The image shows a point and a line. Suppose we create a parabola using the point as the focus and the line as the directrix. Decide whether each point on the list is on this parabola. Explain your reasoning. (-1,5) (3,3) (5,5)
Answers
The parabola is represented by the following expression: y - 3 = (1 / 8) · (x - 3)². Concerning the following points:
(x, y) = (- 1, 5): YES (The point is 4 units left from focus and 4 units up from directrix)
(x, y) = (3, 3): YES (The point is 2 units down from focus and 2 units up from directrix)
(x, y) = (5, 5): NO
Which points does belong to a parabola?
In this question we find a graph of the focus and the directrix of a parabola. The least distance between the focus and the directrix is represented by the following expression:
d = 2p
Where is the distance between the vertex and the focus.
The vertex of the parabola is the midpoint of the line segment of the least distance between focus and directrix. And the equation of the parabola in vertex form is:
y - k = [1 / (4 · p)] · (x - h)²
Where (h, k) are the coordinates of the vertex of the parabola.
First, determine the distance between vertex and focus:
(0, 2 · p) = (3, 5) - (3, 1)
(0, 2 · p) = (0, 4)
p = 2
Second, find the vertex of the parabola:
(h, k) = 0.5 · (3, 5) + 0.5 · (3, 1)
(h, k) = (3, 3)
Third, build the equation of the parabola:
y - 3 = (1 / 8) · (x - 3)²
Fouth, check if each point belongs to the parabola:
(x, y) = (- 1, 5)
y = (1 / 8) · (x - 3)² + 3
y = (1 / 8) · (- 1 - 3)² + 3
y = (1 / 8) · (- 4)² + 3
y = 2 + 3
y = 5
YES (The point is 4 units left from focus and 4 units up from directrix)
(x, y) = (3, 3)
y = (1 / 8) · (x - 3)² + 3
y = (1 / 8) · (3 - 3)² + 3
y = 3
YES (The point is 2 units down from focus and 2 units up from directrix)
(x, y) = (5, 5)
y = (1 / 8) · (x - 3)² + 3
y = (1 / 8) · (5 - 3)² + 3
y = 1 / 2 + 3
y = 4.5
NO
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Related Questions
Which formulas can be used to find an area of a circle?
Answers
Answer:
2nd one
6th one
Step-by-step explanation:
Hope this helps! Pls give brainliest!
Find the area and perimeter. Remember to find any missing measurements first! Round to 3 decimals!
Answers
The area of the given shape above would be = 169m² while the perimeter of the shape = 52m
How to calculate the area of the given shape?To calculate the area of the given shape, the formula that should be used is given below;
Area = a²
where;
a = 13m
area = 13× 13 = 169m²
The perimeter of the shape = 2(a+b)
= 2(13+13)
= 2×26
= 52 m
Therefore, the perimeter and the area of the shape above would be = 52m and 169m² respectively.
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For which graph is the parent function y=x^2?
Answers
Answer:
Sure, I can tell you about the four holy sites in Jerusalem. There are actually many holy sites in Jerusalem, but the four most significant ones are the Western Wall, the Church of the Holy Sepulchre, the Dome of the Rock, and the Al-Aqsa Mosque.
The Western Wall, also known as the Wailing Wall, is the most sacred place in Judaism. It is the last remaining part of the Second Temple, which was destroyed by the Romans in 70 CE. Jews from all over the world come to pray at the wall, and many people write prayers on pieces of paper and stuff them in the cracks between the stones.
The Church of the Holy Sepulchre is one of the most important Christian sites in the world. It is believed to be the place where Jesus was crucified, buried, and resurrected. The church is shared by several Christian denominations, including the Greek Orthodox, Roman Catholic, and Armenian Apostolic Churches.
The Dome of the Rock is a Muslim shrine located on the Temple Mount, which is one of the most contested religious sites in the world. The shrine is built on the spot where Muslims believe the Prophet Muhammad ascended to heaven. The Dome is covered in gold and has a beautiful blue mosaic on the inside.
The Al-Aqsa Mosque is the third holiest site in Islam, after Mecca and Medina. It is located on the Temple Mount, next to the Dome of the Rock. The mosque is believed to be the place where the Prophet Muhammad prayed with the other prophets, and it has a beautiful silver dome.
These four holy sites are all located within a few hundred meters of each other, and they are a testament to the deep religious history and significance of Jerusalem. Each site is unique and beautiful in its own way, and they all attract millions of visitors every year.
Step-by-step explanation:
if x+y=15 and x-y=10,then find the value of x^2-y^2
Answers
Answer:
150
Step-by-step explanation:
Given, x+y=15 and x-y=10
We need to find the value of x^2-y^2
We know that x^2-y^2 = (x+y)(x-y)
Substituting the given values, we get:
x^2-y^2 = (x+y)(x-y) = (15)(10) = 150
Therefore, the value of x^2-y^2 is 150.
what are you supposed to enter in for the individiual data value when trying to calculate standard deviation
Answers
To calculate the standard deviation of a set of data, you need to have the individual data values. The individual data values are the numeric values that make up the data set.
To calculate the standard deviation, you need to perform the following steps:
Calculate the mean (average) of the data set.
For each data value, subtract the mean from the data value.
Square each of the differences calculated in step 2.
Sum the squared differences calculated in step 3.
Divide the sum of the squared differences by the number of data values minus 1 (this is called the "sample" standard deviation) or by the total number of data values (this is called the "population" standard deviation).
Take the square root of the result obtained in step 5 to obtain the standard deviation.
When calculating the standard deviation, it's important to use the correct number of decimal places and units of measurement to ensure accuracy.
What is the definition of accuracy?
Accuracy refers to how close a measured or calculated value is to the true or accepted value.
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Which one of the following situations results in a conventional electric current that flows northward?
Answers
The direction of conventional current flow is opposite to the direction of electron flow. Therefore, to determine the direction of conventional current, we need to know the direction of electron flow in a given situation.
Electrons, which are negatively charged particles, flow from the negative terminal of a battery to the positive terminal, while conventional current flows from the positive terminal to the negative terminal.
In the absence of any further information or context, it is impossible to determine a situation that would result in a conventional electric current that flows northward. The direction of electron flow and, therefore, the direction of conventional current flow, depends on the specific configuration of the electrical circuit.
That being said, it's important to note that conventional current flow is just a conventional convention, and its direction is not fundamental to the operation of electrical circuits. Regardless of the direction of conventional current flow, the actual flow of electrons and the operation of the circuit will be the same.
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For any 30-60-90 triangle, the length of the hypotenuse is 2 times as long as the ____.
A. height
B. diagonal
C. longer leg
D. shorter leg
Answers
if the circumference equals 23.52, what's the radius
Answers
Answer: 3.743324
Step-by-step explanation:
The distance a ball moves is 3 m in 1.5 s. What is the speed of the ball? TYPE THE NUMBER ONLY! .. one hundred point !!
Answers
Answer:
2 m/s
Step-by-step explanation:
To find the speed of the ball, we can use the formula:
Speed = Distance / Time
Given that the distance the ball moves is 3 m and the time it takes is 1.5 s, we can substitute those values into the formula to get:
Speed = 3 m / 1.5 s
Simplifying the expression by dividing 3 by 1.5, we get:
Speed = 2 m/s
Therefore, the speed of the ball is 2 m/s.
The following differential equations appear similar but have very different solutions. Solve both subject to the initial condition y(7) = 6 dy/dx = x
y = ____
dy/dx = y
y = ____
Answers
For the initial condition y(7) = 6,
(i) For dy/dx = x, the function is y = (x²/2) - (19/2),
(ii) For dy/dx = y, the function is y = 6eˣ⁻⁷.
Part (i) : We can solve the differential equation "dy/dx = x" using separation of variables.
⇒ dy/dx = x,
⇒ dy = x dx,
On integrating,
We have,
⇒ y = x²/2 + C, where C is = constant of integration.
To find the value of C, we can use the initial condition which is y(7) = 6;
⇒ 6 = (7²)/2 + C,
⇒ C = 6 - (49/2),
⇒ C = -19/2,
Therefore, the solution to differential equation "dy/dx = x", for initial condition y(7) = 6, is y = (x²/2) - (19/2),
Part (ii) : We solve differential equation "dy/dx = y" using separation of variables.
⇒ dy/dx = y
⇒ dy/y = dx,
On integrating,
We have,
⇒ ln|y| = x + C, where C is = constant of integration.
Solving for y,
We get,
⇒ y = Ceˣ,
To find the value of C, we can use the initial condition y(7) = 6:
⇒ 6 = Ce⁷,
⇒ C = 6/e⁷,
Therefore, the solution to the differential equation dy/dx = y, for initial condition y(7) = 6, is : y = (6/e⁷) × eˣ,
Simplifying, we get:
⇒ y = 6eˣ⁻⁷,
Therefore, the differential-equation is y = 6eˣ⁻⁷.
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The given question is incomplete, the complete question is
The following differential equations appear similar but have very different solutions. Solve both subject to the initial condition y(7) = 6;
(i) dy/dx = x
y = ____
(ii) dy/dx = y
y = ____
The equation of a circle in general form.
x2 + y2 – 18x + 14y + 84 = 0
What is the radius of the circle?
Include all of the following in your work for full credit.
(a) Show all work rewriting this equation from general to standard form
- Be sure to include all work completing the square and factoring the trinomial
(b) Give correct radius of the circle
Answers
The radius of the circle is given as 9
How to solve for the radius of the circleTo rewrite the equation of the circle from general form to standard form, we need to complete the square for both x and y terms.
x^2 - 18x + y^2 + 14y + 84 = 0 (Given)
To complete the square for x, we need to add and subtract the square of half the x coefficient (i.e., (18/2)^2 = 81) inside the parentheses:
x^2 - 18x + 81 + y^2 + 14y + 84 - 81 = 0
(x - 9)^2 + (y + 7)^2 - 81 = 0
(x - 9)^2 + (y + 7)^2 = 81
This is the standard form of the equation of a circle, where the center is located at (9, -7) and the radius is √81 = 9.
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contaminated water: the concentration of benzene was measured in units of milligrams per liter for a simple random sample of five specimens of untreated wastewater produced at a gas field. the sample mean was with a sample standard deviation of . seven specimens of treated wastewater had an average benzene concentration of with a standard deviation of . it is reasonable to assume that both samples come from populations that are approximately normal. can you conclude that the mean benzene concentration is less in treated water than in untreated water? let denote the mean benzene concentration for untreated water and denote the mean benzene concentration for treated water. use the level the -value method with the ti-84 plus calculator.
Answers
The answer is that we can use a hypothesis test to determine if the mean benzene concentration is less in treated water than in untreated water.
: We will use the following hypotheses:
H0: μtreated ≥ μuntreated
Ha: μtreated < μuntreated
where μtreated is the true mean benzene concentration for treated water, and μuntreated is the true mean benzene concentration for untreated water.
We will use a one-tailed t-test with a level of significance of α = 0.05.
Using the Ti-84 Plus calculator, we can find the t-test statistic and p-value. First, we calculate the pooled standard deviation:
Sp =√((n1-1)*s1² + (n2-1)*s2²)/(n1+n2-2))
Sp = √(((5-1)*0.7² + (7-1)*0.6²)/(5+7-2))
Sp = 0.642
Next, we calculate the t-test statistic:
t = (x1 - x2) / (Sp * √(1/n1 + 1/n2))
t = (0.5 - 0.8) / (0.642 *√(1/5 + 1/7))
t = -2.13
Finally, we calculate the p-value:
p-value = tcdf(-100, t, df)
p-value = tcdf(-100, -2.13, 10)
p-value = 0.026
Since the p-value is less than the level of significance, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the mean benzene concentration is less in treated water than in untreated water.
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Layla is the youngest of four siblings whose ages are consecutive even integers. If the sum of their ages is 8484, find Layla's age
Answers
The age of Layla who youngest of four siblings is 18 years old
The age of 4 sibling are in consecutive even integers
Age of 1st sibling = x
Age of 2nd sibling = x + 2
Age of 3rd sibling = x + 4
Age of 4th sibling = x + 6
Sum of the sibling age = 84
x + x + 2 + x + 4 + x + 6 =84
4x + 12 = 84
4x = 84 -12
4x = 72
x = 18
Layla is youngest sibling hence her age will be 18 years
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The question is incorrect the correct question is :
Layla is the youngest of four siblings whose ages are consecutive even integers. If the sum of their ages is 84, find Layla's age
upright vacuum cleaners have either a hard body type or a soft body type. shown are the weights in pounds of a sample of each type. at , can it be concluded that the means of the weights are different? assume the variables are normally distributed and the variances are unequal.
Answers
To determine if the means of the weights of the hard body type and soft body type upright vacuum cleaners are different, a two-sample t-test can be conducted.
However, before conducting the test, it must be checked if the assumptions are met. The variables are assumed to be normally distributed and the variances are assumed to be unequal.
Assuming that the sample sizes are large enough (at least 30), a two-sample t-test with unequal variances can be used. The null hypothesis is that the means of the weights of the two types of vacuum cleaners are equal, while the alternative hypothesis is that they are different.
The t-test calculates a t-statistic, which measures the difference between the sample means in terms of the standard error. The higher the t-value, the more likely it is that the means are different. If the calculated t-value exceeds the critical value at the chosen level of significance (usually 0.05), the null hypothesis is rejected.
In conclusion, to determine if the means of the weights of the hard body type and soft body type upright vacuum cleaners are different, a two-sample t-test can be conducted. The assumptions of normality and unequal variances should be checked before conducting the test.
To determine if the means of the weights of hard body and soft body upright vacuum cleaners are different, we need to conduct a hypothesis test. Since the variances are unequal, we'll use the Welch's t-test. Here are the steps:
1. State the null hypothesis (H0) and the alternative hypothesis (H1):
H0: The means of the weights of hard body and soft body upright vacuum cleaners are equal.
H1: The means of the weights of hard body and soft body upright vacuum cleaners are different.
2. Calculate the t-value and degrees of freedom using the provided sample data of weights in pounds for both hard body and soft body vacuum cleaners. Make sure to use the formula for Welch's t-test, which accounts for unequal variances.
3. Determine the critical t-value for the desired significance level (e.g., 0.05) and the calculated degrees of freedom from Step 2.
4. Compare the calculated t-value with the critical t-value. If the calculated t-value is greater than the critical t-value, reject the null hypothesis in favor of the alternative hypothesis.
In conclusion, by following these steps and using the sample data of weights in pounds for hard body and soft body upright vacuum cleaners, you can determine if there's enough evidence to conclude that the means of the weights are different.
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8. You and your family are preparing to go on a ski trip tomorrow to Lake Tahoe, and you
decide to watch the weather forecast. In the forecast, you can read the probability of
it snowing tomorrow. Match each term to the corresponding probability of it snowing
yaba
there tomorrow.
Likely Impossible Certain
Probability
P(snow)
P(snow) > 1/
P(snow) = 1
P(snow) = 0
Unlikely
OTALugz al sonnige s
wwportileil erh zidW
Suid
sidizzogmi
Answers
Answer:
Step-by-step explanation:
To match each term to the corresponding probability of it snowing tomorrow, we can use the following definitions:
Likely means that the event has a high chance of happening, so the probability is close to 1. For example, P(snow) = 0.9 means that it is very likely to snow tomorrow.
Unlikely means that the event has a low chance of happening, so the probability is close to 0. For example, P(snow) = 0.1 means that it is very unlikely to snow tomorrow.
Certain means that the event will definitely happen, so the probability is exactly 1. For example, P(snow) = 1 means that it will surely snow tomorrow.
Impossible means that the event will never happen, so the probability is exactly 0. For example, P(snow) = 0 means that it will not snow tomorrow at all.
Using these definitions, we can match each term to the corresponding probability as follows:
Likely: P(snow) > 0.5
Unlikely: P(snow) < 0.5
Certain: P(snow) = 1
Impossible: P(snow) = 0
You are given the following statements comparing k-fold cross validation (with k < n) and leave-one-out cross validation (LOOCV), used on a generalized linear model with log link and gamma error. [[Hint: You only need to know this is not a linear regression model fit by least squares.]] I. k-fold cross-validation has a computational advantage over LOOCV. II. k-fold cross-validation has an advantage over LOOCV in bias reduction. III. k-fold cross-validation has an advantage over LOOCV in variance reduction. Determine which of the above statements are true. No explanation needed.
(A) None are true.
(B) I and II only
(C) I and III only
(D) II and III only
(E) The correct answer is not given by (A), (B), (C), or (D)
Answers
Only options I and III are (C)'s correct responses.
I. k-fold cross-validation requires fitting the model k times, whereas LOOCV requires fitting the model n times. Since k > n, k-fold cross-validation is more efficient computationally than LOOCV.
II. k-fold cross-validation has an advantage over LOOCV in bias reduction since each training set is a random sample of the data and tends to reduce bias.
III. LOOCV has an advantage over k-fold cross-validation in variance reduction since LOOCV uses almost all the data to fit the model for each test observation, resulting in lower variance in the test error estimate. In contrast, k-fold cross-validation uses only a subset of the data for each training set, which can lead to higher variance in the test error estimate.
In conclusion, assertions I and III are valid, and the proper response is (C) I and III alone.
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What is the quotient of 2. 408×10^24 divided by 6. 02×10^23
Answers
Answer: 4
Step-by-step explanation:
If m<0=111 degrees find the value of x
Answers
Step-by-step explanation:
The sum of the measures insiqe the quadriateral = 360
one is angle o = 111 and two of them aare 90 degrees each
so 360 - 111 - 90 - 90 = x = 69 degrees
Find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c.
Answers
bcx + acy + abz = abc is another form of the equation of the plane with x-intercept a, y-intercept b, and z-intercept c.
What is the equivalent expression?
Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.
bcx + acy + abz = abc is another form of the equation of the plane with x-intercept a, y-intercept b, and z-intercept c.
To find an equation of the plane with x-intercept a, y-intercept b, and z-intercept c, we can use the intercept form of the equation of a plane:
x/a + y/b + z/c = 1
This equation states that any point (x, y, z) on the plane will satisfy this equation.
To see why this is true, note that if x = a, then the left-hand side of the equation is 1, since y/b + z/c = 0 (since the numerator is 0).
Similarly, if y = b, then the left-hand side of the equation is 1, since x/a + z/c = 0.
Finally, if z = c, then the left-hand side of the equation is 1, since x/a + y/b = 0.
To simplify this equation, we can multiply both sides by the denominators a, b, and c:
hence, bcx + acy + abz = abc is another form of the equation of the plane with x-intercept a, y-intercept b, and z-intercept c.
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find slope of the line
(8,10), (-7,14) m=
Answers
Answer:
-4/15
Step-by-step explanation:
Using the formula for slope of a line
m = ( y2-y1)/(x2-x1)
= ( 14-10) / ( -7-8)
= 4 / -15
= -4/15
Answer:
-0.26667
Step-by-step explanation:
(y1 - y2)/(x1 - x2)
(14 - 10)/(-7 - 8)
4 / -15
-0.26667
ratio problems pls sssss
Answers
Answer:
see explanation
Step-by-step explanation:
ratio of sides = 8 : 5 : 7 = 8x : 5x : 7x ( x is a multiplier )
given the perimeter = 480 , then
8x + 5x + 7x = 480
20x = 480 ( divide both sides by 20 )
x = 24
then
8x = 8 × 24 = 192
5x = 5 × 24 = 120
7x = 7 × 24 = 168
the sides of the triangle are 192 inches, 120 inches, 168 inches
-------------------------------------------------------------------------------------------
ratio of angles = 8 : 15 : 17 = 8x : 15x : 17x ( x is a multiplier )
the sum of the angles in a triangle = 180° , then
8x + 15x + 17x = 180
40x = 180 ( divide both sides by 40 )
x = 4.5
Then
8x = 8 × 4.5 = 36
15x = 15 × 4.5 = 67.5
17x = 17 × 4.5 = 76.5
the angles in the triangle are 36°, 67.5°, 76.5°
I need this now please. Table 1 shows accident data for newly licensed drivers. Use this information
to determine the average rate of change for the percent of drivers in an
accident from states that require between 20 and 100 practice hours. What
does this rate of change represent in the context of this situation?
Answers
The average rate of change for % of drivers in an accident between 20 - 100 practice hrs is [tex]-0.2[/tex].
What is average rate of change?To find the average rate, we will to calculate the slope of the line passing through the points (20, 20%) and (100, 4%).
Using the formula for slope, we get:
Slope = (4% - 20%) / (100 - 20)
Slope = -16% / 80
Slope = -0.2
The average rate of change for the percent of drivers in an accident from states that require between 20 and 100 practice hours is -0.2 which means that for every additional 10 practice hours required by a state, the percent of drivers in an accident decreases by an average of 2%.
Missing table:
Practice hours 20 40 60 80 100
Percent of drivers 20% 19% 18% 10% 4%
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Calculating slopes.
I WILL GIVE BRAINLIEST!!
Answers
The slope of Line 1 is equal to 1.
The slope of Line 2 is equal to 1/2.
Line 1 and Line 2 are neither parallel nor perpendicular.
How to calculate the slope of a line?In Mathematics and Geometry, the slope of any straight line can be determined by using this mathematical equation;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
For line 1, we have the following:
Slope (m) = (2 - 4)/(-2 - 0)
Slope (m) = -2/-2
Slope (m) = 1.
For line 2, we have the following:
Slope (m) = (1 - 2)/(-4 + 2)
Slope (m) = -1/-2
Slope (m) = 1/2
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The ratio of dogs to cats at an animal shelter is 3
to 2
.
How many dogs are at the shelter if there is a total of 30
animals?
Answers
Step-by-step explanation:
Dogs are 3 out of ( 3+2 =5) of the animals
3 out of 5 or 3/5 of the animals are dogs
3/5 * 30 = 18 dogs
(L2) The center of a circle circumscribed around a triangle will also be the circumcenter of the _____.
Answers
The center of a circle that circumscribes a triangle is known as the circumcenter of the triangle.
The circumcenter is the point of intersection of the perpendicular bisectors of the sides of the triangle. This point is equidistant from the three vertices of the triangle, which means that it lies on the perpendicular bisector of each side.
The circumcenter plays an important role in the geometry of triangles. One of its properties is that it is also the center of the circle that passes through the three vertices of the triangle, which is why it is called the circumcenter. This circle is known as the circumcircle of the triangle.
The circumcircle is a unique circle that can be drawn around any triangle, and its center is always the circumcenter. This circle has several important properties, such as:
The circumcircle passes through the three vertices of the triangle.
The circumcircle is centered at the circumcenter of the triangle.
The circumcircle has a radius that is equal to the distance from the circumcenter to any vertex of the triangle.
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Lighthouse B is 8 miles west of lighthouse A. A boat leaves A and sails 5 miles. At this time, it is sighted from B. If the bearing of the boat from B is N65°E, how far from B is the boat? (Please answer in the format of "The boat is either __ miles or ___ miles from lighthouse B")
Answers
The boat is either about 2.334 miles or 2.666 miles from lighthouse B, depending on whether it sailed to the left or to the right of the line AB.
Let's draw a diagram to help visualize the problem. Let A and B be the positions of the lighthouses, and let C be the position of the boat after sailing 5 miles from A. Let x be the distance from C to B that we want to find, and let θ be the angle ACB.
Since the bearing of the boat from B is N65°E, we know that the angle ACB is (180° - 65°) = 115°. Also, since ACB is a straight line, we have
cos(115°) = x/8
Solving for x, we get
x = 8 cos(115°) ≈ 2.334 miles
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explain, using theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. state the domain. 33
Answers
Since f(x) is an increasing function (i.e., its derivative is always positive), it satisfies the conditions of Theorem 9, the Intermediate Value Theorem.
What is a continuous function.?A continuous function is a mathematical function in which arbitrarily small changes in the input result in arbitrarily small changes in the output.
In other words, a function is b if, for every input value in its domain, the output value exists, the limit exists, and the limit equals the function value. It can also be defined as a function whose graph is a single unbroken curve, meaning that there are no abrupt jumps, gaps, or holes.
Using theorems 4, 5, 7, and 9, explain why the function is continuous at every number in its domain and state the domain: f(x) = (3x + 7) / (2x - 5); Domain: R \ {5/2}; Continuous on its domain as it is a quotient of two continuous functions and there are no vertical asymptotes, holes or jumps in the graph.
The domain of f(x) is all real numbers since there are no restrictions on the values of x for which the function is defined.
To prove that f(x) is continuous at every number in its domain, we need to show that it satisfies the conditions of Theorem 4, which states that a function is continuous at a point if and only if the limit of the function exists at that point and is equal to the value of the function at that point.
In this case, we have:
The limit of f(x) as x approaches any number a is given by:
lim(x→a) f(x) = lim(x→a) (2x + 3) = 2a + 3
The value of f(a) at any number a is given by:
f(a) = 2a + 3
Since the limit of f(x) as x approaches a is equal to f(a), we can conclude that f(x) is continuous at every number in its domain, which is all real numbers.
Furthermore, since f(x) is a polynomial, it is continuous everywhere, which means that it satisfies the conditions of Theorems 5 and 7 as well. Finally, since f(x) is an increasing function (i.e., its derivative is always positive), it satisfies the conditions of Theorem 9, the Intermediate Value Theorem.
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Complete Question:
Using Theorems 4, 5, 7, and 9, explain why the function f(x) = 2x + 3 is continuous at every number in its domain. State the domain.
​Recently, two students made headlines by spinning a certain type of coin 260 times and getting 156 heads—​that's 60​%. That makes the 95​% confidence interval ​(54​%,66​%). What does this​ mean? Are the conclusions in parts a through e​ correct? Explain your answers.
a) Between 50% and 62% of all coins of that type are unfair. Choose the correct answer below.
A. This statement is correct.
B. This statement is not correct. At least 50% of all coins are unfair.
C. This statement is not correct. No more than 62% of all coins are unfair.
D. This statement is not correct. The interval is about the proportion of heads, not individual coins
Answers
The confidence interval is about the proportion of heads for the population of coins of that type, not about individual coins.
Therefore, we cannot conclude that any particular coin is unfair or not based on this interval.
The 95% confidence interval (54%, 66%) means that if we were to repeat this experiment many times, using different samples of coins of the same type, and calculate a 95% confidence interval each time, then 95% of those intervals would contain the true proportion of heads for the population of coins of that type.
a) The correct answer is D. The confidence interval is about the proportion of heads, not individual coins.
We cannot conclude that any particular coin is unfair or not based on this interval. It only provides an estimate of the proportion of heads for the population of coins of that type.
b) The correct answer is C. If we spun the same coin many times, we would expect the proportion of heads to be somewhere within the 95% confidence interval (54%, 66%).
c) The correct answer is A. If we randomly selected a coin of that type and spun it many times, we would expect the proportion of heads to be within the 95% confidence interval (54%, 66%).
d) The correct answer is B. The confidence interval does not tell us anything about the fairness of individual coins.
e) The correct answer is A. We can be 95% confident that the true proportion of heads for the population of coins of that type is somewhere within the interval (54%, 66%).
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a firm wants to determine the effect of introducing a new service on customer satisfaction. if the firm surveys a random sample of customers before introducing the new service and measures their level of satisfaction, and surveys the same sample of consumers 60 days after the new service was introduced, what kind of test should be used for comparing the satisfaction scores before and after the new service?
Answers
The appropriate test to compare the satisfaction scores before and after the introduction of the new service is a paired t-test or a dependent t-test.
This is because the same group of customers is being surveyed before and after the introduction of the new service. The two samples are not independent of each other, but are related or dependent because they are measuring the same thing in the same individuals at different points in time.
A paired t-test compares the means of two related samples and determines whether the difference between them is statistically significant or simply due to chance. In this case, the null hypothesis would be that there is no significant difference in customer satisfaction before and after the introduction of the new service, and the alternative hypothesis would be that there is a significant difference.
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Green Tea is holding a coupon event. You can earn Tea coupons
by drawing from a bucket. For each draw, you either earn a $10 coupon or nothing. There
are a total of 100 tickets and 20 of them are coupons. There are two ways to draw: You can
draw with replacement (for each draw, you put the ticket you drew back in the bucket) or
without replacement. Answer the following subproblems.
(a) You draw 5 times with replacement. What is the probability that you get 2 coupons in
total? Show your work. (b) You draw 5 times with replacement. What is the probability that you win exactly $10 in total after the first 2 draws and exactly $30 in total after all the 5 draws? Show your
work. (c) You draw with replacement. What is the probability that you get your first coupon
ticket on the 10th draw? Show your work.
Answers
a. C(5,2) is the number of ways to choose [tex]2[/tex] coupons out of [tex]5[/tex]draws is [tex]0.2048[/tex]
b. the probability that you win exactly [tex]$10[/tex] in total after the first [tex]2[/tex] draws and exactly[tex]$30[/tex] in total after all the[tex]5[/tex]draws is [tex]0.00256[/tex]
c. your first coupon ticket on the[tex]10[/tex] th draw with replacement is[tex]0.0268[/tex]
(a) To get 2 coupons in total out of 5 draws with replacement, we need to calculate the probability of getting exactly 2 coupons in any order, and the probability of not getting a coupon in the other 3 draws. The probability of getting a coupon on any draw is[tex]20/100[/tex] = 1/5, and the probability of not getting a coupon is[tex]80/100[/tex] = 4/5. Therefore, the probability of getting exactly 2 coupons in any order is:
P(2 coupons in any order) =[tex]C(5,2) (1/5)2 (4/5)3 = 0.2048[/tex]
where C(5,2) is the number of ways to choose 2 coupons out of 5 draws.
(b) To win exactly [tex]$10[/tex] after the first 2 draws and exactly [tex]$30[/tex] after all 5 draws with replacement, we need to calculate the probability of getting 1 coupon and 1 non-coupon in the first 2 draws, and then getting 3 coupons in the next 3 draws. The probability of getting a coupon on any draw is [tex]1/5,[/tex] and the probability of not getting a coupon is [tex]4/5.[/tex] Therefore, the probability of getting 1 coupon and 1 non-coupon in the first 2 draws is:
P(1 coupon and 1 non-coupon in the first 2 draws) =[tex]C(2,1) (1/5)1 (4/5)1 = 0.320[/tex]
where C(2,1) is the number of ways to choose 1 coupon and 1 non-coupon out of 2 draws.
After the first 2 draws, there are 18 coupons and 82 non-coupons left in the bucket. The probability of getting 3 coupons in the next 3 draws is:
P(3 coupons in the next 3 draws) =[tex](1/5)³ = 0.008[/tex]
Therefore, the probability of winning exactly $10 after the first 2 draws and exactly $30 after all 5 draws is:
P(win $10 after first 2 draws and $30 after all 5 draws) = P(1 coupon and 1 non-coupon in the first 2 draws) P(3 coupons in the next 3 draws) = [tex]0.320 0.008 = 0.00256[/tex]
(c) To get the first coupon on the 10th draw with replacement, we need to not get a coupon on the first 9 draws and then get a coupon on the 10th draw. The probability of not getting a coupon on any draw is 4/5, and the probability of getting a coupon is 1/5. Therefore, the probability of not getting a coupon on the first 9 draws is:
P(not getting a coupon on the first 9 draws) =[tex](4/5)9 = 0.134[/tex]
The probability of getting a coupon on the 10th draw is 1/5. Therefore, the probability of getting the first coupon on the 10th draw with replacement is:
P(first coupon on the 10th draw) = P(not getting a coupon on the first 9 draws) P(getting a coupon on the 10th draw) =[tex]0.134 1/5 = 0.0268[/tex]
Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty. We can predict only the chance of an event to occur i.e., how likely they are going to happen.
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