For number 3, how do you get the answer? I don't get how you get those roots
Answers
The distance AB is 2√2 units.
The distance ST is 4√2 units.
How to calculate the distance between two points?Distance between two points is the length of the line segment that connects the two points in a plane.
The formula to find the distance between the two points is usually given by d=√((x₂ – x₁)² + (y₂ – y₁)²)
For AB. We have:
A(1, 0) : x₁ = 1 , y₁ = 0
B(3, -2) : x₂ = 3 , y₂ = -2
d = √((3 – 1)² + (-2 – 0)²)
d = √8
d = √(4 * 2)
d = √4 * √2
d = 2 * √2
d = 2√2
Thus, the distance AB is 2√2
For ST. We have:
S(2, 3) : x₁ = 2 , y₁ = 3
T(6, -1) : x₂ = 6 , y₂ = -1
d=√((6 – 2)² + (-1 – 3)²)
d = √32
d = √(16 * 2)
d = √16 * √2
d = 4 * √2
d = 4√2
Thus, the distance ST is 4√2
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Related Questions
The volume of a triangular pyramid is 99 units 3 3 . If the base and height of the triangle that forms its base are 9 units and 6 units respectively, find the height of the pyramid.
Answers
The height of the pyramid is 11 units.
What is a triangular pyramid?
A tetrahedron is a polyhedron in geometry that has four triangular faces, six straight edges, and four vertex corners. It is often referred to as a triangle pyramid. The simplest regular convex polyhedron is the tetrahedron.
Here, we have
Given: The volume of a triangular pyramid is 99 units³. If the base and height of the triangle that forms its base are 9 units and 6 units respectively.
We have to find the height of the pyramid.
The area of the triangular base = (9×6)/2
= 54/2 units
= 27units
The volume of the pyramid with a triangular base = 27units³ = area of base × height/3, or the height of the pyramid.
= 99×3/27
= 11 units
Hence, the height of the pyramid is 11 units.
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a 3.0 kg particle is located on the x-axis at x = −7.0 m and a 5.0 kg particle is on the x axis at x = 3.0 m. what is the center of mass of this two–particle system?
Answers
Answer: At x = -0.75
Step-by-step explanation:
Find the center of mass by plugging into the equation.
(3.0 * -7.0 + 5.0 * 3.0) / (3.0 + 5.0) = -0.75
The question is in the image
Answers
Answer: the answer is f(3)=29
Step-by-step explanation:
Suppose that you must choose a password at your work that is five to seven characters long. How many possible passwords are there if: With 1his
i) each password can be any combination of alphanumeric characters ?
ii) each password must contain at least one digit? (The remaining characters are still able to be any alphanumeric value.)
Answers
The number of possible passwords for a length of 5 to 7 characters, where each character can be any alphanumeric value, is 218,340,105,584. If each password must contain at least one digit, then the number of possible passwords is 577,311,447,520.
There are 62 possible alphanumeric characters (26 uppercase letters + 26 lowercase letters + 10 digits). Therefore, the total number of possible passwords for a length of 5 to 7 characters is:
Total number of passwords = 62^5 + 62^6 + 62^7 = 218,340,105,584,896
If each password must contain at least one digit, then there are 10 choices for the first character, and 62 choices for each of the remaining four to six characters. Therefore, the total number of possible passwords is:
Total number of passwords = 10 * 62^4 + 10 * 62^5 + 10 * 62^6 = 577,311,447,520.
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use integration in cylindrical coordinates in order to compute the volume of: u = {(x, y, z) : 0 ≤ z ≤ 36 − x 2 − y 2 }
Answers
The volume of the given region u in cylindrical coordinates is 72π cubic units.
To compute the volume of the region u using integration in cylindrical coordinates, we first need to express the equation of the region in terms of cylindrical coordinates.
In cylindrical coordinates, x = rcosθ and y = rsinθ. Therefore, we can rewrite the equation of the region as:
0 ≤ z ≤ 36 - [tex]r^2[/tex]
Next, we need to determine the bounds of integration for r and θ. Since the region is bounded by a cylinder of radius √36 = 6, we have 0 ≤ r ≤ 6. The angle θ ranges from 0 to 2π, as it covers the entire region.
The volume of the region u can then be computed as:
V = ∫∫∫ u dV
= ∫0^6 ∫[tex]0^2[/tex]π ∫0^(36-[tex]r^2[/tex]) r dz dθ dr
= ∫0^6 ∫[tex]0^2[/tex]π r(36-[tex]r^2[/tex]) dθ dr
= 2π ∫[tex]0^6[/tex] r(36-[tex]r^2[/tex]) dr
[tex]= 2π [18r^2 - (r^4)/4] evaluated from 0 to 6[/tex]
= 2π (648 - 54)
= 1188π
Therefore, the volume of the region u is 1188π cubic units.
To compute the volume of the region u = {(x, y, z) : 0 ≤ z ≤[tex]36 - x^2 - y^2[/tex]} using cylindrical coordinates, we first need to convert the given Cartesian coordinates into cylindrical coordinates. In cylindrical coordinates, we use variables (ρ, θ, z) with the following relationships:
x = ρ * cos(θ)
y = ρ * sin(θ)
z = z
Now, we can rewrite the inequality for the region u in cylindrical coordinates:
[tex]0 ≤ z ≤ 36 - (ρ * cos(θ))^2 - (ρ * sin(θ))^2[/tex]
Simplifying this inequality, we get:
[tex]0 ≤ z ≤ 36 - ρ^2[/tex]
Next, we need to determine the range for ρ and θ. Since the given region is symmetric about the z-axis, we have:
0 ≤ θ ≤ 2π
For ρ, it ranges from 0 to the maximum value at the outer edge of the region, which is the surface z = 36 - [tex]ρ^2[/tex], so:
0 ≤ ρ ≤ √36 = 6
Now we can set up the triple integral to compute the volume:
Volume = ∫∫∫ ρ * dρ * dθ * dz
The limits of integration for ρ, θ, and z are as follows:
ρ: 0 to 6
θ: 0 to 2π
z: 0 to 36 - [tex]ρ^2[/tex],
Now we can write the full integral:
Volume = ∫(θ=0 to 2π) ∫(ρ=0 to 6) ∫(z=0 to 36-[tex]ρ^2[/tex], , ) ρ * dz * dρ * dθ
Evaluating this triple integral, we get:
Volume = 72π cubic units
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what is the greatest common factor of 400 and 560?
i need an answer asap
Answers
80 is the greatest common factor .
What is a factor in math?
A number or algebraic expression that divides another evenly, i.e. without leaving a residue, is referred to in mathematics as a factor. For instance, the precise values of 12 3 = 4 and 12 6 = 2 show that 3 and 6 are factors of 12. 1, 2, 4, and 12 are additional factors of 12.
the prime factorization of 400
400 = 2 × 2 × 2 × 2 × 5 × 5
the prime factorization of 560
560 = 2 × 2 × 2 × 2 × 5 × 7
the GCF, multiply all the prime factors common to both numbers:
Therefore, GCF = 2 × 2 × 2 × 2 × 5
GCF = 80
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A single die is rolled twice. Find the probability of rolling an odd number the first time and a number greater than 3 the second time. THE Find the probability of rolling an odd number the first time and a number greater than 3 the second time. (Type an integer or a simplified fraction.)
Answers
The probability of rolling an odd number the first time and a number greater than 3 the second time is 1/4
Calculating the probabilityThe probability of rolling an odd number on a single die is 3/6 or 1/2, since there are three odd numbers (1, 3, and 5) out of six possible outcomes.
The probability of rolling a number greater than 3 on a single die is 3/6 or 1/2, since there are three numbers greater than 3 (4, 5, and 6) out of six possible outcomes.
To find the probability of both events happening together (rolling an odd number first and a number greater than 3 second), we need to multiply their individual probabilities:
P(odd number first and number > 3 second) = P(odd number first) * P(number > 3 second)
P(odd number first and number > 3 second) = (1/2) * (1/2) = 1/4
Therefore, the probability is 1/4.
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Find the volume of the solid whose base is the region bounded between the curve y=x² and the x-axis from x=0 to x=2 and whose cross sections taken perpendicular to the x-axis are squares.
Answers
V = ∫[0,2] (x^4) dx
This is because each square cross section has an area of (side length)^2, and since each side length is equal to the height of the function (which is x^2), the area of each square is (x^2)^2 = x^4.
Evaluating this integral, we get:
V = [(1/5) x^5] from 0 to 2
V = (1/5) (2^5 - 0)
V = (1/5) (32) = 6.4
Therefore, the volume of the solid in question is 6.4 cubic units.
Answer this math question for ten points :)
Answers
The trigonometric ratios are given as follows:
sin(A) = 4/5.cos(A) = 3/5.tan(A) = 4/3.sin(B) = 3/5.cos(B) = 4/5.tan(B) = 3/4.What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.5 is the hypotenuse length, while for angle A, we have that the sides are given as follows:
Opposite side of 4.Adjacent side of 3.Hence the ratios are given as follows:
sin(A) = 4/5.cos(A) = 3/5.tan(A) = 4/3.For angle B, we have that 4 is now the adjacent side, while 3 is the opposite side, hence:
sin(B) = 3/5.cos(B) = 4/5.tan(B) = 3/4.More can be learned about trigonometric ratios at brainly.com/question/24349828
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the given curve is rotated about the y-axis. find the area of the resulting surface. y = 1 4 x2 − 1 2 ln(x), 1 ≤ x ≤ 3
Answers
The area of the resulting surface formed by rotating the curve y = 1/4 x^2 − 1/2 ln(x) about the y-axis is approximately 148.81 square units.
To find the area of the surface formed by rotating the curve y = 1/4 x^2 − 1/2 ln(x) about the y-axis, we can use the formula for the surface area of a solid of revolution
S = 2π ∫[a,b] y(x) √(1 + (y'(x))^2) dx,
where a and b are the limits of integration (in this case, 1 and 3), y(x) is the equation of the curve being rotated, and y'(x) is its derivative.
First, we need to find y'(x)
y'(x) = 1/2 x − 1/2x^(-1)
Next, we can substitute y(x) and y'(x) into the formula
S = 2π ∫[1,3] [(1/4 x^2 − 1/2 ln(x)) √(1 + (1/2 x − 1/2x^(-1))^2)] dx
Simplifying the integrand
S = 2π ∫[1,3] [(1/4 x^2 − 1/2 ln(x)) √(1/4 x^2 + 1/4x^(-2))] dx
S = π ∫[1,3] [x^2√(x^2+1) − 2x ln(x)√(x^2+1)] dx
This integral can be evaluated using integration by parts or a suitable substitution. After performing the integration, we get
S = π/6 (54√10 − 5 ln(27) − 6)
= 148.81 square units.
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find the scale factor of the dilation
Answers
bir ankete katılan kişileri yüzde 35 i katılıyorum cevabını vermiştir. Ankete 200 kişi katıldığına göre katılıyorum cevabını veren kaç kişi vardır A 35 B 0 C 70 D105
Answers
Answer:
Cevap C, yani 70 kişi olarak verilmiştir. Ankete katılan 200 kişinin yüzde 35'i, 200 x 0.35 = 70 kişiye denk gelir.
Step-by-step explanation:
Find examples of decimals in a newspaper or magazine write a real world problem in which you could you would divide decimals
Answers
Answer:
Step-by-step explanation:
22
If f(x) = 2x2 – 3x + 5, find f'(o). Use this to find the equation of the tangent line to the parabola y = 2x2 – 3x + 5 at the point (0,5). The equation of this tangent line can be written in the form y = mx + b where m is: and where b is:
Answers
The equation of the tangent line is:
y = -3x + 5
To find f'(x), we need to take the derivative of f(x) with respect to x. Given f(x) = 2x^2 - 3x + 5, the derivative f'(x) is:
f'(x) = 4x - 3
Now, we need to find f'(0):
f'(0) = 4(0) - 3 = -3
So the slope (m) of the tangent line at point (0, 5) is -3. Since the tangent line touches the parabola at (0, 5), we can use this point to find the equation of the tangent line:
y = mx + b
Substitute the point (0, 5) and the slope m = -3:
5 = -3(0) + b
5 = b
Thus, the equation of the tangent line is:
y = -3x + 5
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RS≅ST, m∠RST=7x - 54, m∠STU = 8x
Answers
Answer:
Step-by-step explanation:
Mr. Hedges was tired one night and wrote the (awful, terrible, & atrocious) work below. Identify
his error and correct the mistake:
(Look in the picture)
Answers
Answer:
See below
Step-by-step explanation:
Mr. Hedges added the denominators together when he shouldn't have.
-12/14 + -7/14 DOES NOT EQUAL -19/28, you can't add denominators in fractions. Instead, they stay the same, while the numerators are added together.
The correct answer should be -19/14, by leaving the denominator alone and adding the numerators.
Hope this helps!
Recently, six single-family homes in San Luis Obispo County in California sold at the following prices (in $1,000s): $662, $609, $834, $702, $741, $766. Use Table 2.a.Construct a 90% confidence interval for the mean sale price in San Luis Obispo County.(Round intermediate calculations to 4 decimal places, "sample mean" and "sample standard deviation" to 2 decimal places and "t" value to 3 decimal places, and final answers to 2 decimal places.)
Answers
A 90% confidence interval for the mean sale price in San Luis Obispo County is $635.6 to $802.4.
Constructing the 90% confidence intervalUsing the given data, we can calculate the sample mean and sample standard deviation as follows:
Sample mean (x) = (662 + 609 + 834 + 702 + 741 + 766) / 6 = 719
Sample standard deviation (s) = √[(∑(xi - x)²) / (n - 1)] = √[((662 - 719)² + (609 - 719)² + (834 - 719)² + (702 - 719)² + (741 - 719)² + (766 - 719)²) / (6 - 1)] = 79.44
The degrees of freedom (df) for a 90% confidence interval with n = 6 is df = n - 1 = 5.
The t-value for a 90% confidence interval with df = 5 is found using a t-table or a calculator and is approximately 2.571.
Using the formula for a confidence interval for the population mean with a known standard deviation, we have:
Lower limit = x - (t-value) * (s / √n) = 719 - (2.571) * (79.44 /√6) ≈ 635.6
Upper limit = x + (t-value) * (s / √n) = 719 + (2.571) * (79.44/ √6) ≈ 802.4
Therefore, a 90% confidence interval for the mean sale price in San Luis Obispo County is $635.6 to $802.4.
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5. Tony measured a TV that is approximately 14 inches tall by 24 inches wide. Since the size of TVs are named by the measure of the diagonal, what is the BEST diagonal measure, in inches, for this TV?
A. 772 inches
B. 28 inches
C. 76 inches
D. 9 inches
Answers
a^2 + b^2 = c^2
Plugging the values into that…
14^2 + 24^2 = 772
sqrt772=27.78… which means B is the correct answer.
Answer:
Using this formula, we can calculate the diagonal measure of the TV as follows: (this is the Phytagoras theorem)
diagonal^2 = height^2 + width^2
diagonal^2 = 14^2 + 24^2
diagonal^2 = 196 + 576
diagonal^2 = 772
diagonal ≈ 27.8 inches
Therefore, this TV's best diagonal measure, in inches, is option B, 28 inches (rounded to the nearest inch).
One side of a triangle is 84cm the other two sides are in the ratio 3:8 If the perimeter is 282cm find the the longest and shortest side
Answers
x + 84 + 8x/3 = 282
Multiplying both sides by 3, we get:
3x + 252 + 8x = 846
11x = 594
x = 54
So the shortest side of the triangle is 54 cm. The longest side is 8x/3 = 144 cm.
Problems 11 through 23 and Case Problem 2 require the use of data mining software. If using R/Rattle to solve these problems, refer to Appendix: R/Rattle Settings to Solve Chapter 5 Problems. If using JMP Pro to solve these problems, refer to Appendix: JMP Pro Settings to Solve Chapter 5 Problems.
Association Rules of Browser Histories. Cookie Monster Inc. is a company that specializes in the development of software that tracks web browsing history of individuals. Cookie Monster Inc. is interested in analyzing its data to gain insight on the online behavior of individuals. A sample of browser histories is provided in the files CookieMonsterBinary and CookieMonsterStacked that indicate which websites were visited by which customers. Use a minimum support of 4% of the transactions (800 of the 20,000 total transactions) and a minimum confidence of 50% to generate a list of association rules.
a. Based on the top 14 rules, which three web sites appear in the association rules with the largest lift ratio?
b. Identify the association rule with the largest lift ratio that also has Pinterest as the antecedent. What is the consequent web site in this rule?
c. Interpret the confidence of the rule from part (b). While the antecedent and consequent are not necessarily chronological, what does this rule suggest?
d. Identify the association rule with the largest lift ratio that also has TheEveryGirl as the antecedent. What is the consequent web site in this rule?
e. Interpret the lift ratio of the rule from part (d).
Answers
The lift ratio indicates a strong association between the two websites.
Association rule mining is a data mining technique used to find patterns in datasets. It is particularly useful for finding relationships between variables in large datasets. In this case, the dataset consists of browser histories, and we are interested in finding association rules between the websites that were visited.
The two main measures used in association rule mining are support and confidence. Support is the proportion of transactions that contain both the antecedent and the consequent of a rule, while confidence is the proportion of transactions containing the antecedent that also contain the consequent. Lift ratio is another measure used to evaluate the strength of an association rule. It is the ratio of the observed support to the expected support, assuming the antecedent and the consequent are independent.
To generate a list of association rules, we can use an algorithm like Apriori. The Apriori algorithm starts by finding frequent itemsets, or sets of items that occur together in a sufficient number of transactions. We can then use these frequent itemsets to generate association rules by dividing them into antecedents and consequents and calculating the support and confidence of each rule.
Based on the top 14 rules, we can identify the three web sites that appear in the association rules with the largest lift ratio. Lift ratio is a measure of how much more often the antecedent and consequent appear together than would be expected if they were independent. Therefore, high lift ratios indicate strong associations between the antecedent and consequent. We can sort the rules by lift ratio and identify the three web sites that appear in the rules with the highest lift ratios.
To identify the association rule with the largest lift ratio that also has a specific antecedent, we can filter the rules by the antecedent and sort them by lift ratio. We can then examine the consequent of the rule with the highest lift ratio.
The confidence of a rule indicates how often the consequent is observed in transactions containing the antecedent. In this case, the rule suggests that there is a high probability that customers who visit Pinterest also visit the consequent website. While the antecedent and consequent are not necessarily chronological, the rule suggests that there may be a relationship between the two websites that prompts customers to visit them together.
The lift ratio of a rule indicates the strength of the association between the antecedent and consequent. In this case, the rule suggests that customers who visit TheEveryGirl are much more likely to also visit the consequent website than would be expected if the two websites were independent. Therefore, the lift ratio indicates a strong association between the two websites.
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help!!!
see pic below⬇
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Answer:
10:00 am > 0 (initial time) > 24 gallons10:30 > 30 minutes > 40 gallons 11:00 am > 1 hour (from initial time) > 56 gallonsStep-by-step explanation: every 30 minutes we add 16 to the level
Answer:for 10:am youd put "started" or "0" for time and "24 gallons of water" at 10:30am put "30 minutes" for time because from 10 to 10:30 is 30 minutes. For amount of water put 40 gallons. For the last one put the tine as "1:00pm". Put the time spent as "2 hours and 30 min" and gallons of water put 120
Find the measurement of angle A and round the answer to the nearest tenth. :)
(Show work if you can pleasee)
Answers
The measurement of angle A to the nearest tenth is equal to 40.8 degrees.
How to calculate the magnitude of tan A?In order to determine the magnitude of tan A, we would apply the law of tangent because the given side lengths represent the adjacent side and opposite side of a right-angled triangle.
tan(θ) = Opp/Adj
Where:
Adj represents the adjacent side of a right-angled triangle.Hyp represents the opposite side of a right-angled triangle.θ represents the angle.Based on the information provided in the image, we can logically deduce the following parameters:
Adj = 22 units.Opp = 19 units.By substituting the parameters into the law of tangent formula, we have the following;
TanA = 19/22
A = tan⁻¹(0.8636)
A = 40.8 degrees.
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The value of a certain investment over time is given in the table below. Answer the
questions below to determine what kind of function would best fit the data, linear or
exponential.
Number of
Years Since
Investment
Made, x
Value of
Investment
(8), f(x)
11.486.36
9,181.76
values change
function is approximately
3
6,890.96
4.581.76
function would best fit the data because as x increases, the y
The
of this
Answers
The slope of this function is approximately 4518
How to solveA linear function would best fit the data because as x increases, the y values change values by 4518.
y = mx + c
Linear equation with two variables, when graphed on the cartesian plane with axes of those variables, give a straight line.
the linear function would best fit the data because as x increases, the y values change values by 4518.
The slope of this function is approximately 4518
Slope = change in y values / change in x values
=( 27520.99-23002.99)/(2-1)
= 4518/1
= 4518
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3,200 divided by 1000 in long division
Answers
Answer:
Step-by-step explanation:
1000 | 3200
-3000
----
200
Therefore, 3,200 divided by 1000 is equal to 3.2.
please help i’ll mark brainliest
Answers
The type of quadrilaterals, based on the description of the diagonals are;
a) Isosceles trapezoid
b) Square or rhombus
c) Kite or parallelogram
What is a quadrilateral?A quadrilateral is a four sided polygon.
The properties of the quadrilateral are;
a) The diagonals are congruent but are not perpendicular
b) The diagonals are congruent and perpendicular bisectors
c) One diagonal bisects an angle and the other diagonal
The type of quadrilaterals are;
a) Whereby the diagonals are congruent and the diagonals are not perpendicular, indicates that a possible quadrilateral is an isosceles trapezoid
b) The quadrilaterals with congruent and perpendicular diagonals indicates that the possible quadrilaterals are a square or a rhombus
c) A quadrilateral that bisects an angle and one diagonal indicates that the quadrilateral is a parallelogram or kite
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The number of hours of daylight in Chicago, Illinois, is modeled by the periodic function f(x)=2.95sin(2π365x)+12.21 where x represents the number of days since March 20th. Question 1 What is the period of the function?
Answers
The period of function is approximately 0.6465 days, which means that the pattern of daylight hours repeats approximately every 0.6465 days, or about every 15.55 hours.
What is periodic function?A periodic function is a function that repeats its values in a regular pattern over a specified interval or set of input values.
More specifically, a function f(x) is said to be periodic with period P if, for all values of x in the domain of f(x), we have:
f(x + P) = f(x)
The period of a periodic function is the distance along the x-axis between two consecutive peaks or troughs of the graph of the function. For the given function f(x)=2.95sin(2π365x)+12.21, the coefficient of x in the argument of the sine function is 2π365, which represents the frequency of the oscillation.
The period P of a sine function with frequency f is given by:
P = 1/f
Therefore, for the given function, the period is:
P = 1/f = 1/(2π365) ≈ 0.00177 years
However, since the function is modeling the number of hours of daylight, it is more convenient to express the period in terms of days. One year has 365 days, so we can convert the period to days by multiplying it by 365:
P = 0.00177 × 365 ≈ 0.6465 days
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pls help solve this question!
Answers
Answer:
Yes, by AA, since angle DEF is congruent to HEJ (vertical angles are congruent), and angle DFE is congruent to HJE.
So triangle DEF is similar to triangle HEJ.
Let X be a discrete random variable with probability mass function given byP(x)={c/4 x=0{c/4 x=1{c x=2{0 otherwiseFind the value of that makes p a valid probability mass function.
Answers
In order for a probability mass function (PMF) to be valid, it must satisfy two conditions:
Condition 1- Non-negativity:
c/4, c/4 and c are all non-negative values.
Condition 2- Sum of PMF equals 1:
The sum of the PMF over all possible values of x must be equal to 1.
P(0) + P(1) + P(2) + P(x) for all other x
= c/4 + c/4 + c + 0 (since P(x) = 0 for all other x)
= (c + c + 4c)/4
= 6c/4
In order for this sum to be equal to 1, we must have:
6c/4 = 1
Multiplying both sides by 4/6 to solve for c:
c = 4/6
c = 2/3
So, the value of c that makes P(x) a valid probability mass function is c = 2/3.
To find the value of c that makes P a valid probability mass function, we need to ensure that the sum of all probabilities equals 1.
We can do this by summing the probabilities for all possible values of X:
P(0) + P(1) + P(2) = c/4 + c/4 + c = 1
Simplifying the equation:
c/2 + c = 1
3c/2 = 1
c = 2/3
Therefore, the value of c that makes P a valid probability mass function is 2/3.
To find the value of c that makes P(x) a valid probability mass function, we need to ensure that the sum of probabilities for all possible values of x is equal to 1. Given the probability mass function:
P(x) = {c/4, x=0;
c/4, x=1;
c, x=2;
0, otherwise}
The sum of probabilities for all x values should be:
P(x=0) + P(x=1) + P(x=2) = 1
Substituting the given values:
(c/4) + (c/4) + c = 1
Now, we solve for c:
c/4 + c/4 + c = 1
(2c/4) + c = 1
(1/2)c + c = 1
(3/2)c = 1
To find the value of c:
c = 1 / (3/2)
c = 1 * (2/3)
c = 2/3
So the value of c that makes P(x) a valid probability mass function is 2/3.
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The sum of one positive number and the square of another equals to 30. Find the numbers so that their product is as large as possible. A 2 decimal answer or exact answer is ok. The maximal value of the product is:
Answers
x + y^2 = 30
We want to maximize the product xy. To do this, we can rewrite the equation above as:
y^2 = 30 - x
Now we can use the fact that the square of a number is always non-negative to our advantage. We know that y^2 is always non-negative, so this means that 30 - x must also be non-negative. Therefore, we have:
x <= 30
Now we can maximize the product of xy by finding the value of x that makes y as large as possible. We can do this by completing the square:
y^2 = 30 - x
y^2 = -(x - 30)
This is a downward-facing parabola, so its maximum occurs at its vertex. The vertex occurs at x = 30, so:
y^2 = -(30 - 30) = 0
y = 0
Therefore, the maximum product occurs when x = 15 and y = 0 or x = 0 and y = 15. In either case, the product xy is 0.
Express the area of the region bounded by the given line(s) and/or curve(s) as an iterated double integral.
The coordinate axes and the line x + y = 4
Answers
The area of the region bounded by the line x + y = 4 and the coordinate axes is calculated to be 8/3 square units.
The region bounded by the line x + y = 4 and the coordinate axes is a right triangle with vertices at (0,0), (4,0), and (0,4). To express the area of this region as an iterated double integral, we can integrate over the rectangle R = [0,4] × [0,4] and subtract the integral over the triangle T = {(x,y) : x + y ≤ 4}.
Thus, the area of the region is given by the double integral:
A = ∬R dA - ∬T dA
Since dA = dxdy, we can evaluate this as:
A = ∫0⁴ ∫0⁴ dxdy - ∫0⁴ ∫0⁴-x+y dxdy
Simplifying this, we get:
A = ∫0⁴ ∫0⁴-x dydx
Evaluating the inner integral first, we get:
A = ∫0⁴ (-x)(4-x) dx
Integrating this, we obtain:
A = ∫0⁴ (-4x + x²) dx = [-2x^2 + (1/3)x³]0⁴ = 8/3
Therefore, the area of the region bounded by the line x + y = 4 and the coordinate axes is 8/3 square units.
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