a. the solution for f(x) is f(x) = e^(-x^2+6x+5). b. lim as x approaches infinity of g'(x) is -∞. c. The only solution in the interval 0 <= x < 3 is g(x) = 1 - sqrt(3)/3.
a. Using the given differential equation, we can solve for f(x) by separating variables:
dy/y = 2(3-x)dx
Integrating both sides, we get:
ln|y| = -x^2 + 6x + C
Using the initial condition f(4) = 1, we can solve for C:
ln|1| = -4^2 + 6(4) + C
C = 5 - ln|1| = 5
Therefore, the solution for f(x) is:
f(x) = e^(-x^2+6x+5)
b. Using the given differential equation, we can see that g'(x) = 2g(x)(3-g(x)). Thus, lim as x approaches infinity of g(x) is either 0 or 3. Since g(4) = 1 and g(x) is an increasing function, it follows that lim as x approaches infinity of g(x) is 3.
To find lim as x approaches infinity of g'(x), we take the derivative of g'(x) to get:
g''(x) = 6g'(x) - 4g'(x)^2
Thus, lim as x approaches infinity of g''(x) is 0, and we can use L'Hopital's rule to find lim as x approaches infinity of g'(x):
lim as x approaches infinity of g'(x) = lim as x approaches infinity of (2g(x)(3-g(x)))
= lim as x approaches infinity of (-2g(x)^2 + 6g(x))
= -∞
Therefore, lim as x approaches infinity of g'(x) is -∞.
c. The graph of g has a point of inflection when g''(x) = 0 and changes sign. From part b, we know that lim as x approaches infinity of g(x) is 3, so we only need to consider the behavior of g(x) for 0 <= x < 3. Solving g''(x) = 0, we get:
g''(x) = 6g'(x) - 4g'(x)^2 = 0
g'(x)(3-2g'(x)) = 0
So either g'(x) = 0 or g'(x) = 3/2. The first case corresponds to a local maximum or minimum, while the second case corresponds to a point of inflection. Solving for g(x) in the second case, we get:
2x - ln|3-2g(x)| - ln|g(x)| = C
Using the initial condition g(4) = 1, we can solve for C:
2(4) - ln|3-2(1)| - ln|1| = C
C = 7 - ln|1| = 7
Therefore, the equation for the graph of g(x) in the second case is:
2x - ln|3-2g(x)| - ln|g(x)| = 7
To find the value of y at the point of inflection, we substitute g'(x) = 3/2 into the equation for g''(x) to get:
g''(x) = -9g(x)^2 + 18g(x) - 6 = 0
Solving for g(x), we get two solutions: g(x) = 1 +/- sqrt(3)/3. The only solution in the interval 0 <= x < 3 is g(x) = 1 - sqrt(3)/3.
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Help please!!!!!!!!!!!!!!!!!!!!!!!
find the coordinates of the point p at an angle of −90∘ on a circle of radius 4.3. round your answers to the three decimal places. enter a point as (a,b) including parentheses.
The point p at an angle of -90 degrees on a circle of radius 4.3 is the point where a vertical line intersects the circle.
This is because an angle of -90 degrees is equivalent to a downward vertical direction in the Cartesian coordinate system. To find the coordinates of this point, we can use the equation of a circle in standard form:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) is the center of the circle and r is its radius. Since the circle in this question has a radius of 4.3, we can substitute r = 4.3 into the equation. To find the center of the circle, we would need additional information such as the equation of the circle or another point on the circle.
However, we can still find the coordinates of the point p by realizing that the center of the circle is the origin (0,0) and substituting x = 0 into the equation of the circle. This gives us:
(0 - 0)^2 + (y - 0)^2 = 4.3^2
Simplifying the equation, we get:
y^2 = 4.3^2
Taking the square root of both sides, we get:
y = ± 4.3
Since we are looking for the point p at an angle of -90 degrees, we take the negative square root to get:
y = -4.3
Therefore, the coordinates of the point p are (0, -4.3)
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Prove the following
[tex] {sin}^{2} ( \frac{\pi}{4} - \alpha ) = \frac{1}{2} (1 - sin2 \alpha )[/tex]
Answer:
trig identity proof
Using the trigonometric identity for the sine of the difference of two angles, we have:
sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
Substituting a = π/4 and b = α, we get:
sin(π/4 - α) = sin(π/4)cos(α) - cos(π/4)sin(α)
sin(π/4 - α) = (1/√2)(cos(α) - sin(α))
Squaring both sides, we get:
sin^2(π/4 - α) = 1/2(cos^2(α) - 2cos(α)sin(α) + sin^2(α))
sin^2(π/4 - α) = 1/2(1 - sin(2α))
This proves the first equation.
For the second equation, we use the double angle formula for the sine:
sin(2x) = 2sin(x)cos(x)
Substituting x = 2π - α, we get:
sin(4π - 2α) = 2sin(2π - α)cos(2π - α)
sin(4π - 2α) = 2(-sin(α))(-cos(α))
sin(4π - 2α) = 2sin(α)cos(α)
Dividing both sides by 2sin^2(α), we get:
sin(4π - 2α)/(2sin^2(α)) = cos(α)/sin(α)
csc(4π - 2α) = cot(α)
Using the identity csc(x) = 1/sin(x) and simplifying, we get:
sin(4π - 2α) = (1 - sin^2(α))/sin(α)
sin(4π - 2α) = cos^2(α)/sin(α)
sin(4π - 2α) = (1 - sin^2(α))(1/sin(α))
sin(4π - 2α) = 1/sin(α) - sin(α)
Substituting the value of sin^2(π/4 - α) we found earlier, we get:
sin(4π - 2α) = 1/sin(α) - (1/2)(1 - sin(2α))
sin(4π - 2α) = (1/2)(1 + sin(2α))/sin(α)
This proves the second equation.
The table of values below represent an exponential function. Find the constant ratio of successive y-values.
picture bellow, help asap!!!!!!!!!!
The constant ratio of successive y-values. is 1.5
Finding the constant ratio of successive y-values.From the question, we have the following parameters that can be used in our computation:
The table of values of an exponential function
From the table, we have
x y
0 11.25
1 16.875
Divide the y values
so, we have the following representation
Ratio = 16.875/11.25
Evaluate
Ratio = 1.5
Hence, the constant ratio of successive y-values. is 1.5
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Answer:
Step-by-step explanation:
To find the constant ratio of successive y-values of an exponential function, you need to divide one term by the previous term. That ratio should match for all y values1.
In this case, we can find the constant ratio by dividing each y-value by the previous one. For example, 7.5/5 = 1.5 and 11.25/7.5 = 1.5 and so on. Therefore, the constant ratio of successive y-values is 1.5
option c) 1.5Please help
Find AD length, AE length, BD length and EC length
The solution is: the required length is:
AE = 9 units
Explanation:
We know that the line joining two midpoints in a triangle is parallel to the third side and equals half its length
In the diagram, we are given that:
segment BD // segment AE and that segment BD is a mid-segment of the ΔACE
According the above theorem, we can conclude that:
BD = 0.5 × AE ......................> I
1- getting the length of BD:
Length of segment BD can be calculated using the distance formula:
Formula: distance= √(x_2-x_1)²+(y_2-y_1)²
We are given that:
B is at (3.5,1.5) which means that x₁ = 3.5 and y₁=1.5
D is at (-1,1.5) which means that x₂=-1 and y₂=1.5
Substitute in the formula:
BD = 4.5 units
2- getting the length of AE:
using equation I:
BD = 0.5 × AE
4.5 = 0.5 × AE
AE = 2 × 4.5
AE = 9 units
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complete question:
Find the length of AE if BD AE and BD is a midsegment of ACE
The required length is: AE = 9 units
We know that the line joining two midpoints in a triangle is parallel to the third side and equals half its length
In the diagram, we are given that:
segment BD // segment AE and that segment BD is a mid-segment of the ΔACE
According the above theorem, we can conclude that:
BD = 0.5 × AE ......................> I
1- getting the length of BD:
Length of segment BD can be calculated using the distance formula:
Formula: distance= √(x_2-x_1)²+(y_2-y_1)²
We are given that:
B is at (3.5,1.5) which means that x₁ = 3.5 and y₁=1.5
D is at (-1,1.5) which means that x₂=-1 and y₂=1.5
Substitute in the formula:
BD = 4.5 units
2- getting the length of AE:
using equation I:
BD = 0.5 × AE
4.5 = 0.5 × AE
AE = 2 × 4.5
AE = 9 units
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The question is incomplete complete question is given below
Find the length of AE if BD AE and BD is a midsegment of ACE
what is scientific learning
Scientific learning is the process of acquiring knowledge and understanding of scientific concepts, theories, and principles through observation, experimentation, and analysis of data.
What is scientific learning?Scientific learning involves using the scientific method, which is a systematic approach to investigating and understanding natural phenomena.
Scientific learning includes not only learning about the natural world but also learning how to think critically, analyze data, and make informed decisions based on evidence. It is a dynamic process that involves constantly questioning, investigating, and refining our understanding of the world around us
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The answer to the question
Answer:
Diameter
Step-by-step explanation:
You have to draw the diameter and perpendicularly bisect it. Then, where the bisector touches the circumference, connect them (there should be 4 points of contact).
Hope this helps!
PLEASE ANSWER ASAP
what is 1/4x2−8x+64 when factored completely?
we need to express f(x) = 1 /2 + x in the form 1 / 1 − r and then use the following equation.
f(x) = 1/2 + x in the form 1 / (1 - r) and use the given equation. Here's a step-by-step explanation:
Step 1: Write down the given function:
f(x) = 1/2 + x
Step 2: Rewrite f(x) as a fraction:
f(x) = (1 + 2x) / 2
Step 3: Express f(x) in the form 1 / (1 - r):
To do this, we need to find a value of 'r' such that (1 + 2x) / 2 can be written as 1 / (1 - r).
Since we want to express the function in the form of 1 / (1 - r), we can set the numerators equal:
1 = 1 + 2x
Now, solve for 'x':
-2x = 0
x = 0
So, the value of 'r' that satisfies this condition is:
r = 1 - (1 / (1 + 2x)) = 1 - (1 / 1) = 0
Now, f(x) can be expressed as:
f(x) = 1 / (1 - r) = 1 / (1 - 0) = 1 / 1
Finally, we can use this expression in any given equation, by replacing f(x) with 1 / (1 - 0).
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find the slope of the tangent line to the polar curve r=sin(5) at theta = pi/10
The slope of the tangent line to the polar curve r=sin(5) at theta = pi/10 is -25cos(pi/10).
To find the slope of the tangent line, we need to differentiate the polar curve with respect to theta and then evaluate it at the given value of theta. So, we have r=sin(5) and we can write it in terms of x and y using the conversion formulae x=rcos(theta) and y=rsin(theta). Substituting r=sin(5), we get x=sin(5)*cos(theta) and y=sin(5)*sin(theta). Differentiating both x and y with respect to theta, we get dx/dtheta=-sin(5)*sin(theta) and dy/dtheta=sin(5)*cos(theta).
The slope of the tangent line is given by dy/dx, which is equal to dy/dtheta divided by dx/dtheta. Evaluating this expression at theta = pi/10, we get -25cos(pi/10).
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Write down the iterated integral which expresses the surface area of z=(y^3)[(cos^4)(x)] over the triangle with vertices (-1,1), (1,1), (0,2): Integral from a to b integral from f(y) to g(y) of sqrt(h(x,y) dxdy
The iterated integral that expresses the surface area of the given function over the given triangle is:
∫ from -1 to 0 ∫ from 2x+2 to x+2 √(1 + (9x^4sin^4x)) dy dx + ∫ from 0 to 1 ∫ from 2 to 2x+2 √(1 + (9x^4sin^4x)) dy dx This represents the double integral over the region of the triangle, where the function being integrated is the square root of the sum of the squares of the partial derivatives of the given function with respect to x and y. The limits of integration are determined by the bounds of the triangle in the x and y directions, which are broken up into two regions based on the dividing line x=0. The double integral is evaluated using standard techniques for integrating over regions in two dimensions, such as Fubini's theorem or change of variables. The resulting value represents the surface area of the given function over the given triangle.
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what is the area of a sector of a circle with a radius of 8 inches and formed by a cetnral angle that measures 60
The area of the sector is 16π square inches.
To find the area of a sector of a circle, we need to use the formula:
Area of sector = (central angle/360) x [tex]\pi r^2[/tex]
where r is the radius of the circle.
In this case, the radius is given as 8 inches.
We are also given that the central angle measures from 60 to 150 degrees. To calculate the area of the sector, we need to find the size of the central angle first.
To do this, we subtract the smaller angle from the larger angle:
150 - 60 = 90 degrees
So, the central angle is 90 degrees.
Now, we can substitute the values into the formula:
Area of sector = (90/360) x [tex]\pi 8^2[/tex]
Area of sector = (1/4) x π(64)
Area of sector = 16π square inches
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The volume of a box with a given length varies jointly with its width and height. The original box has a volume of 240 cubic inches and has a width of 8 inches and a height of 2 inches. If the box is modified (keeping the same length) to a width of 6 inches and a height of 3 inches, what is the volume of this new box?
The volume of the second box is 270 in³.
Given that the volume of a box varies jointly with its width and height.
The original box has a volume of 240 cubic inches and has a width of 8 inches and a height of 2 inches.
We can say that the length here works as proportionality constant,
So,
240 = l × 2 × 8
l = 240 / 16
l = 15
Now, when the width of 6 inches and a height of 3 inches, the volume =
15 × 3 × 6 = 270 in³
Hence the volume of the second box is 270 in³.
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in the linear trend equation, ft k = at bt*k, the term that signifies the trend is:
The term that signifies the trend in the linear trend equation, ft k = at bt*k, is the coefficient bt*k. This coefficient represents the slope of the trend line, which indicates the direction and strength of the trend. A positive value of bt*k implies an increasing trend, while a negative value implies a decreasing trend. The magnitude of the coefficient indicates the rate of change in the trend over time. For example, a larger absolute value of bt*k indicates a faster rate of change than a smaller absolute value. Therefore, the bt*k term is crucial in determining the trend in the linear trend equation.
The linear trend equation is a mathematical representation of a trend in data over time. It can be used to identify and quantify the direction and magnitude of a trend. The equation has two components: a constant term (a) and a trend term (bt*k). The constant term represents the intercept of the trend line, while the trend term represents the slope of the trend line. The bt*k term is the coefficient of the trend term and is the primary determinant of the trend.
The bt*k term in the linear trend equation is the coefficient that signifies the trend. It represents the slope of the trend line and indicates the direction and strength of the trend. A positive value implies an increasing trend, while a negative value implies a decreasing trend. The magnitude of the coefficient indicates the rate of change in the trend over time. Therefore, understanding the bt*k term is essential in analyzing trends in data.
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When multiple tests are done in analysis of variance, the family error rate is ______
When multiple tests are done in analysis of variance (ANOVA), the family error rate is the probability of making at least one type I error (rejecting a true null hypothesis) in the family of tests.
To control the family error rate, several methods are available such as the Bonferroni correction, the Holm-Bonferroni method, the Benjamini-Hochberg procedure, among others. These methods adjust the significance level or p-value threshold for each individual test to ensure that the family-wise error rate is below a certain level, such as 0.05.
By controlling the family error rate, we reduce the chances of mistakenly concluding that there is a significant effect in any of the tests, which is important in avoiding false positives and ensuring the validity of the overall analysis.
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Help Please Thank you so much
Step-by-step explanation:The steps are hard to explain. But i did it Hope it helps!
determine whether the given experiment has a sample space with equally likely outcomes. a loaded die is rolled, and the number appearing uppermost on the die is recorded. Yes or No ?
The concept of "equally likely outcomes" refers to the idea that every possible outcome in a given sample space has an equal chance of occurring. In other words, if we were to conduct the experiment multiple times, each possible outcome would have an equal probability of being observed.
In the case of rolling a fair die, the sample space consists of the numbers 1 through 6, and each of these outcomes has an equal probability of occurring. This is because the die is assumed to be fair, meaning that each side has an equal chance of landing face-up.
However, in the case of a loaded die, the sample space does not have equally likely outcomes. This is because the probabilities of each outcome are not equal. A loaded die is one that has been manipulated in some way so that certain outcomes are more likely than others. For example, if the loaded die has been weighted to favor the number 6, then the probability of rolling a 6 would be higher than the probability of rolling any of the other numbers.
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A student swings a 30 centimeters long ruler back and forth which is pivoted at one end on the desk. The ruler tums 135° in a swing. Assuming the arc is circular, what is the distance the tip of the ruler travels
swing(arc length)? round to the nearest hundredth. Use 3.14 for pi
Yvonne leaves school and drives straight to work. If her speed averages 30 km/h, she'll be exactly 18 minutes late for work, whereas if her travel speed averages 45 km/h, she will arrive exactly 8 minutes early to work. What is the distance in km between Yvonne's school and work?
Yvonne leaves school and drives straight to work. So, according to the question the distance between Yvonne's school and work is 15 kilometers.
Let's assume that the distance between Yvonne's school and work is "d" kilometers.
When Yvonne drives at an average speed of 30 km/h, she will cover the distance "d" in (d/30) hours. However, she will be 18 minutes late for work, which is the same as being 0.3 hours late. So, the total time taken by Yvonne to reach work is (d/30) + 0.3 hours.
On the other hand, when Yvonne drives at an average speed of 45 km/h, she will cover the same distance "d" in (d/45) hours. However, this time she will arrive 8 minutes early for work, which is the same as being 0.1333 hours early. So, the total time taken by Yvonne to reach work is (d/45) - 0.1333 hours.
We know that both these times are equal, since Yvonne is covering the same distance "d". So, we can equate them as follows:
(d/30) + 0.3 = (d/45) - 0.1333
Solving this equation gives us the value of "d" as 15 kilometers. Therefore, the distance between Yvonne's school and work is 15 kilometers.
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find the volume of a cone if the perpendicular height is 9cm and radius 4cm
[tex]\textit{volume of a cone}\\\\ V=\cfrac{\pi r^2 h}{3}~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ h=9\\ r=4 \end{cases}\implies V=\cfrac{\pi (4)^2(9)}{3}\implies V\approx 150.80~cm^3[/tex]
let u be an orthogonal matrix, and construct v by interchanging some of the columns of u . explain why v is an orthogonal matrix.
If u is an orthogonal matrix and v is constructed by interchanging some of the columns of u, then v is also an orthogonal matrix. This is because the columns of an orthogonal matrix are orthonormal.
An orthogonal matrix is a square matrix whose columns are orthonormal. This means that each column has a length of 1 and is orthogonal to all the other columns. Formally, this can be written as:
u^T u = u u^T = I
where u^T is the transpose of u and I is the identity matrix.
Now suppose we construct a new matrix v by interchanging some of the columns of u. Let's say we interchange columns j and k, where j and k are distinct column indices of u. Then the matrix v is given by:
v = [u_1, u_2, ..., u_{j-1}, u_k, u_{j+1}, ..., u_{k-1}, u_j, u_{k+1}, ..., u_n]
where u_i is the ith column of u.
To show that v is orthogonal, we need to show that its columns are orthonormal. Let's consider the jth and kth columns of v. By construction, these columns are u_k and u_j, respectively, and we know from the properties of u that:
u_j^T u_k = 0 and u_j^T u_j = u_k^T u_k = 1
Therefore, the jth and kth columns of v are orthogonal and have a length of 1, which means they are orthonormal. Moreover, all the other columns of v are also orthonormal because they are simply copies of the corresponding columns of u, which are already orthonormal.
Finally, we can show that v is indeed an orthogonal matrix by verifying that v^T v = v v^T = I, using the definition of v and the properties of u. This completes the proof that v is an orthogonal matrix.
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what survey concept might explain why statistics show that despite the fact that about 1/2 of marriages eventually end in divorce, the majority of spouses report that their marriage is very happy
The concept that can explain this is called "Survivorship Bias". Survivorship bias occurs when we focus on those who "survived" or made it through a particular event or process, and overlook those who did not. In the case of marriage, those who have divorced are not included in the statistics on happy marriages, so the overall rate of happy marriages appears to be higher than it actually is.
In other words, the statistics on divorce rates only take into account marriages that have ended in divorce, but not the marriages that have remained intact. Therefore, the majority of spouses who report being very happy in their marriage are likely the ones who have successfully stayed married and are still together. The statistics only reflect those who have not been able to maintain a happy marriage.
It's also important to note that happiness is subjective and can vary from person to person. Some individuals may find happiness in their marriage despite facing challenges, while others may not. Therefore, even if a marriage does end in divorce, it does not necessarily mean that it was unhappy throughout its duration.
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Question 2(Multiple Choice Worth 2 points)
(Two-Column Tables MC)
The pharmacist has a 3.6 L bottle of cough syrup. If she fills a bottle that is 1,500 ml, how many ml of cough syrup does the pharmacist have left? (1 L = 1,000 ml)
21 ml
150 ml
1,360 ml
2,100 ml
The pharmacist is left with 2100 ml(milliliter) of cough syrup.
According to the question,
Pharmacists have 3.6 L(liter) of cough syrup.
1 L = 1000 ml (Given)
Therefore, 3.6 L = 3.6 x 1000
= 3600 ml
It’s given in the question that the pharmacist fills a 1500 ml bottle with cough syrup.
To find the quantity of cough syrup left with the pharmacist, we will subtract the quantity of bottle from the total quantity of cough syrup.
Cough syrup left with her after filling the bottle = 3600 – 1500
= 2100 ml
Hence, she is left with 2100 ml of cough syrup after filling up a bottle of 1500 ml quantity.
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Find the volume of a cube whose side measures 9cm. [ V = e³ ]
The volume of a cube is given by the formula V = e³, where e represents the length of the side of the cube. In this case, the length of the side is 9cm. Therefore, the volume of the cube is V = 9³ = 729 cubic centimeters.
To find the volume of the cube, we need to raise the length of one side to the power of 3 since the volume of a cube is given by V = e³. In this case, the side of the cube measures 9cm, so we have e = 9.
Substituting this value into the formula, we get V = 9³ = 729 cubic centimeters. Therefore, the volume of the cube is 729 cubic centimeters. This means that the cube could hold 729 cubic
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in An C?
1, 6, 12
D. 9, 15
avs the results of a survey asking
Key Club watch on television.
n Television
11
Football
16.
The accompanying Venn diagram shows the results of a survey
asking 100 people if they get news by reading newspapers or by
watching television.
Sources of News
Newspapers
15
A. 700
40
B.
Television
20
What is the probability that a person selected at random from
this survey does not claim newspapers as a source of getting the
news?
25
C. f
D.
17. A bag contains 3 red marbles and 4 blue marbles. If one marble
is drawn at random, what is the probability that it is red?
D.
التي
The probability that a person selected at random from the survey does not claim newspapers as a source of getting the news is 85%.
To determine the probability that a person selected at random from the survey does not claim newspapers as a source of getting the news, we need to consider the information provided in the Venn diagram.
According to the diagram, the number of people who get news from newspapers is 15, and the total number of people surveyed is 100. Therefore, the number of people who do not claim newspapers as a source of getting the news would be:
Total number of people surveyed - Number of people who get news from newspapers = 100 - 15 = 85
The probability can be calculated by dividing the number of people who do not claim newspapers by the total number of people surveyed:
Probability = Number of people who do not claim newspapers / Total number of people surveyed = 85 / 100 = 0.85 or 85%
So, the probability that a person selected at random from the survey does not claim newspapers as a source of getting the news is 85%.
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let be a subset of such that no pair of distinct elements in has a sum divisible by . what is the maximum number of elements in ?
The maximum number of elements in the subset is 5. No two distinct elements in the subset have a sum divisible by 5, as the sum of any two elements with distinct remainders is either a multiple of 5 or has a remainder of 1, 2, 3, or 4 when divided by 5.
To maximize the number of elements in the subset, we need to choose elements that have the most possible distinct remainders when divided by 5. Since no pair of distinct elements in the subset has a sum divisible by 5, the remainders of any two distinct elements in the subset must add up to a non-multiple of 5.
The remainders when dividing the first 5 positive integers by 5 are 1, 2, 3, 4, and 0, respectively. Therefore, the maximum number of elements we can choose from the set such that no pair has a sum divisible by 5 is 5.
We can achieve this maximum by selecting one element from each of the following subsets of the original set:
Elements with a remainder of 1 when divided by 5
Elements with a remainder of 2 when divided by 5
Elements with a remainder of 3 when divided by 5
Elements with a remainder of 4 when divided by 5
The element that is a multiple of 5.
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Solve the equation x^2-14x-11=-30 to the nearest tenth
To solve the equation x^2-14x-11=-30, we can first move all the terms to one side of the equation: x^2 - 14x - 11 + 30 = 0Simplifying the left side:
x^2 - 14x + 19 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 1, b = -14, and c = 19. Plugging these o solve the equation x^2-14x-11=-30, we can first move all the terms to one side of the equation:
x^2 - 14x - 11 + 30 = 0
Simplifying the left side:
x^2 - 14x + 19 = 0
To solve for x, we can use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 1, b = -14, and c = 19. Plugging these values into the formula, we get:
x = (14 ± sqrt(14^2 - 4(1)(19))) / 2(1)
Simplifying the square root:
x = (14 ± sqrt(108)) / 2
x = (14 ± 10.39) / 2
x ≈ 12.2 or x ≈ 1.8
Therefore, the solutions to the equation x^2-14x-11=-30 to the nearest tenth are x ≈ 12.2 and x ≈ 1.8.
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integral of e^xdx on x=y^3 from (-1,-1) to (1,1)
The integral of e^x dx on x=y^3 from (-1,-1) to (1,1) is approximately 2.17.
To solve this problem, we need to use substitution. Let y^3 = x, so that dx = 3y^2 dy. Substituting these expressions into the integral, we get:
∫e^x dx = ∫e^(y^3) * 3y^2 dy
We can now integrate this expression using the u-substitution method. Let u = y^3, so that du/dy = 3y^2. Substituting these expressions, we get:
∫e^(y^3) * 3y^2 dy = ∫e^u du
Integrating e^u with respect to u, we get e^u + C, where C is a constant of integration. Substituting back for u and simplifying, we get:
e^(y^3) + C
To find the value of the constant, we can use the limits of integration. Substituting (1,1) and (-1,-1) for (x,y), we get:
e^(1^3) - e^(-1^3) = e - 1/e
So the answer is approximately 2.17.
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In the year 2000, the population of a small city was 45,000. The population grows at a rate of r(t)-1200 people per year t years after 2000. Between 2021 and 2039, is estimated the population will grow by __________ people. (Round to nearest integer.)
To find the estimated population growth between 2021 and 2039, we first need to determine how many years have passed since 2000. The population is estimated to grow by 21,000 people between 2021 and 2039.
Since we are looking at the time period between 2021 and 2039, we know that 21 years have passed since 2000. Therefore, we can use the formula for population growth:
P(t) = P(0) + r(t)
Where P(t) is the population at time t, P(0) is the initial population, and r(t) is the rate of growth per year. We can plug in the values we know:
P(t) = 45,000 + 1200t (since the rate of growth is 1200 people per year)
To find the population in 2021, we need to plug in t=21:
P(21) = 45,000 + 1200(21) = 72,600
To find the population in 2039, we need to plug in t=39:
P(39) = 45,000 + 1200(39) = 93,600
Therefore, the estimated population growth between 2021 and 2039 is:
93,600 - 72,600 = 21,000
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Select the correct answer.
If A = 4
OA.
OB.
O
2 7 5
O
C.
4
-5 7 and AB
D.
B= 3
B =
B =
23
B =
2
3
-5
2
2
24
-46
what is the value of matrix B?
The value of the matrix B is [tex]B =\left[\begin{array}{c}1&3&-5\end{array}\right][/tex]
Calculating the value of matrix B?From the question, we have the following parameters that can be used in our computation:
[tex]A = \left[\begin{array}{ccc}2&4&-2\\4&-5&7\\2&7&5\end{array}\right][/tex]
Also, we have
[tex]AB =\left[\begin{array}{c}24&-46&-2\end{array}\right][/tex]
Represent the matrix B with
[tex]B =\left[\begin{array}{c}a&b&c\end{array}\right][/tex]
So, we have the following product expression
[tex]A = \left[\begin{array}{ccc}2&4&-2\\4&-5&7\\2&7&5\end{array}\right][/tex] * [tex]B =\left[\begin{array}{c}a&b&c\end{array}\right][/tex] = [tex]AB =\left[\begin{array}{c}24&-46&-2\end{array}\right][/tex]
Evaluate the products
[tex]\left[\begin{array}{c}2a+4b-2c\\4a-5b+7c\\2a+7b+5c\end{array}\right] = \left[\begin{array}{c}24&-46&-2\end{array}\right][/tex]
By comparison, we have
2a + 4b - 2c = 24
4a - 5b + 7c = -46
2a + 7b + 5c = -2
When evaluated, we have
a = 1, b = 3 and c = -5
Recall that
[tex]B =\left[\begin{array}{c}a&b&c\end{array}\right][/tex]
So, we have
[tex]B =\left[\begin{array}{c}1&3&-5\end{array}\right][/tex]
Hence, the value of matrix B is [tex]B =\left[\begin{array}{c}1&3&-5\end{array}\right][/tex]
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