value of what is observed or measured is known as the z score or standard score in the z test. The z-score calculates how far an X-score deviates from the mean by standard deviations. Calculating the ACT scores utilizing the Z- score -
Z = X - u/a
Calculating the SAT score -
Where X = 1850, u = 1500 and a = 250
Z = 1850 - 1500/250
= 350/250
= 1.4
Similarly,
Calculating the ACT score -
Where X = 28, u = 20.8 and a = 4.8
Z = 28 - 20.8/4.8
= 7.2/4.8
= 1.5
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n people line up to board a plane. each has a boarding pass with assigned seat. however, the first person has lost the boarding pass and takes a random seat uniformly. after that, each person takes the assigned seat if it is unoccupied, and one of unoccupied seats uniformly at random otherwise. denote by pnthe probability that the last person to board sits in the assigned seat. show that pn
The probability that the last person to board sits in the assigned seat is always 1/2, regardless of the number of people n in the line.
The issue can be tackled utilizing numerical enlistment.
Base Case:
At the point when n = 2, there are two travelers and two seats. The principal traveler takes an irregular seat, and the subsequent traveler will sit in his doled out seat with likelihood 1/2 or take the other seat with likelihood 1/2. Consequently, p2 = 1/2.
Inductive Speculation:
Accept that for some certain number k, pk = 1/2.
Inductive Step:
Think about n = k+1 travelers. The principal traveler takes an irregular seat. There are two cases to consider:
Case 1: The primary traveler sits down relegated to the k+1-th traveler. For this situation, the last traveler will be ensured to sit in his doled out seat, and the leftover k travelers can be considered as a subproblem with similar circumstances. By the inductive speculation, the likelihood that the last traveler in the subproblem sits in his alloted seat is 1/2. Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is 1/(k+1) + (k/(k+1)) * 1/2 = (k+2)/(2(k+1)).
Case 2: The primary traveler sits down other than the one doled out to the k+1-th traveler. For this situation, the issue lessens to the case with k travelers, and the likelihood that the last traveler in the decreased issue sits in his doled out seat is pk = 1/2 (by the inductive speculation). Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is (k/(k+1)) * 1/2 = k/(2(k+1)).
Joining the two cases, we have:
pk+1 = (k+2)/(2(k+1)) + k/(2(k+1)) = (k+1)/(2(k+1)) = 1/2.
Consequently, by numerical acceptance, the likelihood that the last individual to board sits in the appointed seat is dependably 1/2, no matter what the quantity of individuals n in the line.
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Listed below in order are prices in dollars for a Big Mac hamburger in the , , , , , , , , , , , and . Such data are used to compare currency exchange rates and the costs of goods in different countries. Find the range, variance, and standard deviation for the given sample data. What do the measures of variation tell us about the prices of a Big Mac in different countries?
Based on the information, we can infer that the range is an adequate indicator to affirm that the price difference of the Big Mac in different countries is minimal.
How to find the range?To find the range of the given data we must perform the following procedure:
We must find the highest and lowest data, in this case the highest and lowest data are 6.8 and 1.98 respectively. So, to find the range we must find the difference between these two data:
6.8 - 1.89 = 4.91
How to find the variance?To find the variance we must apply the following formula:
S² = 1 / 11 - 1 ( 205.59 - (44.67)²/11)
S² = 2.42
How to find the standard deviation?To find the standard deviation we must apply the following formula:
S = [tex]\sqrt[n]{variance}[/tex]
S = [tex]\sqrt{2.42}[/tex]
S = 1.56
What do measures of variation tell us about the prices of a Big Mac in different countries?Based on the above data, we can infer that the range is an indicator that the difference in the Big Mac price in different countries is minimal.
Note: This question is incomplete. Here is the complete information:
Attached image
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4 friends share 2 granola bars each friend gets _____ of the granola bar
Answer:
1/2 (half)
Step-by-step explanation:
4f over 2g then 1f over x
2/4 to get 1/2
each friend gets half (1/2)
consider the two matricesA = | 1 2 4 | and D = | -5 -6 -2 || 4 2 4 | | 1 -2 1 || 3 4 3 | | 3 4 3 |observe that D is obtained from A using two elementary row operations. find a matrix E such that EA = D (hint: E is the product of the two elementary matrices corresponding to the two row operations).
The value of the matrix E is such that EA = D (E is the product of the two elementary matrices corresponding to the two-row operations) will be:
Matrix E = | 0 1 0 |
| 1 0 0 |
We can find the elementary matrices corresponding to the two-row operations and multiply them together to obtain the matrix E.
First-row operation: Add -4 times the first row to the second row.
This is equivalent to multiplying A on the left by the elementary matrix:
| 1 0 0 |
|-4 1 0 |
| 0 0 1 |
Call this matrix E1.
E1A =
| 1 2 4 | | 1 0 0 | | 1 2 4 |
|-4 -6 -12| * | -4 1 0 | = |-5 -6 -2 |
| 3 4 3 | | 0 0 1 | | 3 4 3 |
Next row operation: Swap the first and third rows.
This is equivalent to multiplying E1A on the left by the elementary matrix:
| 0 0 1 |
| 0 1 0 |
| 1 0 0 |
Call this matrix E2.
E2E1A =
| 3 4 3 | | 0 0 1 | | -5 -6 -2 |
|-4 -6 -12| *| 0 1 0 | = | -4 -2 4 |
| 1 2 4 | | 1 0 0 | | 1 2 4 |
Therefore, the matrix E that satisfies EA = D is given by:
E = E2E1 = | 0 0 1 | | 1 0 0 |
E = | 0 0 1 |
EA = D
| 0 1 0 | * |-4 1 0 | = | 0 1 0 |
| 1 0 0 | | 0 0 1 | | 1 0 0 |
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Which set of ordered pairs does not represent a function?
{(5,-1), (7,-9), (-8,-1), (8,-7)}
O {(-4, 1), (2, -7), (8,-2), (-7, -3)}
O {(4,3), (-6, 2), (-3,5), (9,3)}
O {(-1,7), (7, 1), (5,-7), (7,-8)}
The ordered pair that doesn't represent a function is: D.. {(-1,7), (7, 1), (5,-7), (7,-8)}.
What is a Function?A function is a mathematical concept that associates an input (or argument) to a unique output (or value). In mathematical notation, a function is represented as f(x), where x is the input, and f(x) is the output. The association between the input and output is defined by a set of rules, or a formula, that assigns a unique output value to each input value
In a function, for each unique x-value, there must be exactly one corresponding y-value.
D. {(-1,7), (7, 1), (5,-7), (7,-8)} does not represent a function because the same x-value (7) corresponds to two different y-values (1 and -8).
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According to a 2017 survey conducted by the technology market research firm The Radicati Group, U.S. office workers receive an average of 121 e-mails per day (Entrepreneur magazine website). Assume the number of e-mails received per hour follows a Poisson distribution and that the average number of e-mails received per hour is five.
a. What is the probability of receiving no e-mails during an hour (to 4 decimals)?
b. What is the probability of receiving at least three e-mails during an hour (to 4 decimals)? For this question, if calculating the probability manually make sure to carry at least 4 decimal digits in your calculations.
c. What is the expected number of e-mails received during 15 minutes (to 2 decimals)?
d. What is the probability that no e-mails are received during 15 minutes (to 4 decimals)?
The probability of receiving no e-mails during an hour is approximately 0.0067, or 0.67%. The probability of receiving at least three e-mails during an hour is approximately 0.8754, or 87.54%. The expected number of e-mails received during 15 minutes is 1.25. The probability of receiving no e-mails during 15 minutes is approximately 0.7788, or 77.88%
We know that the average number of e-mails received per hour is 5. Therefore, the parameter λ of the Poisson distribution is also 5, since the Poisson distribution's mean and variance are both equal to λ.
The probability of receiving no e-mails during an hour (or during any other fixed time interval of length t) can be calculated using the Poisson distribution as follows:
P(X = 0) = e^(-λ) * λ^0 / 0! = e^(-5) * 5^0 / 0! ≈ 0.0067
Therefore, the probability of receiving no e-mails during an hour is approximately 0.0067, or 0.67%.
The probability of receiving at least three e-mails during an hour can be calculated as follows:
P(X ≥ 3) = 1 - P(X < 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2))
We already know the value of P(X = 0) from part (a), so we just need to calculate P(X = 1) and P(X = 2) using the Poisson distribution:
P(X = 1) = e^(-5) * 5^1 / 1! ≈ 0.0337
P(X = 2) = e^(-5) * 5^2 / 2! ≈ 0.0842
Substituting these values into the equation above, we get:
P(X ≥ 3) = 1 - (0.0067 + 0.0337 + 0.0842) ≈ 0.8754
Therefore, the probability of receiving at least three e-mails during an hour is approximately 0.8754, or 87.54%.
Since the expected number of e-mails received per hour is 5, the expected number of e-mails received during 15 minutes (i.e., a quarter of an hour) is:
E(X) = λ * t = 5 * (1/4) = 1.25
Therefore, the expected number of e-mails received during 15 minutes is 1.25.
The probability of receiving no e-mails during 15 minutes can be calculated using the Poisson distribution with λ = 1.25 and t = 1/4:
P(X = 0) = e^(-λt) * (λt)^0 / 0! = e^(-1.25/4) * (1.25/4)^0 / 0! ≈ 0.7788
Therefore, the probability of receiving no e-mails during 15 minutes is approximately 0.7788, or 77.88%.
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Wp
Peter bought a sandwich for $4.25, a drink for $2.17, and a cookie for $0.79. If he paid with a $10
bill, how much change did Peter get?
Answer:
$2.79
Step-by-step explanation:
We know
Peter bought a sandwich for $4.25, a drink for $2.17, and a cookie for $0.79.
4.25 + 2.17 + 0.79 = $7.21
If he paid with a $10 bill, how much change did Peter get?
10 - 7.21 = $2.79
So, Peter get $2.79 in change.
Write this ratio as a fraction in simplest form without any units.
54 ounces to 3 pounds
Answer: 8/9
Step-by-step explanation: 3 x 16 = 48 ounces
48 / 54 = 8/9
Shown are graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie. (a) Describe and compare how the runners run the race. (b) At what time is the distance between the runners the greatest? (c) At what time do they have the same velocity?
(a) - A runs the race at a constant speed, never speeding up or slowing down. B accelerates throughout the race, starting out slower than A and, by the end, running faster than A.
(b) - Based on the graph, it appears that they are furthest apart after 8 seconds, when they are approximately 30 meters apart.
(c)- The two graphs appear to have the same slope (i.e., velocity) 9 or 10 seconds into the race.
graph attached below,
constant speed
When an object travels the same distance in the same period of time, it is said to be traveling at a constant speed. At constant speed, an object travels a uniform distance in an equal interval of time. The equation of the speed can be given as: S = d t.
Slope
Slope is a measure of the steepness of a line.
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Find a plane through the points P1(12,3),P2(3,2,1) and perpendicular to the plane 4−y+2z=7.
The equation of the plane passing through P1(12,3), P2(3,2,1), and perpendicular to the plane 4−y+2z=7 is -3x + 51y - 8z = -39.
To find a plane that passes through the points P1(12,3) and P2(3,2,1) and is perpendicular to the plane 4−y+2z=7, we can follow these steps:
Find the normal vector of the given plane by taking the coefficients of x, y, and z in the equation 4−y+2z=7. The normal vector is (1, -1, 2), because the coefficients of x, y, and z are 1, -1, and 2 respectively.
Find the direction vector of the line passing through P1 and P2 by subtracting the coordinates of P1 from P2. The direction vector is (-9, -1, -2), because P2 - P1 = (3-12, 2-3, 1-0) = (-9, -1, -2).
Find the cross product of the normal vector of the given plane and the direction vector of the line passing through P1 and P2 to get the normal vector of the plane we are looking for. The cross product is:
(1, -1, 2) x (-9, -1, -2) = (-3, -17, -8)
Plug one of the given points, say P1(12,3), and the normal vector we found in step 3 into the point-normal form of the equation of a plane:
-3(x - 12) - 17(y - 3) - 8(z - 0) = 0
Simplifying, we get:
-3x + 51y - 8z = -39
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A scientist places a cell in a Petri dish. At the end of 1 hour, the cell divides so
that there are 2 cells in the Petri dish. At the end of 2 hours, those cells divide so
there are 4 cells in the Petri dish. The cells continue to divide this way every hour.
Use the expression 2 to the 10th power to find the number of cells in the Petri dish after 10 hours.
Show your work.
PLEASE HELP DUE ASAP !
The four points on the graph of this function f(x) = √(x + 1) + 4 has been plotted on the graph shown in the image attached below.
What is a graph?In Mathematics, a graph can be defined as a type of chart that is typically used for the graphical representation of data points or ordered pairs on both the horizontal and vertical lines of a cartesian coordinate, which are the x-axis and y-axis respectively.
Next, we would use an online graphing calculator to plot the given absolute value function f(x) = √(x + 1) + 4 as shown in the graph attached below.
By observing critically the graph (see attachment), the four ordered pairs (points) that fit on its axes include the following:
Ordered pair = (-1, 4), which is the leftmost.
Ordered pair = (0, 5).
Ordered pair = (3, 6).
Ordered pair = (8, 7).
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What is the height h for the base that is 5/4 units long?
The height of the triangle is 3/5.
What is a right-angled triangle?A triangle is said to be right-angled if one of its inner angles is 90 degrees, or if any one of its angles is a right angle.
Given:
A right-angled triangle.
Let the values of the hypotenuse in parts be m and n.
So,
(1)² = (5/4)m
m = 4/5
And n = 9/16 x 4/5
n = 9/20
So, the height of the triangle is,
h² = 9/20 x 4/5
h² = 9/25
h = 3/5
Therefore, h = 3/5.
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the length of a rectangle is four times its width. if the area of the rectangle is 256 cm2, find its perimeter.
If the length of a rectangle is four times its width. if the area of the rectangle is 256 cm2, the perimeter of the rectangle is 80 cm.
Let's first use the given information to write two equations relating the length and width of the rectangle:
The length L is four times the width W: L = 4W
The area A is 256 cm²: A = LW = 256
Substituting equation 1 into equation 2, we get
4W^2 = 256
Solving for W, we get:
W^2 = 64
W = 8 (since we're looking for a positive value)
Substituting this value back into equation 1, we get:
L = 4W = 32
Therefore, the length and width of the rectangle are 32 cm and 8 cm, respectively.
The perimeter of the rectangle is twice the sum of its length and width:
P = 2(L + W) = 2(32 + 8) = 80 cm
So the perimeter of the rectangle is 80 cm.
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Why do communist countries use authoritarian methods to maintain their economic and political system
Answer:
Communist countries use authoritarian governments because it requires the absolute obedience to authority figures. Communism is an extreme form of socialism because there is no private ownership or political freedom.
Step-by-step explanation:
Attachment is below:
The numbers that are solutions to the inequality k > -5 are given as follows:
A. -4.
F. -4.99.
G. -0.5.
What are the inequality symbols?The four inequality symbols, along with their meaning, are presented as follows:
> x: greater than x.< x: less than x.≥ x: at least x.≤ at most x.The inequality for this problem is given as follows:
k > -5.
Meaning that the solutions are all the numbers that are greater than -5.
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A forest ranger is watching the progress of a forest fire spreading towards her from the top of a 3334-foot mesa. In 6 minutes,
the angle of depression to the leading edge of the fire changes from 10.5° to 12.3°.
1) How many feet does the fire advance during the 6 minutes that the ranger is observing it?
2) At what speed (in MILES PER HOUR) is the fire spreading towards the ranger?
The speed of the fire is 3.214 miles per hour (rounded to three decimal places)
What do you mean by speed?It is defined as the distance traveled by an object per unit of time, typically in meters per second (m/s) or kilometers per hour (km/h).
Speed can be calculated by dividing the distance traveled by the time taken to cover that distance. It is a scalar quantity, meaning it only has magnitude and no direction.
Let's call the horizontal distance the fire advanced during the 6 minutes "d." We can use the tangent function to find d:
tan(12.3°) = (3334 - h) / d
tan(10.5°) = (3334 - h) / d
Solving for d in each equation, we get:
d = (3334 - h) / tan(12.3°)
d = (3334 - h) / tan(10.5°)
Since the time interval is the same for both observations, the fire advanced a distance equal to the difference between the two distances:
d = [(3334 - h) / tan(12.3°)] - [(3334 - h) / tan(10.5°)]
Plugging in the values and simplifying:
d = 1698.5 feet (rounded to the nearest tenth)
To find the speed at which the fire is spreading towards the ranger, we need to convert the distance from feet to miles, and the time from minutes to hours:
d = 1698.5 feet / 5280 feet per mile = 0.3214 miles
t = 6 minutes / 60 minutes per hour = 0.1 hours
The speed of the fire is then:
s = d / t = 0.3214 miles / 0.1 hours = 3.214 miles per hour (rounded to three decimal places)
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Give the correct notation for the quantity described and give its value.
Mean number of cell phone calls made or received per day by cell phone users. In a survey of
1917 cell phone users, the mean was 13.10 phone calls a day.¹
The notation for this sample is denoted by [tex]\overline{x}[/tex].
What is the difference between the sample and the population?The entire group about whom you want to make conclusions is referred to as a population.
The particular group from which you will gather data is known as a sample. The sample size is always smaller than the population as a whole.
Given, In a survey of 1917 cell phone users, the mean was 13.10 phone calls a day.
This a sample so the notation for the mean is [tex]\overline{x}[/tex].
If it is for population then the notation of mean would have been [tex]\mu[/tex].
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A normal distribution (X) has a mean of 100 and a standard deviation of 10.
What is the probability that X is between 90 and 110?
The probability that X is between 90 and 110 = 0.2 if a normally distributed random variable x has a mean of 100 and P(x < 90) = 0.40.
What is a normal distribution?The majority of the observations are centered around the middle peak of the normal distribution, which is a continuous probability distribution that is symmetrical around its mean.
The probabilities for values that are farther from the mean taper off equally in both directions. Extreme values in the distribution's two tails are likewise rare. Not all symmetrical distributions are normal, even though the normal distribution is symmetrical.
A normally distributed random variable x has a mean of 100
P(x < 90) = 0.4
The probability that X is between 90 and 110
Mean is 100
Hence P (< 100) = 0.5 as Mean is centered
Hence
P (90 < X < 100) = 0.5 - 0.4 = 0.1
The difference between 100 and 90 is 10
The difference between 100 and 110 is 10
and 100 is Mean
Hence P (100 < X < 110) = P (90 < X < 100) = 0.1
Adding Both
P ( 90 < X < 110) = 0.1 + 0.1 = 0.2
The probability that X is between 90 and 110 = 0.2
Using z score table.
Z = ( Value - Mean) /SD
Z score for 0.4 = -0.254
-0.253 = (90 - 100)/SD
SD ≈ 39.53
Z = ( 110 - 100)/39.53
Z = 0.253
For z score < 0.253, the probability is 0.6
Hence Probability in Between is 0.6 - 0.4 = 0.2
The probability that X is between 90 and 110 = 0.2
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