Answer:
We need to find how many number of equivalence relations are on the set {1,2,3}
A relation is an equivalence relation if it is reflexive, transitive and symmetric.
equivalence relation R on {1,2,3}
1.For reflexive, it must contain (1,1),(2,2),(3,3)
2.For transitive, it must satisfy: if (x,y)∈R then (y,x)∈R
3. For symmetric, it must satisfy: if (x,y)∈R,(y,z)∈R then (x,z)∈R
Since (1,1),(2,2),(3,3) must be there is R, (1,2),(2,1),(2,3),(3,2),(1,3),(3,1). By symmetry,
we just need to count the number of ways in which we can use the pairs (1,2),(2,3),(1,3) to construct equivalence relations.
This is because if (1,2) is in the relation then (2,1) must be there in the relation.
the relation will be an equivalence relation if we use none of these pairs (1,2),(2,3),(1,3) . There is only one such relation: {(1,1),(2,2),(3,3)}
we can have three possible equivalence relations:
{(1,1),(2,2),(3,3),(1,2),(2,1)}
{(1,1),(2,2),(3,3),(1,3),(3,1)}
{(1,1),(2,2),(3,3),(2,3),(3,2)}
Equivalence relations on a set satisfy conditions of reflexivity, symmetry, and transitivity. The Bell number counts the number of partitions, or equivalence relations, on a set. Hence, for the set {1, 2, 3}, there are five equivalence relations.
Explanation:The subject of this question relates to equivalence relations on a set which is an important topic in discrete mathematics and set theory. In simple terms, an equivalence relation is a relation on a set that equates certain pair of elements. In your set {1, 2, 3}, an equivalence relation must meet three conditions: reflexivity (each number is equal to itself), symmetry (if 1 is related to 2, then 2 is related to 1), and transitivity (if 1 is related to 2 and 2 is related to 3, then 1 is related to 3).
To find the number of equivalence relations on a set, we refer to the Bell number. Bell numbers count the number of partitions of a set. For a set with 3 elements like yours, the third Bell number gives the number of equivalence relations, which is 5. Therefore, there are 5 equivalence relations on the set {1, 2, 3}.
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Use Definition 7.1.1. DEFINITION 7.1.1 Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = ∞ e−stf(t) dt 0 is said to be the Laplace transform of f, provided that the integral converges. Find ℒ{f(t)}. (Write your answer as a function of s.)
The Laplace Transform of a function f(t) is a tool used to solve differential equations by converting them into simpler algebraic equations. The transform itself is given by the integral ℒ{f(t)} = ∫∞₀ e−stf(t) dt, but without knowing the specific form of f(t), the exact transformation cannot be computed.
Explanation:The Laplace Transform, as defined by Definition 7.1.1, is a mathematical tool often used to handle differential equations, especially in the fields of Physics and Engineering. The main idea behind the transform is to convert the differential equations, which are difficult to solve, into simple algebraic equations. These simple equations are relatively easy to solve. Once solved, the inverse Laplace Transform is employed to obtain the solution to the original differential equation.
The question asks to compute the Laplace Transform ℒ{f(t)} of a function. By definition, the Laplace transform of a function f(t), defined for t ≥ 0 is given by the integral ℒ{f(t)} = ∫∞₀ e−stf(t) dt. The question doesn't give an explicit form of the function f(t). In general, if f(t) = e^(-αt), where α is a constant, then its Laplace Transform is given by ℒ{e^(-αt)} = 1/(s + α) for s > α.
If you could kindly provide me with the specific function f(t), I can better determine the Laplace Transform ℒ{f(t)}
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The Thomas Supply Company Inc. is a distributor of gas-powered generators. As with any business, the length of time customers take to pay their invoices is important. Listed below, arranged from smallest to largest, is the time, in days, for a sample of The Thomas Supply Company Inc. invoices. 13 13 13 20 26 27 31 34 34 34 35 35 36 37 38 41 41 41 45 46 47 47 48 52 53 56 56 62 67 82
Determine the first and third quartiles, determine the seconds decile and the eighth decile,determine the 67th percentile.
Final answer:
Explanation of quartiles, deciles, and percentiles in a dataset of gas-powered generator invoice payment times. Lastly, the 67th percentile in the data set is approximately 46 days.
Explanation:
The first quartile (Q1) is found by locating the median of the lower half of the data set, which results in a value of 34 days. The third quartile (Q3) is the median of the upper half of the data set, yielding a value of 47 days. The second decile corresponds to the 20th percentile, which is approximately 31 days. The eighth decile corresponds to the 80th percentile, which is approximately 47 days. Lastly, the 67th percentile in the data set is approximately 46 days.
A coin is tossed 30 times it lands 12 times on heads and 18 times on tails what is experimental probability of the coin landing on tails?
Answer:
3/5
Step-by-step explanation:
Total tossed : 30
# of times landed on tails : 18
Experimental probability of tails = 18/30 = 3/5
8. Write three other proportions for each given proportion. 35 miles/2 hours=87.5 miles/5 hours
Answer:
218.75 miles / 12.5 hours
437.5 miles / 25 hours
656.25 miles / 37.5 hours
Step-by-step explanation:
35 miles / 2 hours = 87.5 miles / 5 hours
This is the ratio of 2.5. So, the other proportions are
87.5 x 2.5 miles / 5 x 2.5 hours = 218.75 miles / 12.5 hours
87.5 x 5 miles / 5 x 5 hours = 437.5 miles / 25 hours
87.5 x 7.5 miles / 5 x 7.5 hours = 656.25 miles / 37.5 hours
Tangent lines-someone please help explain
The angle between radius and tangent to circle is 90 degrees.
The quadrilateral formed by the two tangents and the two rays has two angles of 90 degrees, an angle of 40 degrees and an unknown angle.
The sum of the angles of a quadrilateral is 360 degrees.
⇒ x = 360° - 90 - 90 - 40 = 140°
x = 140°
Consider the following matrix A={{1,-1,0,0},{2,-1,0,0},{0,0,1,0},{-1,-1,0,1}}Which of the following statements is true? Check the correct answer(s) below.A. The homogeneous system Ax=0 has infinitely many solutionsB. The matrix A has determinant −1C. The matrix A is not invertibleD. The matrix A is singular.E. None of the above
The correct answer is:
Option: E
E. None of the above
Step-by-step explanation:We know that a system of equation i.e.[tex]Ax=0[/tex] has infinite many solutions if det(A)=0
Also, the matrix is singular if det(A)=0and is non-singular or invertible otherwise i.e. when det(A)≠0
We know that when a homogeneous system has i.e. [tex]Ax=0[/tex] is such that: |A|≠0 then the system has a unique solution.The matrix that will be formed by the given set of vectors is:
[tex]A=\begin{bmatrix}-1 &2 &0 &-1 \\ -1&-1 &0 &-1 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1 \end{bmatrix}[/tex]
Also, determinant i.e. det of matrix A is calculated by:
[tex]\begin{vmatrix}-1 &-2 &0 &-1 \\ -1&-1 &0 &-1 \\ 0 &0 &1 &0 \\ 0 &0 &0 &1 \end{vmatrix}=1(1(1+2))=3[/tex]
Hence, determinant is not equal to zero.
This means that the matrix is invertible and non-singular.
A large school district in southern California asked all of its eighth-graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed, with a mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth-graders from this population are randomly selected. Approximately what is probability that the sample mean foot length is less than 23 cm?
Answer:
The probability of the sample mean foot length less than 23 cm is 0.120
Step-by-step explanation:
* Lets explain the information in the problem
- The eighth-graders asked to measure the length of their right foot at
the beginning of the school year, as part of a science project
- The foot length is approximately Normally distributed, with a mean of
23.4 cm
∴ μ = 23.4 cm
- The standard deviation of 1.7
∴ σ = 1.7 cm
- 25 eighth-graders from this population are randomly selected
∴ n = 25
- To find the probability of the sample mean foot length less than 23
∴ The sample mean x = 23, find the standard deviation σx
- The rule to find σx is σx = σ/√n
∵ σ = 1.7 and n = 25
∴ σx = 1.7/√25 = 1.7/5 = 0.34
- Now lets find the z-score using the rule z-score = (x - μ)/σx
∵ x = 23 , μ = 23.4 , σx = 0.34
∴ z-score = (23 - 23.4)/0.34 = -1.17647 ≅ -1.18
- Use the table of the normal distribution to find P(x < 23)
- We will search in the raw of -1.1 and look to the column of 0.08
∴ P(X < 23) = 0.119 ≅ 0.120
* The probability of the sample mean foot length less than 23 cm is 0.120
Solve the following using the crossing-graphs method. (Round your answer to two decimal places.) 5 + 69 × 0.96t = 32
Answer:
The solution is 22.98.
Step-by-step explanation:
Here, the given equation,
[tex]5 + 69\times 0.96^t = 32[/tex],
Let [tex]f(t) = 5 + 69\times 0.96^t[/tex]
And, [tex]f(t) = 32[/tex]
Where, t represents x-axis and f(t) represents y-axis,
Since, [tex]f(t) = 5 + 69\times 0.96^t[/tex] is an exponential decay function having y-intercept (0,74).
Also, f(t) = 32 is the line, parallel to x-axis,
Thus, after plotting the graph of the above functions,
We found that they are intersecting at (22.984, 32)
Hence, the solution of the given equation = x-coordinate of the intersecting point = 22.984 ≈ 22.98
To solve the given equation, 5 + 69 × 0.96t = 32, you start by subtracting 5 from both sides, then divide by 69. Then, divide both sides by 0.96 to solve for t. The solution is t ≈ 0.41 (rounded to two decimal places).
Explanation:To solve the equation 5 + 69 × 0.96t = 32 using the crossing-graphs method, we first simplify the equation:
Start by subtracting 5 from both sides of the equation: 69 × 0.96t = 32 - 5.This results in: 69 × 0.96t = 27.Next, divide both sides by 69: 0.96t = 27/69.Which simplifies to 0.96t ≈ 0.391 (rounded to three decimal places).Finally, divide both sides by 0.96 to solve for t: t ≈ 0.391/0.96 ≈ 0.41 (rounded to two decimal places).Learn more about Crossing-graphs method here:https://brainly.com/question/34386833
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Find the point, P, at which the line intersects the plane. x equals 7 plus 9 t, y equals 3 minus 7 t, z equals 7 minus 5 t; 5 x minus 6 y minus 9 z equals negative 1 \
Answer:
The point of intersection [tex]P\left(\dfrac{1133}{122},\dfrac{149}{122},\dfrac{699}{122}\right)[/tex]
Step-by-step explanation:
Equation of line:
[tex]x=7+9t[/tex]
[tex]y=3-7t[/tex]
[tex]z=7-5t[/tex]
Equation of plane:
[tex]5x-6y-7z=-1[/tex]
We need to find the point of intersection of line and plane.
Point of intersection: When both line and plane meet at single point.
So, put the value of x, y and z into plane.
[tex]5(7+9t)-6(3-7t)-7(7-5t)=-1[/tex]
[tex]35+45t-18+42t-49+35t=-1[/tex]
[tex]122t=-1+32[/tex]
[tex]t=\dfrac{31}{122}[/tex]
Substitute the value of t into x, y and z
[tex]x=7+9\cdot \dfrac{31}{122}=\dfrac{1133}{122}[/tex]
[tex]y=3-7\cdot \dfrac{31}{122}=\dfrac{149}{122}[/tex]
[tex]z=7-5\cdot \dfrac{31}{122}=\dfrac{699}{122}[/tex]
Point of intersection:
[tex]\left(\dfrac{1133}{122},\dfrac{149}{122},\dfrac{699}{122}\right)[/tex]
Hence, The point of intersection [tex]P\left(\dfrac{1133}{122},\dfrac{149}{122},\dfrac{699}{122}\right)[/tex]
A club has 50 members, 10 belonging to the ruling clique and 40 second-class members. Six members are randomly selected for free movie tickets. What is the probability that 3 or more belong to the ruling clique?
Answer: The probability that 3 or more belong to the ruling clique is 0.34.
Step-by-step explanation:
Since we have given that
Number of total members = 50
Number of belonging to ruling clique = 10
Number of belonging to second class member = 40
We need to find the probability that 3 or more belong to the ruling clique.
Let X be the number of outcomes belong to ruling clique.
So, it becomes,
P(X≥3)=1-P(X<3)
[tex]P(X\geq 3)=1-P(X=1)-P(X=2)\\\\P(X\geq 3)=1-\dfrac{^{10}C_1\times ^{40}C_5}{^{50}C_6}-\dfrac{^{10}C_2\times ^{40}C_4}{^{50}C_6}\\\\P(X\geq 3)=1-0.41-0.25\\\\P(X\geq 3)=0.34[/tex]
Hence, the probability that 3 or more belong to the ruling clique is 0.34.
The probability of selecting 3 or more members from the ruling clique when choosing 6 members randomly from a club of 50 members (10 in ruling clique, 40 second-class) is 8.56%.
Explanation:This probability problem can be solved using the concepts of Combinations and Binomial Theorem. You need to determine the number of ways to choose 3, 4, 5, or 6 members from the ruling clique (10 members) and the remaining from the second-class members (40 members). For each case, divide by the total number of ways to choose 6 members from all 50 members to get the probability. Sum up all the probabilities for each case to get the total probability of having 3 or more from the ruling clique.
Calculations:1. Number of ways of choosing 3 from the ruling clique and 3 from the second class: C(10,3)*C(40,3) = 120*9880 = 1,185,600 ways
2. Number of ways of choosing 4 from the ruling clique and 2 from the second class: C(10,4)*C(40,2) = 210*780 = 163,800 ways
3. Number of ways of choosing 5 from the ruling clique and 1 from the second class: C(10,5)*C(40,1) = 252*40 = 10,080 ways
4. Number of ways of choosing 6 from the ruling clique and 0 from the second class: C(10,6)*C(40,0) = 210*1 = 210 ways
Total ways to choose 3 or more from the ruling clique: 1,185,600 + 163,800 + 10,080 +210 = 1,359,690 ways
From 50 members, the total ways to choose 6: C(50,6) = 15,890,700 ways
The Probability of 3 or more from the ruling clique = 1,359,690 / 15,890,700 = 0.0856 or 8.56%
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The volume of water flowing through a pipe varies directly wlth the square of the radius of the pipe. If the water flows at a rate of 80 liters per minute through a pipe with a radlus of 4 cm, at what rate would water flow through a pipe with a radius of 3 cm? (Rigorous) (Competency 007) 11. A) 45 liters per minute B) 6.67 liters per minute C) 60 liters per minute D) 4.5 liters per minute
Answer:
A
Step-by-step explanation:
Volume varies directly with the square of the radius, so:
V = k r²
When V = 80, r = 4.
80 = k (4)²
k = 5
V = 5r²
When r = 3:
V = 5 (3)²
V = 45
The flow is 45 L/min.
Solve the following system of equations
3x - 2y =5
-2x - 3y = 14
Answer:
x = -1 and y = -4
Step-by-step explanation:
It is given that,
3x - 2y = 5 ----(1)
-2x - 3y = 14 ------(2)
To find the solution of equations
(1) * 2 ⇒
6x - 4y = 10 -----(3)
(2) * 3 ⇒
-6x - 9y = 42 ----(4)
eq(3) + eq(4) ⇒
6x - 4y = 10 -----(3)
-6x - 9y = 42 ----(4)
0 - 13y = 52
y = 52/(-13) = -4
Substitute the value of y in eq(1)
3x - 2y = 5 ----(1)
3x - (2 * -4) = 5
3x +8 = 5
3x = 5 - 8 = -3
x = -3/3 = -1
Therefore x = -1 and y = -4
Answer:
The solution is:
[tex](-1, -4)[/tex]
Step-by-step explanation:
We have the following equations
[tex]3x - 2y =5[/tex]
[tex]-2x - 3y = 14[/tex]
To solve the system multiply by [tex]\frac{3}{2}[/tex] the second equation and add it to the first equation
[tex]-2*\frac{3}{2}x - 3\frac{3}{2}y = 14\frac{3}{2}[/tex]
[tex]-3x - \frac{9}{2}y = 21[/tex]
[tex]3x - 2y =5[/tex]
---------------------------------------
[tex]-\frac{13}{2}y=26[/tex]
[tex]y=-26*\frac{2}{13}[/tex]
[tex]y=-4[/tex]
Now substitute the value of y in any of the two equations and solve for x
[tex]-2x - 3(-4) = 14[/tex]
[tex]-2x +12 = 14[/tex]
[tex]-2x= 14-12[/tex]
[tex]-2x=2[/tex]
[tex]x=-1[/tex]
The solution is:
[tex](-1, -4)[/tex]
PLEASE ANSWER WITH AN EXPLANATION! THANK YOU
Answer:
[tex]\large\boxed{A=153\ cm^2}[/tex]
Step-by-step explanation:
Look at the picture.
We have
square with side length a = 9
trapezoid with base lengths b₁ = 9 and b₂ = 6 and the height length h = 6
right triangle with legs lengths l₁ = 3 + 6 = 9 and l₂ = 6
The formula of an area of a square
[tex]A=a^2[/tex]
Substitute:
[tex]A_I=9^2=81\ cm^2[/tex]
The formula of an area of a trapezoid:
[tex]A=\dfrac{b_1+b_2}{2}\cdot h[/tex]
Substitute:
[tex]A_{II}=\dfrac{9+6}{2}\cdot6=\dfrac{15}{2\!\!\!\!\diagup_1}\cdot6\!\!\!\!\diagup^3=(15)(3)=45\ cm^2[/tex]
The formula of an area of a right triangle:
[tex]A=\dfrac{l_1l_2}{2}[/tex]
Substitute:
[tex]A_{III}=\dfrac{(9)(6)}{2}=\dfrac{54}{2}=27\ cm^2[/tex]
The area of the shape:
[tex]A=A_I+A_{II}+A_{III}\\\\A=81+45+27=153\ cm^2[/tex]
State the linear programming problem in mathematical terms, identifying the objective function and the constraints. A firm makes products A and B. Product A takes 2 hours each on machine M; product B takes 4 hours on L and 3 hours on M. Machine L can be used for 8 hours and M for 6 hours. Profit on product A is $9 and $7 on B. Maximize profit.
Final answer:
The problem requires formulating a linear programming model to maximize the profit function Z = 9x + 7y with constraints on machine time for product A and B (2x + 3y ≤ 6 for machine M and 4y ≤ 8 for machine L) and the non-negativity restrictions (x, y ≥ 0).
Explanation:
The linear programming problem can be stated in mathematical terms with an objective function and constraints for a firm making products A and B. The objective is to maximize profit, which is the sum of 9 dollars per unit of product A and 7 dollars per unit of product B. Let the number of products A and B produced be represented by variables x and y, respectively.
The objective function to maximize is Z = 9x + 7y.
Constraints:
Machine M's availability limits product A to 2 hours each, and product B to 3 hours each, with a total available time of 6 hours: 2x + 3y ≤ 6.Machine L can be used for 8 hours and is only required for product B, which takes 4 hours: 4y ≤ 8.Non-negativity constraints: x ≥ 0 and y ≥ 0, since the number of products cannot be negative.A bag contains 2 steel balls and 5 brass balls. The total weight is 13 pounds. If 2 steel balls are added and 2 brass balls are removed, the weight decreases to 12 pounds. How much does each kind of ball weigh?
Answer:
Step-by-step explanation:
The answer above this answer is correct
Angle measures and segment lengths. Someone please explain
ooops, i made a mistake. deleted. Give the other guy brainly
sorry
Check the picture below.
Two automobiles left simultaneously from cities A and B heading towards each other and met in 5 hours. The speed of the automobile that left city A was 10 km/hour less than the speed of the other automobile. If the first automobile had left city A 4 1/2 hours earlier than the other automobile left city B, then the two would have met 150 km away from B. Find the distance between A and B.
Answer:
450 km
Step-by-step explanation:
Equations
We can define 3 variables: a, b, d. Let "a" and "b" represent the speeds of the cars leaving cities A and B, respectively. Let "d" represent the distance between the two cities. We can write three equations in these three variables:
1. The relation between "a" and "b":
a = b -10 . . . . . . . the speed of car A is 10 kph less than that of car B
2. The relation between speed and distance when the cars leave at the same time:
d = (a +b)·5 . . . . . . distance = speed × time
3. Note that the time it takes car B to travel 150 km to the meeting point is (150/b). (time = distance/speed) The total distance covered is ...
distance covered by car A in 4 1/2 hours + distance covered by both cars (after car B leaves) = total distance
4.5a + (150/b)(a +b) = d
__
Solution
Substituting for d, we have ...
4.5a + 150/b(a +b) = 5(a +b)
4.5ab +150a +150b = 5ab +5b^2 . . . . . . multiply by b, eliminate parentheses
5b^2 +0.5ab -150(a +b) = 0 . . . . . . . . . . subtract the left side
Now, we can substitute for "a" and solve for b.
5b^2 + 0.5b(b-10) -150(b -10 +b) = 0
5.5b^2 -5b -300b +1500 = 0 . . . . . . . . eliminate parentheses
11b^2 -610b +3000 = 0 . . . . . . . . . . . . . multiply by 2
(11b -60)(b -50) = 0 . . . . . . . . . . . . . . . . factor
The solutions to this equation are ...
b = 60/11 = 5 5/11 . . . and . . . b = 50
Since b must be greater than 10, the first solution is extraneous, and the values of the variables are ...
b = 50a = b-10 = 40d = 5(a+b) = 5(90) = 450The distance between A and B is 450 km.
_____
Check
When the cars leave at the same time, their speed of closure is the sum of their speeds. They will cover 450 km in ...
(450 km)/(40 km/h +50 km/h) = 450/90 h = 5 h
__
When car A leaves 4 1/2 hours early, it covers a distance of ...
(4.5 h)(40 km/h) = 180 km
before car B leaves. The distance remaining to be covered is ...
450 km - 180 km = 270 km
When car B leaves, the two cars are closing at (40 +50) km/h = 90 km/h, so will cover that 270 km in ...
(270 km)/(90 km/h) = 3 h
In that time, car B has traveled (3 h)(50 km/h) = 150 km away from city B, as required.
Answer:
450km
Step-by-step explanation:
Take it that each automobile travels at 30 km an hour, for 150 km, meaning it will be 450 km apart.
If 2000 dollars is invested in a bank account at an interest rate of 8 per cent per year,
Find the amount in the bank after 12 years if interest is compounded annually:
Answer:
$5036.34
Step-by-step explanation:
Each year, 8% of the existing balance is added to the existing balance, effectively multiplying the amount by 1.08. If that is done for 12 years, the effective multiplier is 1.08^12 ≈ 2.51817. The the amount in the bank at the end of that time is ...
$2000×2.51817 = $5036.34
The amount in the bank after 12 years with an annual interest rate of 8% on a principal amount of 2000 dollars, compounded annually, will be approximately $5025.90.
Explanation:This is a compound interest problem. The formula used to solve this type of problem is A = P(1 + r/n)^(nt), where:
P is the principal amount (initial money),r is the annual interest rate,t is the number of years,n is the number of times that interest is compounded per year.In this case, P = $2000, r = 8% or 0.08, t = 12 years and n = 1 (as interest is compounded annually). Substituting these values in the equation, we get:
A = 2000(1 + 0.08/1)^(1*12)
.
The resulting Amount A after 12 years will be approximately $5025.90.
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A pharmaceutical company receives large shipments of aspirin tablets. The acceptance sampling plan is to randomly select and test 42 tablets, then accept the whole batch if there is only one or none that doesn't meet the required specifications. If one shipment of 6000 aspirin tablets actually has a 5% rate of defects, what is the probability that this whole shipment will be accepted? Will almost all such shipments be accepted, or will many be rejected?
Answer: Hence, the probability that the whole shipment would be accepted is 0.371.
Many would be rejected.
Step-by-step explanation:
Since we have given that
Number of tablets to be tested = 42
Probability of getting a defect = 5% = 0.05
We need to find the probability that this whole shipment will be accepted.
As we have mentioned that if there is only one or none defect, then the whole shipment would be accepted.
P(accepted) = P(either none or one defect) = P(X=0)+P(X=1)
[tex]P(X=0)=(1-0.05)^{42}=(0.95)^{42}=0.115\\\\and\\\\P(X=1)=42\times (0.05)(0.95)^{41}=0.006\times 42=0.256[/tex]
So, P(Accepted) = 0.115+0.256=0.371
Hence, the probability that the whole shipment would be accepted is 0.371.
Many would be rejected.
PLEASE ANSWER
factor the polynomial using the pattern
x^2+9x+8
x^2(a+b)x+ab=(x+a)(x+b)
Answer:
x² + 9x + 8 = (x + 1)(x + 8)
x² + 9x + 8 = (x + 8)(x + 1)
Step-by-step explanation:
* Lets explain how to factorize the polynomial using the pattern
- The form of the quadratic polynomial is x² + px + q, where p is the
coefficient of x and q is the numerical term
∵ x² + (a + b)x + (ab) = (x + a)(x + b)
- From the formula above the coefficient of x is the sum of the two
factors a and b
∴ p = a + b and q = ab
- That means p is the sum of two numbers and q is the product of
the same numbers
* Lets solve the problem
∵ x² + 9x + 8 is a quadratic polynomial
∵ x² + px + q is the form of quadratic polynomial
∴ p = 9 and q = 8
∵ p = a + b and q = ab
∴ a + b = 9 ⇒ (1)
∴ ab = 8 ⇒ (2)
- We must to find two numbers their product is 8 and their sum is 9
∵ The possibility of 8 as a product of two numbers is:
2 × 4 OR 1 × 8
∵ The sum of 1 + 8 = 9
∴ The value of a and b are 1 and 8
- It does't matter which of them = 1 or which of them = 8
∴ x² + (a + b)x + ab = x² + (1 + 8)x + (1)(9)
∵ x² + (a + b)x + (ab) = (x + a)(x + b)
∴ x² + (1 + 8)x + (1)(9) = (x + 1)(x + 8)
∴ x² + 9x + 8 = (x + 1)(x + 8)
- OR
∴ x² + 9x + 8 = (x + 8)(x + 1)
A runner runs around a circular track. He completes one lap at a time of t = 269 s at a constant speed of v = 4.6 m/s. What is the radius, r in meters, of the track?
Answer:
[tex]\boxed{\text{197 m}}[/tex]
Step-by-step explanation:
The formula relating distance (d), speed (s), and time (t) is
d = st
1. Calculate the distance
d = 269 s × 4.6 m·s⁻¹ = 1240 m
2.Calculate the track radius
The distance travelled is the circumference of a circle
[tex]\begin{array}{rcl}C & = & 2 \pi r\\1240 & = & 2 \pi r\\\\r & = & \dfrac{1240}{2 \pi }\\\\& = & 197\\\end{array}\\\text{The radius of the track is }\boxed{\textbf{197 m}}[/tex]
The radius in meters is 196.9 meters.
The runner ran around the track in 269 seconds at a speed of 4.6 m/s. This will enable us to find the distance around the track which is the circumference of the track.
Distance = Speed × time
= 4.6 × 269
= 1,237.4 meters
The distance here is the circumference which can also be found by the formula:
Circumference = π × diameter
1,237.4 = 22/7 × Diameter
Diameter = 1,237.4 ÷ 22/7
= 393.7 meters
Now that we have the diameter, the radius is:
= Diameter / 2
= 196.9 meters
In conclusion, the radius is 196.3 meters
Find out more at https://brainly.com/question/3092498.
A car was valued at $39,000 in the year 1995. The value depreciated to $11,000 by the year 2003.
A)What was the annual rate of change between 1995 and 2003? (Round to 4 decimal places)
B)What is the correct answer to part A written in percentage form?
C)Assume that the car value continues to drop by the same percentage. What will the value be in the year 2007?
Please help ASAP the homework is due Monday!!! :(
Answer:
14.6328% , $5836.03
Step-by-step explanation:
Here we are going to use the formula
[tex]A_{0}(1-r)^n = A_{n}[/tex]
[tex]A_{0}[/tex] = 39000
r=?
[tex]A_{8}[/tex] = 11000
n=8
Hence
[tex]39000(1-r)^8 = 11000[/tex]
[tex](1-r)^8 = \frac{11000}{39000}[/tex]
[tex](1-r)^8 = 0.2820[/tex]
[tex](1-r) = 0.2820^{\frac{1}{8}[/tex]
[tex](1-r) = 0.2820^{0.125}[/tex]
[tex](1-r) = 0.8536[/tex]
[tex](1-0.8536=r[/tex]
[tex]r = 0.1463[/tex]
Hence r= 0.1463
In percentage form r = 14.63%
Now let us see calculate the value of car in 2003 that is after 12 years
we use the main formula again
[tex]A_{0}(1-r)^n = A_{n}[/tex]
[tex]A_{0}[/tex] = 39000
r=0.1463
[tex]A_{12}[/tex] = ?
n=12
[tex]39000(1-0.14634)^{(12} = A_{12}[/tex]
[tex]39000(0.8536)^{12} = A_{12}[/tex]
[tex]39000*0.1497 = A_{12}[/tex]
[tex]A_{12}=5840.34[/tex]
Hence the car's value will be depreciated to $5840.34 (approx) by 2003.
The annual rate of change between 1995 and 2003 is -0.1463
The annual rate of change between 1995 and 2003 is -14.63%
The value of the car in 2007 would be $5,844.24
The value of the car decreases as the years go by. This is referred to as depreciation. Depreciation is the decline in value of an asset as a result of wear and tear.
In order to determine the annual rate of change, use this formula:
g = [tex](FV / PV) ^{\frac{1}{n} } - 1[/tex]
Where:
g = depreciation rate
FV = value of the car in 2003 = $11,000
PV = value of the car in 1995 = $39,000
n = number of years = 2003 - 1995 = 8
[tex](11,000 / 39,00)^{\frac{1}{8} } - 1[/tex] = -0.1463 = -14.63%
The value of car in 7 years can be determined using this formula:
FV = P (1 + g)^n
$39,000 x (1 - 0.1463)^12
$39,000 x 0.8537^12 = $5,844.24
A similar question was answered here: https://brainly.com/question/12980665?referrer=searchResults
if cos θ = -0.6, and 180° < θ < 270°, find the exact value of sin 2θ.
Answer:
sin(2θ) = 0.96
Step-by-step explanation:
In the third quadrant, both sin(θ) and cos(θ) are negative. Then the double-angle trig identity tells us ...
sin(2θ) = 2·sin(θ)·cos(θ) = -2cos(θ)√(1 -cos(θ)²) . . . . using the negative root
Filling in the given value, we have
sin(2θ) = -2·(-0.6)(√(1-(-0.6)²) = 2·0.6·0.8 = 0.96
Solve for x in the equation
Answer:
[tex]\large\boxed{x=6\pm3\sqrt{10}}[/tex]
Step-by-step explanation:
[tex]x^2-12x+36=90\\\\x^2-2(x)(6)+6^2=90\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(x-6)^2=90\iff x-6=\pm\sqrt{90}\\\\x-6=\pm\sqrt{9\cdot10}\\\\x-6=\pm\sqrt9\cdot\sqrt{10}\\\\x-6=\pm3\sqrt{10}\qquad\text{add 6 to both sides}\\\\x=6\pm3\sqrt{10}[/tex]
3 balls are randomly drawn from a box containing 4 yellow 2 blue and 5 red balls. Youwin a prize if all three are yellow. What is the probability of winning?
Answer: [tex]\dfrac{4}{165}[/tex]
Step-by-step explanation:
Given : The number of yellow balls in the box = 4
The total number of balls = [tex]4+2+5=11[/tex]
Since the given situation has dependent events.
Then, the probability of that all three balls are yellow is given by :-
[tex]\text{P(YYY)}==\dfrac{^4P_3}{^{11}P_3}=\dfrac{4\times3\times2}{11\times10\times9}\\\\\Rightarrow\ \text{P(YYY)}=\dfrac{4}{165}[/tex]
Hence, the probability of winning =[tex]\dfrac{4}{165}[/tex]
A Membership to the Gym Costs $ 25 Per Person in 1995 .The Membership Cost has Increased by an average of $ 6 per Person for each Year Since 1995 .Write a Linear Equation for the Cost of a GYM Membership For One Person Since 1995. So , What is the Cost of A GYM Membership in 2009 ????
Step-by-step explanation:
Let n = number of years since 1995
Let C = cost of gym membership in any particular year
Initial cost in 1995 = $25
Additional cost each year = $6
We can say the following:
Cost at any given year = cost in 1995 + ($6 x number of years after 1995)
Or expressed as the following : C = 25 + 6n (Ans)
In 2009, the number of years since 1995,
= 2009 - 1995
= 14 years
Hence, cost in 2009,
= $25 + ($6 x 14 years)
= $109 (Ans)
$109 is the Cost of A GYM Membership in 2009.
What does "cost" mean to you?
The cost is the sum of money that a business spends on the development or manufacturing of goods or services. It excludes the profit margin markup. The sum of money spent to create a good or product, as seen from the seller's perspective, is the cost.given,
Let n = number of years since 1995
Let C = cost of gym membership in any particular year
Initial cost in 1995 = $25
Additional cost each year = $6
formula ,
Cost at any given year = cost in 1995 + ($6 x number of years after 1995)
put value in formula
C = 25 + 6n
now,
In 2009, the no of member in years since 1995,
= 2009 - 1995
= 14 years
Hence, cost of gym membership in 2009,
= $25 + ($6 x 14 years)
= $109
Therefore, cost of gym membership in 2009 = $109
Learn more about cost
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........Help Please.......
Answer:
b = 1.098
Step-by-step explanation:
Each year, the GDP is 9.8% higher than the year before, so the multiplier each year is 1 + 9.8% = 1.098. This is the value of b.
b = 1.098
In a particular course, it was determined that only 70% of the students attend class on Fridays. From past data it was noted that 95% of those who went to class on Fridays pass the course, while only 10% of those who did not go to class on Fridays passed the course. If a student passes the course, what is the probability that they did not attend on Fridays?
Answer: Probability that students who did not attend the class on Fridays given that they passed the course is 0.043.
Step-by-step explanation:
Since we have given that
Probability that students attend class on Fridays = 70% = 0.7
Probability that who went to class on Fridays would pass the course = 95% = 0.95
Probability that who did not go to class on Fridays would passed the course = 10% = 0.10
Let A be the event students passed the course.
Let E be the event that students attend the class on Fridays.
Let F be the event that students who did not attend the class on Fridays.
Here, P(E) = 0.70 and P(F) = 1-0.70 = 0.30
P(A|E) = 0.95, P(A|F) = 0.10
We need to find the probability that they did not attend on Fridays.
We would use "Bayes theorem":
[tex]P(F\mid A)=\dfrac{P(F).P(A\mid F)}{P(E).P(A\mid E)+P(F).P(A\mid F)}\\\\P(F\mid A)=\dfrac{0.30\times 0.10}{0.70\times 0.95+0.30\times 0.10}\\\\P(F\mid A)=\dfrac{0.03}{0.695}=0.043[/tex]
Hence, probability that students who did not attend the class on Fridays given that they passed the course is 0.043.
g. How many different 6-letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once? a. 6 b. 36 c. 720 d. 46,656
Answer:
720
Step-by-step explanation:
Given : The word ABSENT
To Find: How many different 6-letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once?
Solution:
Number of letters in ABSENT = 6
So, No. of arrangements can be formed using the letters in the word ABSENT, if each letter is used only once = 6!
= [tex]6 \times 5 \times 4\times 3 \times 2 \times 1[/tex]
= [tex]720[/tex]
So, Option C is true
Hence there are 720 different 6-letter arrangements can be formed using the letters in the word ABSENT.
Final answer:
There are 720 different 6-letter arrangements that can be formed from the word ABSENT, by applying the permutation formula 6! = 720. The correct option is c.
Explanation:
The question asks: How many different 6-letter arrangements can be formed using the letters in the word ABSENT, if each letter is used only once? To answer this, we need to calculate the number of permutations of 6 letters taken from 6. This is a simple permutation problem where we use the formula for permutations which is n!, where n is the total number of items to choose from and ! denotes a factorial, meaning the product of all positive integers up to n.
Given that the word ABSENT has 6 letters and we are arranging all 6, we have 6! = 6×5×4×3×2×1 = 720. Therefore, there are 720 different 6-letter arrangements that can be formed using the letters in ABSENT, with each letter used only once.
please help asap!!!!!
Answer:
The volume of the prism is 27√3/2
Step-by-step explanation:
* Lets revise the triangular prism properties
- The triangular prism has five faces
- Two bases and three side faces
- The two bases are triangles
- The three side faces are rectangles
- The rule of its volume is Area of its base × its height
* Lets solve the problem
- The triangular prism has two bases which are equilateral triangles
- The length of each side of the triangular base is 3"
- The height of the prism is 6"
∵ The volume of the prism = area of the base × its height
∵ The base is equilateral triangle of side length 3"
- The area of any equilateral triangle is √3/4 s²
∴ The area of the base of the prism = √3/4 × (3)² = 9√3/4
∵ The length of the height of the prism is 6"
∴ The volume of the prism = 9√3/4 × 6 = 27√3/2
* The volume of the prism is 27√3/2