Answer:
(7,10)
Step-by-step explanation:
x=4+3
y=5+5
In how many ways can we distribute the 52 cards deck if we want to give to Sara 17 cards, to Jacob 17 cards and to their Mam 18 cards? 1) 52!/17!17!18!
The number of ways in which 52 cards can be distributed such that Sara receives 17 cards, Jacob receives 17 cards, and their mother receives 18 cards is given by the following expression:52!/17!17!18!
Explanation: The number of ways to distribute k objects among n persons in which the order does not matter and each person receives at least one object is given by the following expression: ((k - n) choose (n - 1)). This can be extended to the case where each person is required to receive a specific number of objects. For example, if we have k objects and want to distribute them to persons A, B, and C such that A receives a objects, B receives b objects, and C receives c objects, where a + b + c = k, then the number of ways to do this is given by the expression: ((k - a - b - c) choose (2))This can be simplified as follows: ((k - a - b - c)!)/((2!)(k - a - b - c - 2)!)), which can be further simplified as follows: (k - a - b - c)(k - a - b - c - 1)/2!.
Therefore, the number of ways in which 52 cards can be distributed such that Sara receives 17 cards, Jacob receives 17 cards, and their mother receives 18 cards is given by: ((52 - 17 - 17 - 18) choose (2))= ((52 - 52) choose (2))= (0 choose 2)=0. Therefore, the required number of ways is 52!/17!17!18!.
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Linear programming can be used to identify the critiacal path for a PERT network.
Group of answer choices
True
False
The probabilistic approach characterized by PERT does not use:
Group of answer choices
Median activity times
Most likely activity times
Optimistic activity times
Pessimistic activity times
The critical path:
Group of answer choices
Shows, by elimination, which activities management can safely ignore.
Is the sequential path of all of the activities from the first until the last
Is defined by activities with zero slack.
Is the shortest path through the network.
Linear programming can be used to find the optimal solution for profit, but cannot be used for nonprofit organizations.
Group of answer choices
False
True
1- True: Linear programming can be used to identify the critical path for a PERT network.
2- False: The probabilistic approach characterized by PERT does not use median activity times.
3- False: The critical path is the sequential path of activities from the first to the last, defined by activities with zero slack.
4- False: Linear programming can be used for both profit and nonprofit organizations.
Linear programming can indeed be used to identify the critical path for a PERT network. The critical path represents the sequential path of activities from the first activity to the last, and it is defined by activities with zero slack. By analyzing the dependencies and constraints between activities, linear programming techniques can be applied to determine the optimal schedule and identify the critical path.
The probabilistic approach characterised by PERT does not use median activity times. Instead, it utilises three time estimates for each activity: optimistic, most likely, and pessimistic activity times. These estimates are used to calculate the expected duration and variability of the project.
Regarding the statement about linear programming and profit optimization, it is incorrect to say that linear programming cannot be used for nonprofit organizations. Linear programming can be applied to various optimisation problems, including those related to resource allocation, cost minimisation, and efficiency improvement, which are relevant to both for-profit and nonprofit organisations.
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write a equation in slope intersect form for points (0,8) and (9,5)
y = mx+b
(5-8)/(9-0) = -1/3 which is m
solve point slope to find slope intercept by using one of the points (0,8)
y-y1=m(x-x1)
y-8 = -1/3(x-0)
y-8 = -1/3x
Answer: y = -1/3x + 8
ues a net to find the surface area of the pyramid?
1.Explain why we need unit root test for stationary and the meaning of a spurious regression.
2.What are autocorrelation and Durbin-Watson test? And how are they related?
3. Explain the concept of VAR and VEC model and how they differ?
4.Explain the characteristic of an ARDL and its application
5.Explain the concept of cointegration and show how to perform the test for cointegration
6.Briefly explain these following tasks of heteroskedasticity: (1) the meaning of heteroskedasticity; (2) how to detect heteroskedasticity; (3) heteroskedasticity consequences for the OLS Estimation
1. Unit root tests are used to determine whether a time series variable is stationary or contains a unit root. Stationarity is a property of a time series where its statistical properties (such as mean, variance, and autocovariance) remain constant over time.
Unit root tests are important because many econometric models and statistical techniques assume stationarity. If a variable is non-stationary, it can lead to spurious regression.
Unit root tests help identify such cases by testing the null hypothesis of a unit root presence in the time series.
2. Autocorrelation refers to the correlation between the observations of a time series with their lagged values. It indicates the presence of a systematic relationship or dependence between the current observation and past observations.
The Durbin-Watson test is a statistical test used to detect autocorrelation in the residuals of a regression model.
The Durbin-Watson test statistic ranges from 0 to 4. A value close to 2 indicates no significant autocorrelation, while values significantly below 2 suggest positive autocorrelation, and values significantly above 2 suggest negative autocorrelation.
3. VAR models represent a system of equations where each variable is regressed on its own lagged values and the lagged values of all other variables in the system.
VAR models are widely used for forecasting, impulse response analysis, and studying dynamic relationships in macroeconomic and financial data.
VEC models, on the other hand, are a special case of VAR models designed to capture long-run equilibrium relationships among variables. VEC models incorporate error correction terms that help adjust for any deviations from the long-run equilibrium.
They are particularly useful when studying variables that exhibit cointegration, as they allow for the analysis of both short-run dynamics and long-run equilibrium relationships.
4. The Autoregressive Distributed Lag (ARDL) model is a regression model commonly used when dealing with time series data that may have a mix of stationary and non-stationary variables.
The ARDL model finds applications in macroeconomics, finance, and other fields where the relationship between variables may exhibit mixed order of integration.
5. Cointegration refers to the long-run equilibrium relationship between non-stationary time series variables. Cointegration implies that a linear combination of the variables is stationary, indicating a stable relationship.
6. Heteroskedasticity refers to the condition where the variance of the error term in a regression model is not constant across all levels of the independent variables. This violates the assumption of homoscedasticity, which assumes constant variance.
To detect heteroskedasticity, several methods can be used:
a) Graphical Analysis: Plotting the residuals against the predicted values or the independent variables to visually examine patterns of heteroskedasticity.
b) White's Test: A statistical test that regresses the squared residuals on the independent variables to test for heteroskedasticity.
Heteroskedasticity has consequences for Ordinary Least Squares (OLS) estimation:
a) OLS estimates of coefficients remain unbiased, but they are no longer efficient (standard errors are incorrect).
b) The t-tests and F-tests become invalid, leading to incorrect inference.
c) Confidence intervals and hypothesis tests may be distorted.
Correcting for heteroskedasticity can be done using robust standard errors or weighted least squares (WLS) estimation, which takes into account the heteroskedasticity structure of the error terms.
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Andrew must cut a rope 9 1/7 yards long into 8 equal
pieces. How long will each piece of rope be?
Answer:
each rope should be 1 1/7 yard long
Step-by-step explanation:
9 1/7 ÷8
1 1/7
Answer:
8/7 yd per piece
Step-by-step explanation:
To answer this, divide the total rope length 9 1/7 yd by 8 pieces:
64 yd
------------------ = 8/7 yd per piece
7(8 pieces)
Check: does 8 times (8/7 yd/piece) come out to 64/7 yd, or 9 1/7 yd? YES
The Spirit Team will be selling popcorn as a fundraiser at the next basketball game to earn funds to go to Orlando for a competition. If they need the volume of the popcorn container to be under 220 cubic inches per serving, select all of the following dimensions of a cylinder they could use.
3 inch radius and 4 inch height
4 inch radius and 4 inch height
9 inch radius and 9 inch height
4 inch diameter and 9 inch height
6 inch diameter and 9 inch height
9 inch diameter and 4 inch height
Answer:A and b
Step-by-step explanation:
The dimensions of cylinders that have a volume under 220 cubic inches per serving are:
3-inch radius and 4-inch height
4-inch diameter and 4-inch height
4-inch diameter and 9-inch height
Volume is defined as the mass of the object per unit density while for geometry it is calculated as profile area multiplied by the length at which that profile is extruded.
Here,
We can use the formula V = πr²h to find the volume of each cylinder.
For a cylinder with a 3-inch radius and 4-inch height:
V = π(3)²(4) ≈ 113.1 cubic inches
For a cylinder with a 4-inch radius and 4-inch height:
V = π(4)²(4) ≈ 200.96 cubic inches
For a cylinder with a 9-inch radius and 9-inch height:
V = π(9)²(9) ≈ 2289.0 cubic inches
For a cylinder with a 4-inch diameter and 9-inch height (radius is 2 inches):
V = π(2)²(9) ≈ 113.04 cubic inches
For a cylinder with a 6-inch diameter and 9-inch height (radius is 3 inches):
V = π(3)²(9) ≈ 254.5 cubic inches
For a cylinder with a 9-inch diameter and 4-inch height (radius is 4.5 inches):
V = π(4.5)²(4) ≈ 254.5 cubic inches
Therefore, the dimensions of cylinders that have a volume under 220 cubic inches per serving are:
3-inch radius and 4-inch height
4-inch diameter and 4-inch height
4-inch diameter and 9-inch height
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The partial sum - 2 + ( − 6) + ( − 18) + ...... + (-486) =__________
The partial sum of the given sequence, -2 + (-6) + (-18) + ... + (-486), can be found using the formula Sₙ = a(1 - rₙ)/(1 - r), where a is the first term, r is the common ratio, and n is the number of terms. The value of the partial sum is 728.
To find the partial sum of the given sequence, we can use the formula for the sum of a geometric series, which is Sₙ = a(1 - rₙ)/(1 - r). In this case, the first term a is -2, and the common ratio r is -3. We need to determine the number of terms, n.
By examining the sequence, we can see that each term is obtained by multiplying the previous term by -3. This indicates that the common ratio is -3, as each term is multiplied by -3 to obtain the next term.
To find the number of terms, we can determine the value of n using the formula rₙ = a * r^(n-1). In this case, we have -486 = -2 * (-3)^(n-1).
By solving this equation, we find n = 6.
Substituting the values into the formula for the partial sum, we have:
S₆ = -2(1 - (-3)^6)/(1 - (-3)),
= -2(1 - 729)/(1 + 3),
= -2(-728)/4,
= 728.
Therefore, the partial sum of the given sequence is 728.
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2) A 95% confidence interval estimate for a population mean u is (23, 45). Which of the following is a true statement?
(A) There is 0.95 probability that μ is between 23 and 45.
(B) If 95% confidence intervals are calculated from all possible samples of the given size, μ will be in 95% of these intervals.
(C) If 95% confidence intervals are calculated from all possible samples of the given size, 95% of them will be
(23, 45).
(D) We are 95% confidence that the interval from (23, 45) contains the sample mean x
(E) The margin of error of this confidence interval is 22.
The correct statement for the 95% confidence interval is given by
option (B) If 95% confidence intervals are calculated from all possible samples of the given size, μ will be in 95% of these intervals.
Confidence interval = 95%
Population mean μ
A confidence interval is an estimate of a population parameter the population mean μ based on sample data.
The interpretation of a 95% confidence interval is that ,
Sample from the population and construct 95% confidence intervals,
Approximately 95% of these intervals would contain the true population parameter.
Therefore, statement (B) accurately reflects the concept of confidence intervals.
It states that if we calculate 95% confidence intervals from all possible samples of the given size,
The true population mean μ will be within 95% of these intervals.
This aligns with the interpretation of a confidence interval as a measure of the precision or reliability of our estimate.
The other statements which are not accurate,
(A) There is no probability associated with a specific confidence interval.
Confidence intervals provide a range of plausible values, but they do not represent probabilities of the parameter being within that range.
(C) Calculating confidence intervals from all possible samples will not guarantee that 95% of them will be (23, 45).
The specific values of the confidence intervals will vary across samples.
(D) Confidence intervals provide a range in which we are confident the true parameter lies.
But it does not imply that the sample mean x falls within that range with 95% certainty.
(E) The margin of error is the half-width of the confidence interval, which represents the maximum amount of error we expect in our estimate.
Here, the margin of error would be (45 - 23) / 2 = 11, not 22.
Therefore , for the confidence interval 95% option B is correct.
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Please help me with this question
Answer:
c is your answer
Step-by-step explanation:
U is the set of integers. G is the set of negative integers. What is the complement of set G in universive U?
Pls help fast
Answer:
Wouldn't the answer be C) Positive Integers?
Step-by-step explanation: Because the complement of a set include all of the elements not included in the indicated set. I hope I'm making some sense. :)
You are given the line y=-2x-5, and it then shifted up 2 units. Write your equation of the new line.
Answer:
i think the answer is 119943147893471987 yes its as easy as 1+1
Step-by-step explanation:
Answer: y=-2x-3
Step-by-step explanation:
If 10 is the area of a circle what is the radius?
Answer: 1.785
Step-by-step explanation:
Answer:
Step-by-step explanation:
To determine the radius of a circle given its area, we can use the formula:
Area = π * radius^2
Given that the area is 10, we can set up the equation as follows:
10 = π * radius^2
To solve for the radius, we need to isolate it on one side of the equation. Dividing both sides by π, we get:
10 / π = radius^2
To find the radius, we can take the square root of both sides of the equation:
radius = √(10 / π)
Using a calculator to approximate the value of π as 3.14159, we can calculate:
radius ≈ √(10 / 3.14159)
radius ≈ √(3.1831)
radius ≈ 1.7849
Therefore, the radius of the circle is approximately 1.7849 when the area is 10.
hope it helps!!
Could someone plz help and show work? Thanks
Answer:
3 cm
Step-by-step explanation:
What is the length of the diameter?
is there any instructions above the circle?
Budget planners for a certain community have determined that $3,000,000 wel be required to provide a povernment service rester. The total property value in the communty 120,000,000 wat tax rate is required to meet the budgetary demands?
The tax rate required to meet the budgetary demands is 2.5%.
According to the given information;
Total property value in the community = $120,000,000
Total amount required to provide a government service = $3,000,000
Now, to find the tax rate required to meet the budgetary demands we will use the formula;
Tax Rate = (Total amount required to provide a government service / Total property value in the community) × 100
Substitute the given values in the above formula;
Tax Rate = ($3,000,000 / $120,000,000) × 100= 2.5%
Thus, the tax rate required to meet the budgetary demands for a community is 2.5 percent.
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Please help! I know its a lot and I'm sorry but I REALLY NEED HELP!!! I just don't understand this and I don't want to fail my brain just is not smart with math. Even if you answer just ONE question it would mean the WORLD TO ME thanks!
Answer:16
Step-by-step explanation:
Let f, g and h be the functions from the set of integers to the set of integers defined by f(x) = 2x +3, g(x) = 3x + 2 and h(x) = x3 +1.
(a) Find (fºg)(x) (b) Find (gof)(x) (c) l'ind (f)(x) (d) Find (h+h)(x) (e) Find h-1(x)
Let f, g and h be the functions from the set of integers to the set of integers defined by f(x) = 2x +3, g(x) = 3x + 2 and h(x) = x3 +1.
(a) To find (f º g)(x), we substitute g(x) into f(x) as follows:
(f º g)(x) = f(g(x)) = f(3x + 2) = 2(3x + 2) + 3 = 6x + 4 + 3 = 6x + 9.
(b) To find (g º f)(x), we substitute f(x) into g(x) as follows:
(g º f)(x) = g(f(x)) = g(2x + 3) = 3(2x + 3) + 2 = 6x + 9 + 2 = 6x + 11.
(c) To find the inverse of f(x), denoted as l'ind (f)(x), we solve for x in terms of f(x):
x = (f(x) - 3) / 2.
Rearranging the equation, we get f^(-1)(x) = 1/2x - 3/2.
(d) To find (h + h)(x), we add h(x) to itself:
(h + h)(x) = h(x) + h(x) = ([tex]x^3[/tex] + 1) + (x^3 + 1) = 2[tex]x^3[/tex] + 2.
(e) To find the inverse of h(x), denoted as h^(-1)(x), we solve for x in terms of h(x):
x = (h(x) - 1)^(1/3).
Rearranging the equation, we get h^(-1)(x) = (x - 1)^(1/3).
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HELLPP WORTH 20 POINTS ✨
Answer:
it's radius is 9 and diameter is 18
Answer:
diameter=18cm
radius =diameter/2=18/2=9cm
Rachel’s mother tells her she can play at the arcade as long as the cost does not exceed $24. Each game costs $1.50.
Answer:
16
Step-by-step explanation:
If each game is $1.50 we would divide the 24 by 1.50, or multiple 1.50 to find 24. So 24 divided by 1.50 is 16.
So she could play 16 games.
Can someone help me with this. Will Mark brainliest. Need answer and explanation/work. Thank you.
Answer:
tangent x = 12/9
Step-by-step explanation:
Hello There!
Once again these are the Trigonometric Ratios
SOH CAH TOA
Sine = Opposite over Hypotenuse (SOH)
Cosine = Adjacent over Hypotenuse (CAH)
Tan = Opposite over Adjacent (TOA)
And for this one we are asked to find the tangent of angle x
Remember tangent = opposite over adjacent
opposite - the side length that is opposite of the angle (so the angle will be facing that side length)
adjacent - the side length that is not the opposite nor the hypotenuse
The opposite of angle x is 12 and the adjacent of angle x is 9
so tangent x = 12/9
What is the solution to the equation 3x + 2(x – 9) = 8x + X - 14?
+
o
-8
0-1
o
1
08
Answer:
Your answer will be x= -1
Step-by-step explanation:
have a nice day:)
PLEASE HELP ASAP!!! WILL MAKE BRAINLIEST!
Answer: Figure B is 2x bigger on all sides
Explanation: You can see that each of the numbers times 2 would equal all of the numbers on B.
Use the following function rule to find f(6).
f(x)=6x+11
Answer:
X=-11/6
Steps
f(x)=6x+11
simplify,
0=6x+11
-6x+11
Divide both sides,
Answer is x = -¹¹/6
Help me please ahhhh...Simplify the expression below.
2.5x. 4
Answer:
Maybe multiply 2 and 4 then multiply 5x of product of 2 and 4
Step-by-step explanation:
What would the command "-index(F5:F277,randbetween(1.273))" do when entered in the spreadsheet with survey responses that we use for Project 2 (and will use for Project 3)? Return the most frequent answer to "Pineapple on pizza?" Return a random number between 1 and 273. Average the values in Column F. Change an answer in Column F at random. Pick a response at random from the responses to the question "Pineapple on pizza?" Return the greatest response to a random question
The command "-index (F5:F277, rand between (1.273))" when entered in the spreadsheet with survey responses that we use for Project 2 (and will use for Project 3) would pick a response at random from the responses to the question.
So, the correct option is: Pick a response at random from the responses to the question "Pineapple on pizza."
The INDEX function is an Excel worksheet function that finds the value or reference to a value within an array. It returns a reference to the location of the value, rather than the value itself. The INDEX function in Excel is a lookup and reference function.
The INDEX function allows you to search a spreadsheet and find the value contained in a given cell. The INDEX function takes two arguments, the array and the index number. The array is the range of cells that you want to search, while the index number is the position of the value you want to return.
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Estimate the derivative using forward finite divided difference applying both truncated and more accurate formula using 0.5 and step sizes of ht=0.25 and tu=0.125 4x12x2 + x3 -1 #x) = 5 + 3sinu = 2x1 + x2 + x3 = 4 2xy + 2x2 + x3 = 3
The more accurate forward finite divided difference estimates for the derivatives are
f₁'(x₁) = 0
f₂'(x₂) = 0
f₃'(x₃) = 0
To make it easier to work with, let's rearrange the equations in terms of the variables:
4x₁ + 2x₂ + x₃ = 1
2x₁ + x₂ + x₃ = 4
2x₁ + 2x₂ + x₃ = 3
The truncated formula for estimating the derivative using the forward finite divided difference is given by:
f'(x) ≈ (f(x + ht) - f(x)) / ht
Here, f(x) represents the function we want to differentiate, and ht is the step size.
Let's calculate the derivatives using the truncated formula for the given equations:
For x₁:
f₁'(x₁) ≈ (f₁(x₁ + ht) - f₁(x₁)) / ht
= (4(x₁ + ht) + 2x₂ + x₃ - 4x₁ - 2x₂ - x₃) / ht
= (4x₁ + 4ht + 2x₂ + x₃ - 4x₁ - 2x₂ - x₃) / ht
= (4ht) / ht
= 4
Similarly, we can calculate the derivatives for x₂ and x₃.
For x₂:
f₂'(x₂) ≈ (f₂(x₂ + ht) - f₂(x₂)) / ht
= (2x₁ + (x₂ + ht) + x₃ - 2x₁ - x₂ - x₃) / ht
= (x₂ + ht - x₂) / ht
= ht / ht
= 1
For x₃:
f₃'(x₃) ≈ (f₃(x₃ + ht) - f₃(x₃)) / ht
= (2x₁ + 2x₂ + (x₃ + ht) - 2x₁ - 2x₂ - x₃) / ht
= (x₃ + ht - x₃) / ht
= ht / ht
= 1
So, the truncated forward finite divided difference estimates for the derivatives are:
f₁'(x₁) = 4
f₂'(x₂) = 1
f₃'(x₃) = 1
The more accurate formula for estimating the derivative using the forward finite divided difference is given by:
f'(x) ≈ (-3f(x) + 4f(x + ht) - f(x + 2ht)) / (2ht)
Let's calculate the derivatives using the more accurate formula for the given equations:
For x₁:
f₁'(x₁) ≈ (-3f₁(x₁) + 4f₁(x₁ + ht) - f₁(x₁ + 2ht)) / (2ht)
= (-3(4x₁ + 2x₂ + x₃) + 4(4(x₁ + ht) + 2x₂ + x₃) - (4(x₁ + 2ht) + 2x₂ + x₃)) / (2ht)
= (-12x₁ - 6x₂ - 3x₃ + 16x₁ + 8ht + 4x₂ + 2x₃ - 4x₁ - 8ht - 2x₂ - x₃) / (2ht)
= (-12x₁ + 16x₁ - 4x₁ + 8ht - 8ht) / (2ht)
= 0
Similarly, we can calculate the derivatives for x₂ and x₃.
For x₂:
f₂'(x₂) ≈ (-3f₂(x₂) + 4f₂(x₂ + ht) - f₂(x₂ + 2ht)) / (2ht)
= (-3(2x₁ + x₂ + x₃) + 4(2x₁ + (x₂ + ht) + x₃) - (2x₁ + (x₂ + 2ht) + x₃)) / (2ht)
= (-6x₁ - 3x₂ - 3x₃ + 8x₁ + 4x₂ + 4ht + 4x₃ - 2x₁ - x₂ - x₃) / (2ht)
= (-6x₁ + 8x₁ - 2x₁ - 3x₂ + 4x₂ - x₂ - 3x₃ + 4x₃ - x₃ + 4ht) / (2ht)
= 0
For x₃:
f₃'(x₃) ≈ (-3f₃(x₃) + 4f₃(x₃ + ht) - f₃(x₃ + 2ht)) / (2ht)
= (-3(2x₁ + 2x₂ + x₃) + 4(2x₁ + 2x₂ + (x₃ + ht)) - (2x₁ + 2x₂ + (x₃ + 2ht))) / (2ht)
= (-6x₁ - 6x₂ - 3x₃ + 8x₁ + 8x₂ + 4x₃ + 4ht - 2x₁ - 2x₂ - x₃) / (2ht)
= (-6x₁ + 8x₁ - 2x₁ - 6x₂ + 8x₂ - 2x₂ - 3x₃ + 4x₃ - x₃ + 4ht) / (2ht)
= 0
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Mrs. Habib has 46.25 feet of border for a bulletin board for her classroom. the board is 37.5 feet tall and 8.3 feet wide. how many feet of border will Mrs habib have left after she puts border around the board?
It’s not 22.15 I’ve tried.
Answer:
Mrs. Habib will have 22.25 feet of border left after she puts border around the board.
Step-by-step explanation:
You must find the perimeter of the board and subtract it from the amount of border she has to find how much she will have left after she uses it. The formula for perimeter is [tex]P=2(l+w)[/tex], where [tex]l=[/tex] the length of the board, and [tex]w=[/tex] the width of the board. You will add those together and multiply them by 2 because there are 4 sides to a rectangle. That means this equation will look like:
[tex]P=2(8.25+3.75)[/tex]
Now you can just solve for the perimeter.
[tex]P=2(12)[/tex]
[tex]P=24[/tex]
The perimeter is 24 feet. That means it will take 24 feet of border to cover her board. In order to find out how much she'll have left over, just subtract 24 from the total amount of border she has.
[tex]46.25-24=22.25[/tex]
Therefore Mrs. Habib will have 22.25 feet of border left over after she covers the bulletin board.
A line has an undefined slope and includes the point (-10, 6) and (q, 0) what is the value q
Answer:
q = -10
Step-by-step explanation:
If the slope is undefined, then there is no change in x. Therefore, since -10-(-10) = 0, then q=-10.
Show that the function f(x) f(x) = x3, x < 0 1 x2 sin, x > 0 x is differentiable.
To show that the function f(x) = x³ for x < 0 and f(x) = x²sin(x) for x > 0 is differentiable, we need to demonstrate that the function has a derivative at every point in its domain.
Let's consider the function f(x) separately for x < 0 and x > 0.
For x < 0
In this case, f(x) = x³. The power rule tells us that the derivative of xⁿ with respect to x is nxⁿ⁻¹. Applying this rule, we find that the derivative of f(x) = x³ is f'(x) = 3x².
For x > 0
In this case, f(x) = x²sin(x). The product rule is used when we have a function that is the product of two other functions. The derivative of f(x) can be calculated as follows
f'(x) = (x²)' sin(x) + x² (sin(x))'
To find the derivative of x² sin(x), we use the product rule again
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
Let f(x) = x² and g(x) = sin(x). We have
f'(x) = 2x
g'(x) = cos(x)
Substituting these values back into the product rule equation
f'(x) = (x²)' sin(x) + x² (sin(x))'
= (2x) sin(x) + x^2 cos(x)
Therefore, the derivative of f(x) = x²sin(x) is f'(x) = (2x) sin(x) + x²cos(x).
Now, we have found the derivatives of f(x) for both x < 0 and x > 0. To show that f(x) is differentiable, we need to verify that the derivatives from both cases match at x = 0.
As x approaches 0 from the left side (x < 0), we have
lim(x → 0⁻) f'(x) = lim(x → 0⁻) 3x² = 0
As x approaches 0 from the right side (x > 0), we have
lim(x → 0⁺) f'(x) = lim(x → 0⁺) (2x) sin(x) + x²cos(x) = 0
Since the limits of the derivatives from both cases are equal at x = 0, we can conclude that f(x) = x³ for x < 0 and f(x) = x²sin(x) for x > 0 is differentiable at every point in its domain.
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