Answer:
[tex] I = \dfrac{1}{3}sin(3x) - 3cos(x) + C[/tex]
Step-by-step explanation:
We need to integrate the given expression. Let I be the answer .
[tex]\implies\displaystyle I = \int (cos(3x) + 3sin(x) )dx \\\\\implies\displaystyle I = \int cos(3x) + \int sin(x)\ dx [/tex]
Let u = 3x , then du = 3dx . Henceforth 1/3 du = dx .Rewrite using du and u .[tex]\implies\displaystyle I = \int cos\ u \dfrac{1}{3}du + \int 3sin \ x \ dx \\\\\implies\displaystyle I = \int \dfrac{cos\ u}{3} du + \int 3sin\ x \ dx \\\\\implies\displaystyle I = \dfrac{1}{3}\int \dfrac{cos(u)}{3} + \int 3sin(x) dx \\\\\implies\displaystyle I = \dfrac{1}{3} sin(u) + C +\int 3sin(x) dx \\\\\implies\displaystyle I = \dfrac{1}{3}sin(u) + C + 3\int sin(x) \ dx \\\\\implies\displaystyle I = \dfrac{1}{3}sin(u) + C + 3(-cos(x)+C) \\\\\implies \boxed{\boxed{\displaystyle I = \dfrac{1}{3}sin(3x) - 3cos(x) + C }}[/tex]
Cooper has 35 video games in his collection, and Maria has 15 in hers. Cooper decides to add 10 video games to his collection each month. Maria decides to add 8 video games to her collection each month. Part A Drag the values to the positions in the table to show how many video games each of them will have at the ends of months 1, 2, 3, and 4. 314533436529 Month Cooper Maria Start 35 15 1 23 2 55 3 39 Part B Which of the following is a true statement about the relationship between the number of video games collected by Cooper and Maria after each month? A. Cooper will always have 20 more video games than Maria. B. Maria will always have 20 more video games than Cooper. C. Cooper will always have 28 more video games than Maria. D. There is no constant relationship between the number of video games collected by Cooper and Maria.
From the values that we have here, the statement that can be said to be true here would be that: There is no constant relationship between the number of video games collected by Cooper and Maria. Last option
What is a constant relationship?A relationship with a fixed ratio between two quantities is referred to as proportionate. The graph will therefore be linear, or straight.
When two variables are directly or indirectly proportional to one another, their relationship can be expressed using the formulas y = kx or y = k/x, where k specifies the degree of correspondence between the two variables. The proportionality constant, k, is often used.
The values between these two does not show any constant relationship hence the last option
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Answer: part a
cooper- 35 45 55 65
maria 15 23 31 39
part B
there is no constant relationship between the number of video games collected by coopeer and maria
Step-by-step explanation:
Ravi sells real estate. Based on previous data, he knows that 5% of home tours result in a sale. Assume that the results of these tours are independent from each other. Which of the following choices are binomial random variables? Choose all answers that apply: A. Take a random sample of 30 tours and let L = the number of tours that result in a sale. B. Take a random sample of 3 tours and let K = the number of tours that result in a sale. C. Take a random sample of 3 tours and let M = the amount of sales (in dollars) generated by the tours
The following choices are binomial random variables: A. Take a random sample of 30 tours and let L = the number of tours that result in a sale. B. Take a random sample of 3 tours and let K = the number of tours that result in a sale.
Choice A is a binomial random variable because it represents the number of tours that result in a sale in a random sample of 30 tours. The trials are independent, as the results of one tour do not affect the results of other tours, and the probability of success (a sale) is constant at 5%.
Choice B is also a binomial random variable because it represents the number of tours that result in a sale in a random sample of 3 tours. The trials are independent and the probability of success is constant at 5%.
Choice C is not a binomial random variable because it represents the number of sales generated by a random sample of 3 tours, which is a continuous variable. Binomial random variables are always discrete, as they represent the number of successes in a sequence of trials.
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NO LINKS!!
The inverse function of the exponential function f(x) = a^x is the (a. transcendental, b. logarithmic, c. rational, d. polynomial, e. algebraic) function with base a.
Answer:
The inverse function of the exponential function f(x) = a^x is the logarithmic function with base a.
The inverse function of a function f is a function that "undoes" the original function, meaning that it reverses the transformation applied by the original function. In the case of the exponential function, the inverse function is the logarithmic function, which "undoes" the transformation applied by the exponential function.
For example, suppose we have the exponential function f(x) = 2^x. The inverse function of this function is the logarithmic function with base 2, which is written as y = log_2 x. If we apply the inverse function to 2^x, we get:
y = log_2 (2^x)
Solving for x gives:
x = 2^y
This means that the inverse function of the exponential function f(x) = a^x is the logarithmic function with base a, which is written as y = log_a x.
The other options (a. transcendental, c. rational, d. polynomial, e. algebraic) are not correct, since they do not describe the inverse function of the exponential function.
Answer:
b. logarithmic
Step-by-step explanation:
Given exponential function:
[tex]f(x)=a^x[/tex]
The inverse of the given function is the logarithmic function with base a.
To find the inverse of a function, replace f(x) with y:
[tex]\implies y=a^x[/tex]
Swap the x and y:
[tex]\implies x=a^y[/tex]
Take logs with base a of both sides of the equation:
[tex]\implies \log_ax=\log_aa^y[/tex]
Apply the log power law: logₐ xⁿ = n logₐ x
[tex]\implies \log_ax=y\log_aa[/tex]
Apply the log law: logₐ a = 1
[tex]\implies \log_ax=y[/tex]
[tex]\implies y=\log_ax[/tex]
Replace y with f⁻¹(x):
[tex]\implies f^{-1}(x)=\log_ax[/tex]
Thus proving that the inverse of the exponential function f(x) = aˣ is the logarithmic function with base a.
9. If point Q is reflected across x = 1, what are the coordinates of its reflection image?
The coordinates of reflection image are (-1, -2).
What is point reflection?
A particular kind of Euclidean space isometry is known as a point reflection in geometry. A thing is said to have point symmetry if it is invariant under a point reflection; if it is invariant under a point reflection through its center, it is said to have central symmetry or to be centrally symmetric.
Given:
point Q is reflected across x = 1.
From graph the point Q is (3, -2).
We have to find the coordinates of its reflection image.
x = 1 is a vertical line passing through all points with an x- coordinate of 1.
(3, - 2) is 2 units to the right of x = 1 ( 3 - 1 ), thus
The reflection is 2 units to the left of x = 1 ( 1 - 2 = - 1 )
Q(3, - 2 ) → Q'(- 1, - 2 )
This is because the x-coordinate goes 2 units left to the line x = 1 and the y-coordinate remains the same.
Hence, the coordinates of reflection image are (-1, -2).
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How do I find the value of a + b + c?
The expression of (a + b + c) is equivalent to 16.5.
What is a mathematical function, equation and expression?Function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function
Expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators
Equation : A mathematical equation is used to equate two expressions.
Given are two circles intersecting at each others centers at D and E.
We can write the area of the common region as -
[A] = 4 x {(θ/180) x (πr²/2) - (1/2)r²sinθcosθ}
Now, we can write θ as -
cos(θ) = (3/6) = 1/2
θ = 60°
[A] = 4 x {(60/180) x (π(6)²/2) - (1/2)(6)²sin(60)cos(60)}
[A] = 4 x {(1/3 x 36π/2 - (1/2) x 36 x [tex]\sqrt{\frac{3}{2} }[/tex] x 1/2}
[A] = 4 x {6π - 9 [tex]\sqrt{\frac{3}{2} }[/tex] }
[A] = 24π + (- 9 [tex]\sqrt{\frac{3}{2} }[/tex])
So, on comparing, with the expression below -
aπ + b[tex]\sqrt{c}[/tex]
We get -
a = 24
b = - 9
c = 3/2
So, we get -
a + b + c
24 - 9 + 3/2
24 - 9 + 1.5
24 - 7.5
16.5
Therefore, the expression of (a + b + c) is equivalent to 16.5.
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Evaluate (if possible) the six trigonometric functions of the angle (0).
-5pi/6
Answer:
sin(-5π/6) = -1/2cos(-5π/6) = -√3/2tan(-5π/6) = √3/3csc(-5π/6) = -2sec(-5π/6) = -2√3/3cot(-5π/6) = √3Step-by-step explanation:
You want the 6 trig functions of the angle -5π/6.
QuadrantThe angle is in the 3rd quadrant, so all trig functions are negative except the tangent and cotangent. The reference angle is -5π/6 +π = π/6.
Trig functionsSince you have memorized the values of the trig functions for multiples of π/6, you know ...
sin(-5π/6) = -1/2
cos(-5π/6) = -√3/2
tan(-5π/6) = sin/cos = √3/3
csc(-5π/6) = 1/sin = -2
sec(-5π/6) = 1/cos = -2√3/3
cot(-5π/6) = 1/tan = √3
Answer/Step-by-step explanation:
Use a Unit Circle, you may have access to (hopefully) or have been asked to memorize.
To find -5pi/6 count clockwise the sixths to 5. Or add:
-5pi/6 + 2pi
= -5pi/6 + 12pi/6
= 7pi/6
-5pi/6 is the same as 7pi/6 on the Unit Circle.
Use the coordinates at that spot
(-root3/2 , -1/2)
to find the six trig ratios.
The Unit Circle is a big, giant, Answer Key. It has all the trig answers on it all in one place. It looks like a scary math pizza, but really its all the answers all in one place.
Once you find your angle, the sine is the y-coordinate.
sin theta = -1/2
The cosine is the x-coordinate.
cos theta = -root3/2
Tangent theta is the sine over the cosine, sin/cos,
= y/x
= (-1/2) / (-rt3/2)
Use Keep-Change-Flip to simplify this.
see image.
= root3/3
Sec is cos flipped over. Csc is sin flipped over. Cot is tan flipped over (the math word is reciprocal)
Yes, just flip over sin to get csc:
sintheta = -1/2
csctheta = - 2/1 = -2
Flip over cos to get sec:
costheta = -rt3/2
sectheta = 2/-rt3
Then rationalize, see image.
= -2rt3/3
Flip over tan to get cot.
tantheta = 1/rt3
Use a form that's easy to flip.
cottheta = rt3/1 = rt3
see image.
Two members of the Math Competition Team solve 13 problems in 1 hour. Assume all team members solve problems at the same rate. How many team members are needed to solve in 1 hour 39 problems
Two members of the maths competition team solve 13 problems in 1 hour.
Person=13
Time=1 hour
Then.
Rate=13/1 (person/time)
With the help of question,
Let us consider,
x team members are needed to solve in 1 hour 39 problems,
2M×1/13=1×xM/39
2M/13=x M/39
2M=x M/13
M=26
26 team members are needed to solve in 1 hour 39 problems,
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If x = 29°, find the measures of angles 1, 2, and 3.
Answer:
75.5
Step-by-step explanation:
division answer in feet and inches 8 divided by 34ft 8in
Answer:
since 12inches=1foot
x inches=34feet
34feet converted to inches =408inches
total inches=408+8
T=416inches
dividing 416 by 8 we have
52 inches left
converting back to feet we get
4ft 4in
find the interval of convergence of the power series. (be sure to include a check for convergence at the endpoints of the interval.) (7x)^n/(10n)!
The interval of convergence of the power series is (-10/7, 10/7).
What does Interval of convergence tell you?
A group of x-values on which a power series converges is known as the interval of convergence. In order to create a convergent series, you can plug in this interval of x-values.
The interval of convergence of a power series is the set of values of x for which the series converges. To find the interval of convergence of the given power series, we can use the ratio test.
The ratio test states that if the series [tex]∑ a_nx^n[/tex] has a positive radius of convergence R, then the series converges for |x| < R and diverges for |x| > R. The radius of convergence is given by the formula
[tex]R = 1/limsup|a_n+1/a_n|.[/tex]
In the given power series, the coefficient [tex]a_n[/tex] is given by [tex]a_n = (7x)^n/(10n)!.[/tex]
We can rewrite this as [tex]a_n = x^n(7/10)^n/n!.[/tex]
Taking the limit of the ratio [tex]a_n+1/a_n[/tex] as n goes to infinity, we get:
[tex]limsup|a_n+1/a_n| \\\\= limsup|x^(n+1)(7/10)^(n+1)/(n+1)!/(x^n(7/10)^n/n!)|\\\\= limsup|x(7/10)/(n+1)|\\\\= |x(7/10)|[/tex]
Therefore, the radius of convergence R is [tex]1/|x(7/10)|[/tex].
This means that the interval of convergence is [tex]|x| < 1/|7x/10| = 10/7.[/tex]
We can check the endpoints of the interval to see if the series converges at these points. At x = 10/7, the series becomes [tex](7(10/7))^n/(10n)! = 1^n/(10n)!.[/tex]
This series is a geometric series with first term 1 and common ratio 1/10, and it converges because [tex]|1/10| < 1.[/tex]
At x = -10/7, the series becomes[tex](7(-10/7))^n/(10n)! = (-1)^n/(10n)![/tex]. This series is an alternating series with first term 1 and common ratio (-1/10), and it also converges because [tex]|-1/10| < 1.[/tex]
Therefore, The interval of convergence of the power series is (-10/7, 10/7).
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A composite figure is represented in the image.
What is the total area of the figure?
A: 192 m2
B: 216 m2
C: 288 m2
D: 336 m2
The total area of the figure is equal to 192 square meters. The correct option is A.
What is an area?The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the rectangle in a two-dimensional plane is called the area of the rectangle.
Calculate the area of the rectangle by the formula written below,
Area of the rectangle = L x W
Area of the rectangle = 8 x 18
Area of the rectangle = 144 square meters
The area of the triangle will be calculated by the formula written as,
Area of triangle = 1/2 x B x H
Area of triangle = 1/2 x ( 18 - 6 ) x 8
Area of triangle = 1 / 2 x 12 x 8
Area of triangle = 48 square meters
Total area = 144 + 48
Total area = 192 square meters
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-7 x 3^0.25x = -10
Which of the following is the solution of the equation?
Is (6, 8) a solution to this system of equations?
y =1/6x+5
y=5/6x+3
Yes
No
Answer:No
Step-by-step explanation:
When we substitute x=6 into the first equation we get
y=1/6 *6+5=6 so we get (6, 6)
When we substitute x=6 into the second equation we get
y=5/6*6+3=8 so we get (6, 8) . it works for the second equation but not the first. It must be true for both equations
Does meditation cure insomnia? Researchers randomly divided 400 people into two equal-sized groups. One group meditated daily for 30 minutes, the other group attended a 2-hour information session on insomnia.
At the beginning of the study, the average difference between the number of minutes slept between the two groups was about 0. After the study, the average difference was about 32 minutes, and the meditation group had a higher average number of minutes slept.
To test whether an average difference of 32 minutes could be attributed to chance, a statistics student decided to conduct a randomization test. She wrote the number of minutes slept by each subject in the study on an index card. She shuffled the cards together very well, and then dealt them into two equal-sized groups, representing those who meditated and those who attended the information session.
Which of the following best describes the outcome of the randomization test.
O The average difference between the two values on the two stacks of cards is expected to be about 0 minutes.
O If meditation is effective, the average difference between the values on the two stacks of cards is expected to be more than 32 minutes.
O The average difference between the two values on the two stacks of cards is expected to be about 32 minutes.
1. The average difference between the two values on the two stacks of cards is expected to be about 0 minutes best describes the outcome of the randomization test.
This randomization test was conducted to test whether the average difference of 32 minutes between the two groups could be attributed to chance. The test essentially involved shuffling the index cards containing the minutes slept by each subject and then dealing them into two equal-sized groups.
This process allows for a fair comparison between the two groups and ensures that any differences in the average minutes slept are due to chance. Since the cards are shuffled randomly, the average difference between the two stacks of cards is expected to be about 0 minutes.
If meditation is effective, then the difference between the values on the two stacks of cards would be expected to be more than 32 minutes.
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James was also measured at 23 inches long. How many centimeters long was he at birth
The length of the birth in centimeters will be 58.42 centimeters.
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
James was also measured 23 inches long.
We know that, in one inch, there are 2.54 centimeters.
The length of the birth in centimeters is calculated as,
⇒ 23 inches
⇒ 23 x 2.54 centimeters
⇒ 58.42 centimeters
The number of birth in centimeters will be 58.42 centimeters.
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on monday therease went to the doctor and got an antibiotic for strep throat. The doctor told her take a dose of 4.8 ml every 12 hours for 7 days. If thereas took her first dose at 9:00 AM on Monday, what day and time should she take her 7th dose?
Therease should take her 7th dose on Friday at 9:00AM
What is time ?
Time can be described in mathematics as an ongoing and continuous series of events that take place one after another, from the past through the present, and into the future. The duration of events or the gaps between them can be measured, compared, or even ordered using time.
Time is the ongoing progression of existence and things that happen in what seems to be an irrevocable order from the past, present, and forward into the future.
Time is defined by physicists as the flow of events from the past through the present and into the future. In essence, a system is timeless if it is unchanging. When describing events that take place in three-dimensional space, time can be thought of as the fourth dimension of reality.
The doctor told her to take a dose of 4.8 ml every 12 hours for 7 days.
If Theresa took her first dose at 9:00 AM on Monday,
Simply add 12 hours every time up to the 7th dose,
First dose ----> 9:00 AM on Monday,
Second dose ----> 9:00 PM on Monday,
Third dose ----> 9:00 AM on Tuesday,
Fourth dose ----> 9:00 PM on Tuesday,
Fifth dose ----> 9:00 AM on Thursday,
Sixth dose ----> 9:00 PM on Thursday,
Seventh dose ----> 9:00 AM on Friday,
Hence, Theresa should take her 7th dose at 9:00 AM on Friday.
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Trey makes 8 dollars for each hour of work. Write an equation to represent his total pay p after working h hours.
Answer:
p=8h
Step-by-step explanation:
P is the total pay, so it will be on one side of the equation. 8 represents the fact that Trey receives 8 dollars per hour. It's multiplied by h to represent the number of hours he works.
In the article, Attitudes About Marijuana and Political Views (Psychological Reports, 1973), researchers reported on the use of cannabis by liberals and conservatives during the 1970's.
To test the claim (at 1% significance) that the proportion of voters who smoked cannabis frequently was lower among conservatives, the hypotheses were
In the hypothesis test about cannabis use by conservatives and liberals, the test statistic was z = -4.27, with a corresponding p-value of about 0.00001.
Which conclusion is most appropriate in the context of this situation?
The data do not support the claim that a lower proportion of conservatives smoke cannabis when compared to liberals.
The data support the claim that the proportion of conservatives who smoke cannabis is no different that the proportion for liberals.
The data support the claim that a lower proportion of conservatives smoke cannabis when compared to liberals.
The z-test statistic is used to test the claim that the proportions of the two populations differ. A normal test statistic, the z-test statistic can also be used to measure the proportion of a single population.
Let's say that:
p1: Liberal voters' proportion of the population who smoked cannabis, p. 2:
Conservative voters' proportion of the population who smoked cannabis The null hypothesis is H 0:
The alternative hypothesis is H 0: p 1 = p 2.
p 1 p 2 P-value = 0.00001 The P-value is less than the significance level of 1%. The null hypothesis is rejected at the 1% significance level.
We can conclude that conservative voters had a lower percentage of voters who regularly smoked cannabis.
What is theory trying?Two distinct hypotheses are tested by all analysts using a random population sample: the alternative hypothesis and the null hypothesis. Typically, the null hypothesis is a hypothesis that all population parameters are equal; For instance, a null hypothesis might assert that the population mean return is zero.
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Use the model to help you find 1/(8/12)
1/(8/12)=___
Answer:
3/2 or 1.5 or 1 1/2 in mix number
What is the area of the triangle with vertices at A(2, 2), B(4, 5) and C(7, 5)? Express your
answer as a decimal to the nearest tenth.
Answer:
Therefore, the area of the triangle with vertices at A(2, 2), B(4, 5), and C(7, 5) is approximately 4.5, to the nearest tenth.
Step-by-step explanation:
To find the area of a triangle with vertices at A(2, 2), B(4, 5), and C(7, 5), you can use the Shoelace Theorem.
The Shoelace Theorem states that the area of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) is given by the following formula:
A = 1/2 * |(x1y2 + x2y3 + ... + xn-1yn + xny1) - (y1x2 + y2x3 + ... + yn-1xn + ynx1)|
To apply the Shoelace Theorem to a triangle with vertices at A(2, 2), B(4, 5), and C(7, 5), you can plug in the coordinates of these vertices into the formula:
A = 1/2 * |(25 + 45 + 72) - (24 + 57 + 52)|
= 1/2 * |(10 + 20 + 14) - (8 + 35 + 10)|
= 1/2 * |44 - 53|
= 1/2 * |-9|
= 1/2 * 9
= 4.5
You are packing books into a box. The box can hold at most 10 books. The function y=5.2x represent the weight y (in pounds) of x books
Is 52 the range?
Is 45 the range?
Is 15 the domain?
Is the domain discrete or continuous?
Answer:
52 is range
Step-by-step explanation:
Range is the 'y' values the function can have
from 0 to 10 books would give a range of 0 - 52
Domain is the x-values 0, 1,2,3,4,56,7,8,9,10
discrete, because you cannot have anything but the whole numbers listed for the values of 'x'
Answer:
The domain is the input. It is the number of books. You can have 0-10 books. These would be the domains.
The range is the outputs. The outputs can be from 0 to 52 pounds.
The domain is discrete. The domain is the number of books. We cannot have half or a book or 3/4 of a book. So this will be discrete. The number of books can only be whole numbers up to 10.
Step-by-step explanation:
Mathematical models are used as tools to describe reality. These models are supposed to characterize the important features of the analyzed phenomena and provide insight. The normal distribution is an example of a random variable that is widely used by researchers to model real data.
Researchers often model real observations using the normal distribution, but sometimes the real distribution is a bit different from the perfect, normal distribution. List some reasons why researchers might make approximations like this and describe at least one situation when researchers should not use this approximation.
When forming your answer to this question you may give an example of a situation from your own field of interest for which a random variable can serve as a model.
There are several reasons why researchers might not use the normal distribution to model real data which are due to assumptions of normal distribution.
First, the normal distribution assumes that the data is symmetric and is distributed around the mean. However, in some cases, the data may be skewed, making the normal distribution an inappropriate model.
Second, the normal distribution assumes that the data is homoscedastic, meaning that the variance is constant across the sample. However, in some cases, the variance may be heteroscedastic, meaning that the variance changes across the sample. This makes the normal distribution an inappropriate model for this type of data.
One situation in which researchers should not use the normal distribution approximation is when modeling financial data. Financial data often follows a power-law distribution, which is not normal. Therefore, researchers should use a power-law distribution to model this type of data.
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Write an inequality for the graph
The inequality that represents the line in the figure will be y > - (2/3) x + 3.
What is inequality?Inequality is defined as an equation that does not contain an equal sign. Inequality is a term that describes a statement's relative size and can be used to compare these two claims.
From the figure, the points of the line are (0, 3) and (3, 1). And the region above the line is a consideration. Then the equation of the line is given as,
(y - 3) > [(3 - 1) / (0 - 3)](x - 0)
Simplify the equation, then we have
(y - 3) > [(3 - 1) / (0 - 3)](x - 0)
(y - 3) > - (2/3)x
y > - (2/3) x + 3
The disparity that addresses the line in the figure will be y > - (2/3) x + 3.
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The question is in the picture
Answer:
-6/6 or -1
Step-by-step explanation:
To find the equation of the line (or the slope, assuming that's what equation means, ) we need to write the formula Y1-Y2/X1-X2
So, we need to fill in the numbers, which gives us the fraction -6-0/-2-(-8)
When we solve for the numerator (-6-0), we get -6.
When we solve for the denominator (-2 - {-8}), we get 6.
So, we have -6/6, which, when simplified, is -1.
(If the problem is not asking for slope when it says equation, then sorry!)
Solve for x -13x<65 simplify
The value of x in the inequality, -13x < 65, is simplified as, x > 5.
How to Solve an Inequality?Any inequality given can be solved by find the value of the variable in the inequality, which will make it a true statement. To do this, isolate the given variable to make it stand alone on one side.
Given the inequality, -13x < 65, solve as shown below:
-13x < 65 [given]
Divide both sides by -13:
-13x/-13 < 65/-13
x > -65/13 {the sign changes from < to > because we divided both sides by a negative quantity]
x > 5
The value of x is simplified as x > 5.
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If four vertices of a regular octagon are chosen at random then the probability that the quadrilateral formed by them is a rectangle is
A.1/8
B.2/21
C.1/32
D.1/35
The correct answer is option D. 1/35. To count the number of rectangles that can be formed, we need to consider the two cases where the diagonals of the rectangle are parallel to the sides of the octagon, and where the diagonals are perpendicular to the sides of the octagon.
In the first case, there are four choices for which side the diagonals are parallel to, and once that side is chosen, there are two choices for which pair of vertices lie on that side. Therefore, there are $4 \cdot 2 = 8$ rectangles of this type.
In the second case, there are four choices for which vertex the diagonals intersect, and once that vertex is chosen, there are two choices for which pair of vertices lie on the same side of the intersection point as the chosen vertex. Therefore, there are $4 \cdot 2 = 8$ rectangles of this type.
Altogether, there are $8 + 8 = 16$ rectangles. Therefore, the probability that the quadrilateral formed by the four vertices is a rectangle is
$\frac{16}{70} = \frac{16}{{8 \choose 4}}= \frac{1}{{8 \choose 4}/16}
= \frac{1}{{8 \choose 4}/2^4}
= \frac{1}{{8 \choose 4}/2^4}
= \boxed{\text{(D) } 1/35}$.
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PLEASE HELP ASAP
A line that passes through the points (-1, -4) and (4, 2) is shown.
Which equation is the equation of the line that is perpendicular to the given line and passes through the point (-3, 7) ?
The equation of the line that is perpendicular to the line that passes through the points (-1, -4) and (4, 2): D. 5x + 6y = 27.
How to Write the Equation of Perpendicular Lines?Perpendicular lines have slopes with a product of -1, or are negative reciprocals.
First find the slope of the line that passes through (-1, -4) and (4, 2):
Slope (m) = change in y / change in x = (2 -(-4)) / (4 -(-1))
m = 6/5
The negative reciprocal of 6/5 is -5/6. This means that the perpendicular line is, m = -5/6.
To write the equation of the line that is perpendicular to the given line, substitute m = -5/6 and (a, b) = (-3, 7) into y - b = m(x - a):
y - 7 = -5/6(x - (-3))
y - 7 = -5/6(x + 3)
6(y - 7) = -5(x + 3)
6y - 42 = -5x - 15
6y = -5x - 15 + 42
6y = -5x + 27
5x + 6y = 27
The answer is: D. 5x + 6y = 27.
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Question
Solve the equation.
−2w=−9
w=
w=9/2 =4,5
Step-by-step:
:)
To start dividing 126 by 23, Miranda used the estimate 120÷20 = 6. How could you tell six is too high?
Answer:
the product of 6 and 23 is more than 126
Step-by-step explanation:
You want to know how to tell that 6 is too high an estimate for the first digit of the quotient of 126 and 23.
Trial dividendWhen the trial quotient value of 6 is multiplied by the actual divisor of 23, we are computing 6(20 +3) = 120 +18. This is more than 126, so the trial quotient value is too large.
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Additional comment
Another way to tell is to consider the dual problem that 120/20 = 6 represents: 120/6 = 20. It is easy to see that 126/6 = 21, so we know that a divisor of 23 (larger than 21) will give a quotient less than 6.
find the k-value so that the point is on the line
kx+3y=7; (1,-k)
Answer:
k = -7/2
Step-by-step explanation:
You want the k-value so that the point (1, -k) is on the line kx+3y=7.
EquationThe point will be on the line when its coordinates make the equation true. Using (x, y) = (1, -k), we have ...
k(1) +3(-k) = 7
-2k = 7 . . . . . . . . simplify
k = -7/2 . . . . . . . divide by -2
The value of k = [tex]\frac{-7}{2}[/tex] so the point is on the line kx+3y=7
what is line?A line is an endlessly long object without breadth, depth, or curvature in geometry. Thus, despite the fact that they can exist in two, three, or higher dimensional environments, lines are one-dimensional things. The term "line" can also refer to a line segment that has two locations that serve as its ends in daily life.
given
the point of the line = ( 1 ,-k )
the line = kx + 3y = 7
the coordinates make the equation true when its points are on the line
so putting the values of x and y in equation of line
k( 1 ) + 3 ( -k ) = 7
k - 3k = 7
-2k = 7
k = [tex]\frac{-7}{2}[/tex]
The value of k = [tex]\frac{-7}{2}[/tex] so the point is on the line kx+3y=7
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