(a) The differential Equation that is satisfied by y is dy/dt = k × y × (1 - y) ,
(b) Solution of the differential equation assuming y(0) = c is y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex] .
In the question ,
Part (a)
let the fraction of people who heard the rumor is = y
So , the fraction who have not heard the rumor is = 1 - y .
the rate of rumor spread is ⇒ dy/dt = k×y(1 - y)
dy/y(1-y) = k.dt ...where k is the constant of proportionality .
So , the differential equation is ..
dy/dt = k × y × (1 - y)
Part (b)
So , 1/y(1-y) = 1/y + 1/(1 - y) ....equation(1)
integrating equation(1) , we get
∫dx/(1 + ax) = ㏑(1 + ax)/a ,....where a is the constant
㏑y + ㏑(1-y)/(-1) = kt + d ,.....where d is the constant
By using , ㏑a - ㏑b = ㏑(a/b) and taking exponential . we get ,
y/(1 - y) = c₁[tex]e^{k\times t}[/tex]
for t = 0 and y(0) = c
solving further , we get
c₁ = c/(1 - c)
So , y = (1-y)c₁[tex]e^{k\times t}[/tex]
y(1 + c₁[tex]e^{k\times t}[/tex]) = c₁[tex]e^{k\times t}[/tex]
y = c₁[tex]e^{k\times t}[/tex]/(1 + c₁[tex]e^{k\times t}[/tex])
taking c₁[tex]e^{k\times t}[/tex] common , and substituting the value of c₁ we get ,
the solution as , y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex] .
Therefore , (a) the differential equation is dy/dt = k × y × (1 - y) and
(b) the solution is y = [tex]\frac{1}{(\frac{1-c}{c})e^{-kt} +1 }[/tex] .
The given question is incomplete , the complete question is
One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor.
(a) Write a differential equation that is satisfied by y. (Use k for the constant of proportionality.)
(b) Solve the differential equation. Assume y(0) = C.
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Suppose that the daily number of miles driven per school bus driver in the United States is normally distributed with a mean of 300 mi and standard deviation equal to 26 mi. A local school bus budget committee believes the daily number of miles driven per bus driver in their community is different from 300 mi, but they are not sure if it is greater than or less than the national average.
To test their hypothesis, the committee chooses a random sample of 40 school bus drivers and records how many miles each of them drives per day during a typical week. On average, the bus drivers from the sample drive 309 mi per day. To test the hypothesis that the average number of miles driven by local bus drivers is different from the national average, the budget committee calculates the z-statistic to be 2.19 standard deviations above the population mean.
Use the standard normal z-distribution table to calculate the P-value that represents the probability of mistakenly rejecting the claim that the daily number of miles driven by local bus drivers is not statistically different from the national average. Give your answer as a decimal rounded to four places.
P-value:
The p-value for the z-distribution table is 0.4854.
What is z-distribution?
A unique type of normal distribution with a mean of 0 and a standard deviation of 1 is known as the standard normal distribution, or z-distribution. By transforming the values of any normal distribution into z scores, it is possible to standardise it. Z scores provide the number of standard deviations from the mean that each value falls within.
Given that the z statistic to be 2.19 standard deviation above the population mean.
2.19 can be written as 2.1 + 0.09
Z₂.₁₉ = Z₂.₁₊₀.₀₉ = 0.4854
Therefore p-value is 0.4854.
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60 POINTS!!! As a special promotion for its 12-ounce cans of cold coffee, a coffee drink company printed a message on the bottom of each can. Some of the bottoms read, "Better luck next time!" whereas others read, "You won!" The company advertised the promotion with the slogan "One in five wins a prize!" Suppose the company is telling the truth and that every 12-ounce can of coffee has a one-in-five chance of being a winner. Six friends each buy one 12-ounce can of coffee at a local convenience store. Let X equal the number of friends who win a prize.
Part A: Explain why X is a binomial random variable.
Part B: Find the mean and standard deviation of X. Interpret each value in context.
Part C: The store clerk is surprised when two of the friends win a prize. Is this group of friends just lucky, or is the company's one-in-five claim inaccurate? Compute P(X ≥ 2) and use the result to justify your answer.
A) Yes, X is a binomial random variable because all of the requirements for a binomial distribution have been established
B) The mean and standard deviation are respectively; 1.2 and 0.6
C) The probability given as P(X ≥ 2) is 0.34464
How to solve binomial probability distribution?The binomial probability is defined as the probability of exactly x successes on n repeated trials, with p probability.
The general formula of binomial probability distribution is;
P(X = x) = nCr * p^(x) * (1 - p)^(n - x)
where;
p = probability of "success"
n = number of trials, or the sample size
x = the number of "successes" in the probability we are trying to calculate.
A) We are told that 6 friends each buy one 12-ounce bottle of the soda at a local convenience store. Let X = the number who win a prize.
Probability of success; p = 1/5
Number of events; n = 6 which is the random variable being binomial random variable:
There are two types of accomplishments: successes and failures.
Candies are unrelated to one another.
There is a set number of buddies available.
In addition, the chance of success is fixed.
Because all of the requirements for a binomial distribution have been established, then it is a binomial probability distribution.
B) The formula for the mean here is;
μ = np
Plugging in the relevant values gives;
μ = 6(1/5) = 1.2
Formula for the standard deviation is;
σ = √(np(1 - p))
Thus;
σ = √(6 * 1/5)(1 - (1/5)))
σ = 0.6
C) The probability P(X ≥ 2) calculated with online binomial probability calculator gives;
P(X ≥ 2) = 0.34464
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When you are testing hypotheses by using proportions, what are the necessary requirements? OA. np 25 and nq 25 ?B.np < 5 and nq < 5 C. The population standard deviation is unknown. D. p
The following conditions must be met in order to test hypotheses using proportions: np ≥ 5 and nq ≥ 5
To test hypotheses using proportions, the following requirements are necessary:
np ≥ 5 and nq ≥ 5: It is necessary to have a sufficient number of samples in each group being compared.
The rule of thumb is to have at least 5 observations in each group, where n is the sample size and p is the proportion in that group.
This ensures that the sampling distribution of the proportion is approximately normal, which is necessary for hypothesis testing.
Hence, the correct answer would be an option (A).
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A. A manufacturer knows that their items have a normally distributed length, with a mean of 19.7 inches, and standard deviation of 4 inches.
If 25 items are chosen at random, what is the probability that their mean length is less than 17.4 inches?
Round to 4 decimal places
A manufacturer knows that their items have a normally distributed lifespan, with a mean of 12.1 years, and standard deviation of 1.9 years.
If you randomly purchase 12 items, what is the probability that their mean life will be longer than 11 years?
Round to 4 decimal places.
B. A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 98.4-cm and a standard deviation of 0.9-cm. For shipment, 15 steel rods are bundled together.
Find the probability that the average length of the rods in a randomly selected bundle is between 97.9-cm and 98.6-cm.
P(97.9-cm < X¯¯¯ < 98.6-cm) = Round to 4 decimal places.
The probability that their mean length is less than 17.4 inches is 28.26%
Given :
A manufacturer knows that their items have a normally distributed length, with a mean of 19.7 inches, and standard deviation of 4 inches , If 25 items are chosen at random .
z = ( X - μ ) / σ, where X = date , μ = mean , σ = standard deviation .
substitute the values
z = 17.4 - 19.7 / 4
= -2.3 / 4
= -0.575
P - value at z = -0.575 is 0.2826
Converting into percentage :
= 0.2826 * 100%
= 28.26 %
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A random sample of 115 observations results in 46 successes. Use Table 1.
a. Construct a 90% confidence interval for the population proportion of successes. (Round intermediate calculations to 4 decimal places, "z" value to 2 decimal places, and final answers to 3 decimal places.)
Confidence interval to
b. Construct a 90% confidence interval for the population proportion of failures. (Round intermediate calculations to 4 decimal places, "z" value to 2 decimal places, and final answers to 3 decimal places.)
Confidence interval to
Option a) 90% confidence interval for the population proportion of successes = 0.402
Option b) 90% confidence interval for the population proportion of failures = 0.598
According to the question given,
Critical values of Standard deviations are to be used to construct the proportions for the confidence interval.
The formula of a confidence interval that we will be using for the solution will be,
p = [tex]z_{a} * \sqrt{\frac{p(1-p)}{n} }[/tex]
In the above expression,
p = observed population.
n = sample size
[tex]z_{a}[/tex] = critical value of confidence interval
Now the given observations in question = 115
That result in success = 46
The resultant sample proportion = [tex]\frac{46}{115}[/tex]
⇒ 0.40 and it is for both success and failure
Here we can see that the answer for both parts will be identical.
The critical value of z at 90% confidence = 1.64
a) Lower bound = [tex]0.4 - 1.64 * \sqrt{\frac{0.4(1-0.4)}{90} }[/tex]
⇒ 0.402
b) Upper bound = [tex]0.4 + 1.64 * \sqrt{\frac{0.4(1-0.4)}{90} }[/tex]
⇒ 0.598
Confidence interval = 0.402 to 0.598
Therefore,
Option a) 90% confidence interval for the population proportion of successes = 0.402
Option b) 90% confidence interval for the population proportion of failures = 0.598
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Data is collected from the US Geologic Survey about the flow rate of the Poudre river. After some study, and unit conversions, a scientist makes a first model of the historical flow rate data as F(t) = 1.5 + 1.2 cos(2 t) in billions of cubic feet per year, while t is measured in fractions of a year since May 1, 2000. Use the Fundamental Theorem of Calculus to find the net amount of water from t = 0) to t = using this flow rate F"(t).First, F(t) = +C The model predicts the net amount of water discharged from the Poudre is F112 -Filo 0.75 What does the model predict the net discharge will be from t = 0 to t = 12 Net discharge = 1.5 Billions of ft 3
The model predicts that the net discharge will be -0.8 + 1.2 cos(24) billions of cubic feet from t = 0 to t = 12.
The net amount of water discharged from the Poudre river from t = 0 to t = 12 can be found using the Fundamental Theorem of Calculus, which states that the net change in a function over an interval is equal to the function evaluated at the endpoints of the interval minus the function evaluated at the beginning of the interval.
In this case, the net discharge from t = 0 to t = 12 can be calculated as follows:
Net discharge = F(12) - F(0)
= (1.5 + 1.2 cos(212)) - (1.5 + 1.2 cos(20))
= 1.5 + 1.2 cos(24) - 1.5 - 1.2
= -0.8 + 1.2 cos(24)
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Can someone please explain step by step on how I stole this problem?
-1 - ( -0.25 )
The final answer of the problem is -0.75.
What is the order of operations?
The order of operations is also known as the hierarchy of operations. The order of operations tells us the order in which we should perform mathematical operations in order to get the correct result. The order of operations is usually represented by the acronym PEMDAS:
P: Parentheses first
E: Exponents (ie Powers and Square Roots, etc.)
MD: Multiplication and Division (left-to-right)
AS: Addition and Subtraction (left-to-right)
We have,
-1 - (-0.25)
Using the order of operations, we can solve this problem as follows:
Parentheses: The first step is to perform any calculations inside parentheses. In this case, there are no parentheses, so we can move on to the next step.
Exponents: There are no exponents in this problem, so we can move on to the next step.
Multiplication and Division: There is no multiplication or division in this problem, so we can move on to the next step.
Addition and Subtraction: Finally, we can perform addition and subtraction in the problem. We start from left to right, so we start with the first operation: -1 - (-0.25). To subtract a negative number, we can use the rule that subtracting a negative number is the same as adding a positive number.
Therefore, we have -1 + 0.25.
The final step is to simplify the expression by performing the addition: -1 + 0.25 = -0.75.
Therefore, the final result of the problem is -0.75.
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Awnser if this is a function or not yes or no please.
Answer: NO
Step-by-step explanation:
This is not a function, since there are multiple outputs for ONE input.
Answer:
no
Step-by-step explanation:
to be a function, the x's can only have one partner. Here, 7 has two partners (7 goes to 20 and also to 18) and the 11 has two partners also. This relation is not a function.
Andreas has $18. Bonnie has 1 2/3 times as much as Andreas. Cheryl has 1 3/5 times as much as Bonnie.
How much money do
they have altogether?
Select the correct symbol.
9.48 ? 9.480
The 2 3/4 is greater than 1 5/6, hence 1 1/4 × 2 1/5 > 5/6 × 2 1/5
What is meant by Inequality expressions?An algebraic inequality is a mathematical statement that uses the inequality sign to connect an expression to a value, a variable, or another expression.
Relationships between two expressions that aren't equal to one another are known as inequalities. Inequalities are represented by the symbols, >,,, and. has the connotation "7 is greater than" (or "is less than 7," when read left to right).
Given the incomplete expression
1 1/4 × 2 1/5 _ 5/6 × 2 1/5
We are to fill it with the necessary inequality sign
For the expression:
1 1/4 × 2 1/5 = 5/4 * 11/5
1 1/4 × 2 1/5 = 55/20
1 1/4 × 2 1/5 = 2 3/4
For the expression 5/6 × 2 1/5
5/6 × 2 1/5 = 5/6 * 11/5
5/6 × 2 1/5 = 55/30
5/6 × 2 1/5 = 1 25/30
Since 2 3/4 is greater than 1 5/6, hence 1 1/4 × 2 1/5 > 5/6 × 2 1/5
The complete question is : Select the correct symbol to compare the expressions below.
1 1/4 × 2 1/5 _ 5/6 × 2 1/5
Symbols: <, >, =
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determine the number of different sets of quantum numbers possible for each of the following shells. n2
Answer:
Step-by-step explanation:
Explanation:
As you know, each electron that's part of an atom has its own unique set of four quantum numbers that describes its position and spin.
This means that looking for the number of unique sets of quantum numbers is equivalent to looking for the number of electrons that can occupy the third energy level.
As you know, the number of orbitals you get per energy level is given by the equation
no. of orbitals
=
n
2
, where
n
- the principal quantum number, the ones that gives the energy level.
So, if you're dealing with the third energy level, you can say that it will contain a total of
no. of orbitals
=
3
2
=
9
distinct orbitals.
Now, each orbital can contain a maximum of
2
electrons of opposite spins. This means that the third energy level will contain a maximum number of
no. of electron
=
9
⋅
2
=
18 e
−
So, if each electron is described by an unique set of quantum numbers, you can conclude that
18
sets of quantum numbers are possible for the third energy level.
ok maybe that will help!
Label a ray with endpoint Y. Determine an appropriate name for a ray whose endpoint is Y. Choose the correct ray whose endpoint is Y. A. Y X B. X Y C. X Y D. Y x Choose an appropriate name for a ray whose endpoint is Y. A. XY B. XY C. YX
Label a ray with endpoint Y. An appropriate name for a ray whose endpoint is XY displays a ray with the endpoint Y.
A line with a single defined endpoint that continuously moves away from another is known as a ray (endpoint B in this case).
A segment of a line that lacks an endpoint but has a specified starting point. It can go on forever in a single direction. A ray cannot be measured since it has no end point.
In mathematics, a ray is a segment of a line with a definite starting point but no ending. It can go on forever in a single direction. A ray cannot be measured since it has no end point. Two rays with the same terminal are united to form an angle. The vertex of the angle is referred to as the common endpoint of the rays, while the sides of the angle are referred to as the rays themselves.
The ray in the hypothetical situation has one endpoint B and an endlessly extending endpoint A. (as shown by the arrowhead).
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Review 2 HW Jack and Jill compare their monthly text plans. Jack's plan cost $9.00 plus $0.30 per text. Jills plan cost $0.60 per text. The equations below show the cost, c for each plan depending on how many texts, t, are sent Jack's Plan: c = 0.30t+9 Jills's Plan: c = 0.60t In one month, Jack and Jill paid the same amout. Enter the number of text messages the each sent.
Jack's plan: c=0.30t+9
Jill's plan: c=0.60t
I one month, Jack and Jill paid the same amount. enter the number of text messages they each sent
Answer: To solve this problem, we need to set the two equations equal to each other and solve for the number of text messages that were sent by each person. We know that the cost of Jack's plan is c = 0.30t + 9 and the cost of Jill's plan is c = 0.60t. To find the number of text messages they each sent, we can set these two equations equal to each other and solve for t:
0.30t + 9 = 0.60t
0.30t = 0.60t - 9
0.30t = 0.60t - 9
-0.30t = -9
t = 30
Therefore, if Jack and Jill paid the same amount for their text plans, they each sent 30 text messages.
Which points have a distance of 6 units between them?
Select all that apply.
A-(-12,2) and B-(-6,2)
B-(-6,2) and C-(-6-4)
C-(-6-4) and D-(6,2)
D-(6.2) and E=(6,4)
E-(64) and F=(-12.4) please help me with this
The points that have a distance of 6 units between them are A(-12, 2) and B(-6, 2); B(-6, 2) and C(-6, -4).
Distance between two points:The distance between two points P(x₁, y₁) and Q(x₂, y₂) is given by
Distance PQ = [tex]\sqrt{(x_{2} -x_{1})^{2} +(y_{2} - y_{1})^{2} }[/tex]Here we have
A (-12, 2) and B(-6, 2)
B (-6, 2) and C(-6, -4)
C (-6, -4) and D(6, 2)
D (6, 2) and E(6, 4)
E (6, 4) and F(-12, 4)
Here we will find the distance between given points to find the point which has 6 units between them.
As we know Distance = [tex]\sqrt{(x_{2} -x_{1})^{2} +(y_{2} - y_{1})^{2} }[/tex]
For Option A
A (-12, 2) and B(-6, 2)
Distance = √(-6 + 12)² + (2 - 2)²
= √[(6)² + (0)²] = √36 = 6
For Option B
For B (-6, 2) and C(-6, -4)
Distance = √(-6 - (-6))² + ( - 4 - 2)²
= √[(0)² + (-6)²] = √36 = 6
For Option C
For C (-6, -4) and D(6, 2)
Distance =√(6 - (-6))² + (2 - (-4))²
= √(12)² + (6)² = √180 = 6√5
For Option D
For D (6, 2) and E(6, 4)
Distance = √(4 - 2)² + ( 6 - 6)²
= √[(2)² + (0)²] = √4 = 2
For Option E
For E (6, 4) and F(-12, 4)
Distance = √(-12 - 6)² + (4 - 4)²
= √[(18)² + (0)²] = √18² = 18
Therefore,
The points that have a distance of 6 units between them are A(-12, 2) and B(-6, 2); B(-6, 2) and C(-6, -4).
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What is the rate of change of the linear function whose graph includes the points (5, 2) and (10, 0)?
gravel is being dumped from a conveyor belt at a rate of 40 cubic feet per minute. it forms a pile in the shape of a right circular cone whose base diameter and height are always equal. how fast is the height of the pile increasing when the pile is 16 feet high? recall that the volume of a right circular cone with height h and radius of the base r is given by v=(1/3)xpix(r^2)xh
The height of the pile increasing when the pile is 16 feet high is 5/32 ft/min.
We are given dV/dt=10 cubic ft/min and we want to find dh/dt. Therefore, first we need to write a formula relating v and h.
V=pi*r^2*h/3 since d=h at any moment, we know that r=h/2 at any moment and if we substitute h/2 for r, we get
V=pi(h/2)^2*h/3 and if we simplify this, we get
V=pi*(h^3)/12
Now we need to take the derivative of both sides with respect to t
dV/dt = pi * ( 3 * h^2 ) */12 * dh/dt
= pi * h^2 / 4 * dh/dt, substitute 40 for dV/dt and 16 for h and solve for dh/dt and we get
40 = pi * 16^2/4*dh/dt
dh/dt = 40 * 4 / ( 256 * pi )
= 40/(256*pi) ft/min,
= 10/64
= 5/32 ft/min
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help!!
find the measures of each angle
The measures of angles x, y and z are 40°, 50° and 130° and measures of the segments are AY = 16, IY = 9, FG = 30, AP = 24
What is centroid?Centroid is the point in a triangle where all the medians of the triangle intersects.
Given are two triangles, in first we have to find the missing angles and in the second one we need to find the asked segments,
1) We know that, in a right triangle, one angle is 90° and the sum of other two sides is 90°,
Therefore, 40°+y = 90°
y = 50°
x+y = 90°
x = 90°-50°
x = 40°
z = 180°-50° (linear pair)
z = 130°
2) we know that, the centroid divides the medians into 2:1
AY = 2YP
AY = 16
IY = TY/2
IY = 9
FG = GY+YF (segment addition postulate)
GY = YF/2
GY = 10
FG = 10+20 = 30
PA = AY+YP (segment addition postulate)
PA = 8+16
PA = 24
Hence. the measures are x, y and z are 40°, 50° and 130° and AY = 16, IY = 9, FG = 30, AP = 24
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52% of 70
what is the answer
Answer: 36.4
Step-by-step explanation:
Just convert the percentage to decimal and multiply.
70 x 0.52 = 36.4
Solve
y=-2 x
y=-9 x+21
Answer:
X = 3, Y = -6
Step-by-step explanation:
y = -2x
y = -9x + 21
Bring Y and X on the same side (alone).
0 = -2x – y
-21 = -9x – y
Multiply the top equation by -1 so we can cancel out y later on (to make it easier to solve for x).
0 = 2x + y
-21 = -9x – y
Add the equations.
-21 = -7x
Divide by -7.
X = 3
Solve for Y and check.
Y = -2(3)
Y = -6
Use the other equation to make sure.
(-6) = -9(3) + 21
-6 = -27 + 21
Have a nice day :D
Drag and drop the choices into the boxes to correctly complete the table.
corresponding alternate interior alternate exterior none of these
Definitions:
Corresponding angles are the angles at matching corners.Alternate interior angles lie on the inner side of the parallel lines but on the opposite side of the transversal.Alternate exterior angles lie on the outer side of the parallel lines but on the opposite side of the transversal.The given angles are:
Top left figure ⇒ Alternate interiorTop middle figure ⇒ CorrespondingTop right figure ⇒ CorrespondingBottom left figure ⇒ Alternate exteriorBottom right figure ⇒ None of thesee the problem.
1) Scores on a test are approximately normally distributed with a mean of 70 and a standard deviation of 9. The
teacher wants to give A's to the top 10% of students, B's to the next 25%, and C's to the next 42%. What is the
bottom cutoff for a C grade? Round your answer to the nearest whole number..
A) 68
B) 63
C) 77
D) 65
z = -1.34 < (a - 70)/9
And if we solve for we got
a = 70 - 1.34 * 9 = 57.95 = 98
So, the value of height that separates the bottom 9% of data from the top 91% is 58.
And the answer for this case would be:
a) 58
What is the Normal distribution and Z-score?
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
The solution to the problem
Let X be the random variable that represents the scores of a population, and for this case we know the distribution for X is given by:
X ~ N (70, 9)
Where μ = 70, and σ = 9
For this case, the figure attached illustrates the situation for this case.
We know from the figure that the lower limit for D accumulates 9% or 0.09 of the area below and 0.91 or 91% of the area above.
we want to find a value a, such that we satisfy this condition:
P(X > a) = 0.91 (a)
P(X < a) = 0.09 (b)
Both conditions are equivalent in this case. We can use the z score again in order to find the value a.
As we can see in the figure attached the z value that satisfies the condition with 0.09 of the area on the left and 0.91 of the area on the right it's z=-1.34. On this case P(Z<-1.34)=0.09 and P(z>-1.34)=0.91
If we use condition (b) from the previous we have this:
P(X < a) = P(X - μ)/σ < (a - μ)/σ) = 0.09
P(z < (a - μ)/σ) = 0.09
But we know which value of z satisfies the previous equation so then we can do this:
z = -1.34 < (a - 70)/9
And if we solve for we got
a = 70 - 1.34 * 9 = 57.95 = 98
Hence, the value of height that separates the bottom 9% of data from the top 91% is 58.
And the answer for this case would be:
a) 58
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Forest Fires and Acres Burned Numbers (in thousands) of forest fires over the year and the number (in hundred thousands) of acres burned for 7 recent years are shown. Number of fires x 69 58 47 84 62 57 72 Number of acres burned y 64 53 42 79 57 52 67 The correlation coefficient for the data is r=1 and α=0.05. Should regression analysis be done? The regression analysis should not be done. The regression analysis should be done. Find the equation of the regression line. y′=a+bx a= Find y' when x=50
The equation of the regression line is y = 4 + x and the regression analysis of the given data need not be done.
Straight line equation is y = a + bx.
The normal equations are
∑y = a·n + b ∑x ∑ xy = a ∑x + b∑x² Hence the regression analysis of the data :
The values are calculated using the following table
x y x² x⋅y
58 62 3364 3596
47 51 2209 2397
84 88 7056 7392
62 66 3844 4092
57 61 3249 3477
72 76 5184 5472
69 73 4761 5037
hence :
∑x = 449
∑y = 477
∑x² = 29667
∑x⋅y = 31463
Substituting these values in the normal equations
7a+449b=477
449a+29667b=31463
Solving these two equations using Elimination method,
7a+449b=477
and 449a+29667b=31463
7a+449b=477→(1)
449a+29667b=31463→(2)
equation(1)×449
⇒3143a+201601b=214173
equation(2)×7
⇒3143a+207669b=220241
Substracting
⇒-6068b = -6068
⇒6068b = 6068
⇒b = 1
Putting b = 1 in equation (1), we have
7a+449(1) = 477
⇒7a = 477 - 449
⇒7a = 28 or a = 4.
Now Estimate y for x=50
y = a + bx , we get
y = 4 + 1×50
y = 4 + 50
y = 54
At x equals 50 we get the value of y as 54 .
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Please help me with this question
The average rate change of the function is -4.
What is a function?
A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
Given function is f(x) = 1x² + 1x - 8.
Find the average rate change of the function in the interval [-4, -1]
For this compute the value of f(x) at x = -4 and x = -1.
Then find the difference the value of f(x) and divide it by the difference the value of x.
The average rate change of function f(x) in the interval [a,b] is [f(b) - f(a)]/(b-a).
Putting x = -4 in the given function:
f(-4) = 1(-4)² + 1(-4) - 8
f(-4) = 4
Putting x = -1 in the given function:
f(-1) = 1(-1)² + 1(-1) - 8
f(-1) = -8.
The rate change of x is -1 - (-4) = 3
The average rate change of the function is
[f(-1) - f(-4)] /3
= -8 - 4/3
= - 12/3
= -4
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I'm not sure what the reasoning is
The reasons that complete the statements are:
Statement 2: GivenStatement 4: SAS congruence propertyHow to determine the reasons that complete the statementsFrom the question, we have the following parameters that can be used in our computation:
Triangles = RST, and TUR
From the question, we have the following given parameters
∠SRT ≅ ∠UTR
This means that
Statement 2: ∠SRT ≅ ∠UTR ------ Given
At this point, we have
RS ≅ TU --- Congruent side (S)
∠SRT ≅ ∠UTR ------ Congruent angle (A)
RT ≅ TR --- Congruent side (S)
When these congruent sides and angles are combined, we have
SAS congruence property
The SAS congruent theorem implies that the corresponding sides of the triangles in question are congruent and the angles between the corresponding sides are also congruent
Hence, the additional information on the congruency of the triangles are Given and SAS congruence property
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2. Of the total pounds, x, of fudge at the fudge shop on Monday morning, {3}/{4} contain nuts. Helena works at the fudge shop and prepares 12 {3}/{4} more pounds of fudge containing nuts to fulfill a special order Monday afternoon. With this additional amount of fudge, there are at least 29 {5}/{8} pounds of fudge containing nuts at the fudge shop on Monday.
(a) Write an inequality that represents the scenario. Begin by defining your variable.
(b) Solve your inequality from Part (a). Show all work.
(c) Fudge is packaged into 1-pound boxes. What is the fewest number of boxes of fudge at the fudge shop Monday morning?
Answer:
At least means less than or equal to.
Fewest means the smallest number of boxes.
12 3/4 +x ≥ 29 5/8
x ≥ 16 7/8
The fewest number of boxes would be almost ≅ 42 boxes
Part A:
Let x denote the amount of fudge at the shop.
Helena prepares 12 3/4 pounds of fudge containing nuts.
Mathematically, the total amount of fudge on Monday will be
12 3/4 +x
Of this amount of fudge there are at least 29 5/8 pounds of fudge containing nuts at the fudge shop on Monday
12 3/4 +x ≥ 29 5/8
Part B:
The inequality can be solved
12 3/4 +x ≥ 29 5/8
x ≥ 29 5/8- 12 3/4
x ≥ 29- 12 5/8- 3/4
The whole numbers are subtracted and the fractions are solved separately to avoid long calculations.
x ≥ 17 (5-6)/8
x ≥ 17 - 1/8
x ≥ (136-1)/8
x ≥ (135)/8
x ≥ 16 7/8
Part C :
To find the number of boxes we divide the equal number of pounds .
12 3/4 + 29 5/8
12 + 29 3/4 + 5/8
41 11/8
42 3/8
The fewest number of boxes would be almost ≅ 42 boxes
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3. Which of the following is the solution to lx + 12| = 3x.
A. x = 3
C. x = 6
b. x = -6
d. x = - 3
Find the LCM of the numbers by using the GCF.
5. 22,38
The lowest common multiple (LCM) of the numbers 22 and 38 is 418.
What is L.C.M and H.C.FThe L.C.M. defines the least common multiple number which is exactly divisible by two or more numbers, while H.C.F. defines the highest common factor present in between given two or more numbers,
Given the numbers 22 and 38, we can express them as products of their prime factors and thus derive their LCM as follows;
22 = 2 × 11
38 = 2 × 19
LCM = 2 × 11 × 19
LCM = 418
Therefore, the LCM of 22 and 38 is 418 using prime factor method.
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Suppose that in an election voter preference is sharply divided along gender lines. 60% of women will vote for Candidate J, the rest will vote for Candidate K; 30% of men will vote for Candidate J, the rest for Candidate K. Which of the following represents the lowest percentage of voters that must be women on election day, in order that Candidate J wins the election?
* 65%
* 67%
* 69%
* 71%
* 73%
By using proportions, it is discovered that 67% of voters who are female and vote for Candidate J to win.
What exactly is a ratio ?A percentage is a component of a total amount, and the three-step rule is used to relate the measurements.
The percentage of voters who select Candidate J is:
60% of x are voters are female
30% of (1 - x), who vote are male
If the total of these percentages is larger than 50% = 0.5, Candidate J will be elected; therefore:
0.6x + 0.3(1 - x) > 0.5
0.6x + 0.3 - 0.3x > 0.5
0.3x > 0.2
x > 2/3
x > 0.667
x > 67%
By using proportions, it is discovered that 67% of voters who are female and vote for Candidate J to win.
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Dae and Eric are working together on a paper. Dae typed 333 words in 9 minutes, and Eric typed 252 words in 6 minutes. How many more words per minute did Eric type?
Eric typed 15 words more than Dae per minute.
What is a Linear Equation?Linear equations are equations whose highest power of the variables is 1. The graph of a linear equation is a straight line. A nonlinear equation is an equation whose highest power of the variables is not 1 and the graph is not a straight line.
To find the amount of words typed in 1 minute for both;
Dae = 333/9
Dae = 27 words per minute
Eric = 252/6
Eric = 42 words per minute
Therefore 42 - 27 = 15 words per minute
Eric typed 15 more words per minute.
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A student completes ⅘ of a math project in ⅔ hour. If the student continues at the same rate, how long will it take the student to complete the project?
The number of hours it will take the student to complete the project is 5 / 6 hours.
How to find the time the project will be completed?A student completes ⅘ of a math project in ⅔ hour. If the student continues at the same rate, the time it will take the student to complete the project can be calculated as follows:
Therefore,
let
x = time it will take the student to complete the project
Hence
if 4 / 5 x = 2 / 3 hours
x = ? hours
cross multiply
Therefore,
number of hours to complete the whole math project = x × 2 / 3 ÷ 4 / 5 x
number of hours to complete the whole math project = 2 / 3 x × 5 / 4x
number of hours to complete the whole math project = 10x / 12x
number of hours to complete the whole math project = 10 / 12
number of hours to complete the whole math project = 5 / 6 hours
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