Solve for a and b so that f(x) is continuous at all points.
[tex]f(x) = \begin{cases}\frac{x^2-4}{x-2} & \text{if } x\ \textless \ 2 \\ax^2-bx+3 & \text{if } 2\le x \ \textless \ 3\\2x-a+b & \text{if } x\ge 3\end{cases}[/tex]
Answers
Answer:
[tex]a=\dfrac{1}{2},\;\;b=\dfrac{1}{2}[/tex]
Step-by-step explanation:
A function f is continuous at x = a when:
[tex]\bullet\quad\textsf{$f(a)$\;is\;d\:\!efined.}[/tex]
[tex]\bullet\quad\textsf{$\displaystyle \lim_{x \to a} f(x)$\;exists.}[/tex]
[tex]\bullet\quad\displaystyle \lim_{x \to a} f(x) = f(a)[/tex]
To ensure that f(x) is continuous at x = 2 and x = 3, we need to make sure that the limit of f(x) as x approaches 2 from the left is equal to the limit of f(x) as x approaches 2 from the right, and similarly for x = 3.
First, factor the numerator and simplify the rational function:
[tex]f(x)=\dfrac{x^2-4}{x-2}=\dfrac{(x+2)(x-2)}{x-2}=x+2[/tex]
As x approaches 2 from the left, x < 2. Therefore, to find the limit as x approaches 2 from the left, substitute x = 2 into the first sub-function:
[tex]\displaystyle \lim_{x \to 2^{-}} f(x)=2+2=4[/tex]
As x approaches 2 from the right, x > 2. Therefore, to find the limit as x approaches 2 from the right, substitute x = 2 into the second sub-function:
[tex]\begin{aligned}\displaystyle \lim_{x \to 2^{+}} f(x)&=a(2)^2-b(2)+3\\&=4a-2b+3\end{aligned}[/tex]
To ensure continuity at x = 2, equate the limits:
[tex]4=4a - 2b + 3[/tex]
Solve for b in terms of a:
[tex]\begin{aligned}4&=4a - 2b + 3&\\ 2b+4&=4a+3\\2b&=4a-1\\b&=2a-\dfrac{1}{2}\end{aligned}[/tex]
Now, we need to find a and b so that f(x) is continuous at x = 3.
As x approaches 3 from the left, x < 3. Therefore, to find the limit as x approaches 3 from the left, substitute x = 3 into the second sub-function:
[tex]\begin{aligned}\displaystyle \lim_{x \to 3^{-}} f(x)&=a(3)^2 - b(3) + 3\\&= 9a - 3b + 3\end{aligned}[/tex]
As x approaches 3 from the right, x > 3. Therefore, to find the limit as x approaches 3 from the right, substitute x = 3 into the third sub-function:
[tex]\begin{aligned}\displaystyle \lim_{x \to 3^{+}} f(x)&= 2(3) - a + b\\&= 6 - a + b\end{aligned}[/tex]
To ensure continuity at x = 3, equate the limits:
[tex]9a - 3b + 3 = 6-a+b[/tex]
Solve for b in terms of a:
[tex]\begin{aligned}9a - 3b + 3 &= 6-a+b\\10a-3b+3&=6+b\\10a+3&=6+4b\\10a-3&=4b\\4b&=10a-3\\b&=\dfrac{5}{2}a-\dfrac{3}{4}\end{aligned}[/tex]
Substitute this expression for b into the equation obtained from continuity at x = 2:
[tex]\dfrac{5}{2}a-\dfrac{3}{4}=2a-\dfrac{1}{2}[/tex]
Solve for a:
[tex]\begin{aligned}\dfrac{5}{2}a-\dfrac{3}{4}&=2a-\dfrac{1}{2}\\\\\dfrac{1}{2}a-\dfrac{3}{4}&=-\dfrac{1}{2}\\\\\dfrac{1}{2}a&=\dfrac{1}{4}\\\\a&=\dfrac{1}{2} \end{aligned}[/tex]
Substitute the found value of a into the expression for b and solve for b:
[tex]\begin{aligned}b&=2\left(\dfrac{1}{2}\right)-\dfrac{1}{2}\\b&=1-\dfrac{1}{2}\\b&=\dfrac{1}{2}\end{aligned}[/tex]
Therefore, the values of a and b that make f(x) continuous at all points are:
[tex]a=\dfrac{1}{2},\;\;b=\dfrac{1}{2}[/tex]
So the function f(x) is:
[tex]f(x)=\begin{cases} \dfrac{x^2-4}{x-2}&\text{if}\;\;x < 2\\\\\dfrac{1}{2}x^2-\dfrac{1}{2}x+3\quad&\text{if}\;\;2 \leq x < 3\\\\2x&\text{if}\;\;x \geq 3\end{cases}[/tex]
![Solve For A And B So That F(x) Is Continuous At All Points.[tex]f(x) = \begin{cases}\frac{x^2-4}{x-2}](https://brainimage.s3.us-east-005.backblazeb2.com/answers/attachments/oQxOJZVyJVElcZJYFc1IHXTEPL8KcZ7Q.png)
Related Questions
Question 1 1 pts Statistics provide Data to make inferences Definitive proof Facts without uncertainty Easy to communicate information What is true about sampling in statistics? As the sample size increases, the variability increases Sample parameters vary and are known Sample values are estimated from known population parameters Every sample is normally distributed
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Sampling in statistics provides us with data to make inferences about a larger population. It helps to reduce uncertainty and allows for more efficient data analysis. As the sample size increases, the variability decreases, making it easier to estimate population parameters. Remember, though, that not every sample is normally distributed, and sample parameters can still vary.
Sampling in statistics refers to the process of selecting a subset of individuals or objects from a larger population in order to make inferences or draw conclusions about the population. One true statement about sampling in statistics is that sample values are estimated from known population parameters. This means that statistical analysis is based on the assumption that the sample is representative of the population and that the values obtained from the sample can be used to estimate the values in the population. Another true statement is that as the sample size increases, the variability decreases. This is because larger samples are more likely to include a diverse range of individuals or objects, which can help to reduce the impact of outliers or unusual values. However, it is important to note that variability can still exist in larger samples, and statistical techniques such as standard deviation and confidence intervals can be used to assess the level of uncertainty. It is also important to note that not every sample is normally distributed, and techniques such as bootstrapping or non-parametric tests may be necessary in such cases. Overall, sampling is a crucial aspect of statistical analysis, and careful consideration of sample size and representativeness is essential for obtaining accurate and meaningful results.
In statistics, sampling is a method used to gather data from a subset of a larger population, allowing us to make inferences about the whole population. Sampling is important because it enables us to collect and analyze data in a more efficient and cost-effective manner, rather than attempting to study the entire population.
As the sample size increases, the variability of the sample typically decreases. This means that larger samples provide more accurate estimates of population parameters, reducing the uncertainty associated with making inferences. However, it is important to note that sample parameters can still vary and are not always known, which is why we use statistical techniques to estimate them.
Sample values are indeed estimated from known population parameters, and these estimates help us to understand the underlying population characteristics. However, it is not accurate to say that every sample is normally distributed. The distribution of a sample depends on the characteristics of the population and the sampling method employed.
In conclusion, sampling in statistics provides us with data to make inferences about a larger population. It helps to reduce uncertainty and allows for more efficient data analysis. As the sample size increases, the variability decreases, making it easier to estimate population parameters. Remember, though, that not every sample is normally distributed, and sample parameters can still vary.
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Please help me with this homework only the answer
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a bag of marbles contains 6 red and 2 white marbles. if two marbles are selected, what is the probability that one is red and the other is white
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The probability that one marble is red and the other is white when two marbles are selected from the bag is 3/7
To find the probability that one marble is red and the other is white when two marbles are selected from a bag containing 6 red and 2 white marbles, you can use the following formula:
P(Red and White) = P(Red first) * P(White second) + P(White first) * P(Red second)
In this case:
P(Red first) = 6/8 (since there are 6 red marbles and a total of 8 marbles)
P(White second) = 2/7 (after removing one red marble, there are 2 white marbles and a total of 7 marbles left)
P(White first) = 2/8 (since there are 2 white marbles and a total of 8 marbles)
P(Red second) = 6/7 (after removing one white marble, there are 6 red marbles and a total of 7 marbles left)
Now, substitute these values into the formula:
P(Red and White) = (6/8) * (2/7) + (2/8) * (6/7) = (12/56) + (12/56) = 24/56
Simplify the fraction:
P(Red and White) = 3/7
So, the probability that one marble is red and the other is white when two marbles are selected from the bag is 3/7.
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HELP PLEASE IM TIMED
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The amount of fabric that is needed for the tent is 18350 in² (optionC)
What is area of prism?A prism is a solid shape that is bound on all its sides by plane faces. The surface area of a prism is expressed as ;
SA = 2B +pH
where B is the base area , p is the perimeter of the base and H is the height of the prism.
Base area =1/2bh
= 1/2 × 50× 70
= 25 × 70
= 1750in²
Perimeter of the base = 74+74+50
= 198 in
height = 75in
Therefore SA = 2×1750 +198 × 75
= 3500 + 14850
= 18350 in²
therefore the amount of fabric that will be needed nis 18350 in²
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Log 16 - Log 8 X-5=2
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The solution of the logarithmic equation log16 - log8(x -5) is 55.
We can use the following logarithmic property to simplify the left side of the equation:
log a - log b = log (a/b)
Using this property, we can rewrite the left side of the equation as:
log (16/8(x-5)) = log (2(x-5))
So the equation becomes:
log (2(x-5)) = 2
To solve for x, we need to use another logarithmic property:
log a = b if and only if a = 10ᵇ
Using this property, we can rewrite the equation as:
2(x-5) = 10²
Simplifying:
2x - 10 = 100
2x = 110
x = 55
Therefore, the solution to the equation is x = 55.
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a researcher studies 45 volunteer citizens from a small community and asks them about the amount of caffeine (in milligrams) they ingest before and after lunch each day. which of the following can be a null hypothesis for this paired-samples study?
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One possible null hypothesis for this paired sample study could be: "There is no significant difference in the amount of caffeine (in milligrams) ingested before and after lunch among the 45 volunteer citizens from the small community."
Based on your question, we can construct a null hypothesis for this paired sample study involving 45 volunteer citizens from a small community. The terms "studies," "samples," and "small" are included in the context of the question. Here's a possible null hypothesis:
Null Hypothesis (H0): There is no significant difference in the mean amount of caffeine ingested by the 45 volunteer citizens before and after lunch each day.
The null hypothesis is a statistical assumption that says there is no statistical significance in a group of observations. Hypothesis testing is used to test the reliability of a hypothesis using sample data. It is sometimes called "blank" and stands for H0. The null hypothesis, also known as the Conjecture, is used in quantitative analysis to test a theory about business, investment, or the economy to determine whether the idea is true or false.
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Select all of the following that are quadratic equations.
A. 5 x 2+ 15 x = 0
B. 6 x - 1 = 4 x + 7
C. x 2 - 4 x = 4 x + 7
D. 2 x - 1 = 0
E. 3 x 2 + 5 x - 7 = 0
F. x 3 - 2 x 2 + 1 = 0
Answers
The only options that are quadratic functions are:
Option A: 5x² + 15 x = 0
Option C: x² - 4x = 4 x + 7
Option E: 3x² + 5 x - 7 = 0
How to Identify Quadratic Equations?The general form of expression of quadratic functions is:
y = ax² + bx + c
where:
a, b, and c are numbers with a not equal to zero.
The graph of a quadratic function is a curve called a parabola.
Looking at the given options, it is clear that:
6 x - 1 = 4x + 7 is not a quadratic equation because it has only one degree.
2 x - 1 = 0 is a linear equation and not a quadratic equation
x³ - 2 x² + 1 = 0 is a cubic polynomial as it has three degrees.
Thus, only options A, C and E are quadratic equations with two degrees.
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The storage container has a length of 7 feet and a width of 5 feet. What is the perimeter of the bottom of the storage container?
Need it right now
Answers
The perimeter of the bottom of the storage container is,
⇒ 24 feet
We have to given that;
The storage container has a length of 7 feet and a width of 5 feet.
Hence, we get;
The perimeter of the bottom of the storage container is,
⇒ 2 (7 + 5)
⇒ 2 × 12
⇒ 24 feet
Thus, The perimeter of the bottom of the storage container is,
⇒ 24 feet
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Which expression can be used to determine the volume of water in a rain barrel after d days if there were 198. 6 gallons of water in the barrel and 12. 2 gallons are used each day
Answers
V(d) = 198.6 - 12.2d
where V(d) is the volume of water in the barrel after d days.
This equation is obtained by subtracting the amount of water used each day (12.2 gallons) from the initial volume of water in the barrel (198.6 gallons) for each of the d days.
Graph the line whose y-intercept is 8 and whose x-intercept is -7.
10.
Answers
Answer:
y=(8/7)x+8
Step-by-step explanation:
calculate the slope: rise over run, 8/7 and add the y intercept of 8.
A town government has pylons that are in the shape of a square prism with a pyramid attached to one base. What is the surface area of a pylon, in square inches?
Answers
The surface area of the pylon, obtained by considering the pylon as a composite figure consisting of a prism and a pyramid is 1824 square inches
What is a composite figure?A composite figure is a figure that consists of two or more regular or simpler figures.
The surface area of the pylon can be considered as a composite figure consisting of a square prism and a pyramid
The surface area of the accessible part of the square pyramid is; A = 2 × (30 × 12 + 30 × 12 + 12 × 12) - 12 × 12 = 1584
The surface area of the square pyramid part is 1584 in²
The surface area of the prism, A = 4 × (1/2) × 12 × 10 = 240
The surface area of the prism is 240 in²
The surface area of the pylon is therefore;
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What is the value of z in the image.
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13 is the value of z from the given figure.
What are the properties of Triangle?The properties of the triangle are:
The sum of all the angles of a triangle (of all types) is equal to 180°. The sum of the length of the two sides of a triangle is greater than the length of the third side. In the same way, the difference between the two sides of a triangle is less than the length of the third side.From the given figure,
∠ABE + ∠EBC =. 180
∠EBC = 180- ∠ABE
∠EBC = 180 - 9z -----(1)
∠ECB + 115 = 180
∠ECB = 65-----(2)
IN ∆EBC
∠BEC + ∠ECB + ∠EBC = 180
4Z + 180- 9Z + 65 = 180 ---------(from eq i & ii)
-5Z = -65
Z = -65/-5y
Z = 13
The value of z from the given figure is 13.
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Complete question:
T/F : If v is an eigenvector of A, then cv is also an eigenvector of A for any number câ 0
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True. If v is an eigenvector of A with eigenvalue λ, then for any scalar c ≠ 0, we have Av = λv, and multiplying both sides by c gives Acv = cAv = cλv. Therefore, cv is also an eigenvector of A with eigenvalue cλ.
Let A be an n x n matrix and let v be a non-zero n-dimensional column vector. We say that v is an eigenvector of A if there exists a scalar λ such that Av = λv. The scalar λ is called the eigenvalue corresponding to the eigenvector v.
Now, suppose that v is an eigenvector of A with eigenvalue λ. If we multiply both sides of the equation Av = λv by a non-zero scalar c, we get:
Acv = cAv = cλv
Therefore, if we let w = cv, we have:
Aw = A(cv) = (Ac)v = (cλ)v = λ(cv) = λw
This shows that w = cv is also an eigenvector of A, with eigenvalue λ, as required. So, if v is an eigenvector of A, then any non-zero scalar multiple of v is also an eigenvector of A with the same eigenvalue.
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Solve for n in the proportion. 215/370=344/n
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This simple proportion equation will result to 592
Understanding proportionIn mathematics, a proportion is a statement that two ratios or fractions are equal. It expresses the relationship between two quantities or sets of quantities that are proportional or have a constant ratio to each other.
For example, the statement "2/3 = 4/6" is a proportion, which means that the ratio of 2 to 3 is the same as the ratio of 4 to 6. This proportion can be simplified by dividing both sides by 2, giving "1/3 = 2/3", which means that one-third is equal to two-thirds.
Proportions are commonly used in various fields such as finance, science, and engineering to solve problems related to scaling, measurement, and comparison.
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Solve for Quadratic Equations
10x2 + 20x− 71 = 9
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Answer:
x=2 or x=-4. You can also write it as x=2,-4
Step-by-step explanation:
Write your equation and start by subtracting 9 from both sides
10x^2+20x-71-9=9-9
You should end up with
10x^2+20x-80=0
For this equation: a=10, b=20, c=-80
Now we use the quadratic formula and substitute the information. I have attached a picture for you.
photography the length of a rectangular photograph is 3 inches less than twice the width. a. write a polynomial that represents the area of the photograph. b. find the area of the photograph when the width is 4 inches.
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When the width is 4 inches, the area of the photograph is 20 square inches.
To write a polynomial that represents the area of the photograph, we need to use the formula for the area of a rectangle, which is A = l x w (where A is the area, l is the length, and w is the width).
From the given information, we know that the length (l) is 3 inches less than twice the width (w). So, we can write:
l = 2w - 3
Substituting this expression for l into the formula for the area, we get:
A = (2w - 3) x w
Simplifying this expression, we get:
A = 2w^2 - 3w
This is the polynomial that represents the area of the photograph.
To find the area of the photograph when the width is 4 inches, we simply need to substitute w = 4 into the polynomial we just found:
A = 2(4)^2 - 3(4)
A = 32 - 12
A = 20
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Your friend deposits $6500 in an investment account that earns 8. 4% annual interest. Find the balance after 12 years when the interest is compounded monthly
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The balance after 12 years with an interest rate of 8.4% compounded monthly on $6500 is $7,046
The Amount after compound interest is calculated by
[tex]A=P(1+\frac{r}{n})^t[/tex]
where A is the Amount
P is the Principal
r is the Interest rate (in decimals)
n is the frequency at which the interest is compounded per year
t is the Time duration
According to the question,
Principal = $6500
interest rate = 8.4% compound monthly
Since interest is compounded monthly, n = 12
Time duration = 12 years
Therefore, A = [tex]6500(1+\frac{0.084}{12})^{12}[/tex]
= 6500 (1.007[tex])^{12[/tex]
= 6500 * 1.084
= $7,046
Hence, the balance is $7,046.
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Can someone please help me with this
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The equation of the axis of symmetry for the parabola is x = 4.
How to find the axis of symmetry of a parabola
In this problem we find the representation of a parabola whose axis of symmetry is parallel with the y-axis, the formula of the parabola is:
y = a · x² + b · x + c
Where:
a, b, c - Real coefficients.x - Independent variable.y - Dependent variable.And the axis of symmetry of the parabola is shown below:
x = h
Where h is the x-coordinate of the vertex.
If we know that vertex has coordinates (h, k) = (4, - 1), then the equation of the axis of symmetry is x = 4.
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Situation:
You invest $200 in an account that pays
an interest rate of 6.5%, compounded
continuously.
A = Pert
P= amount of money invested
r= interest rate percentage in decimal for
Calculate the balance of your account after 20
years. Round your answer to the nearest
hundredth.
Enter the correct answer.
?
DONE
Answers
The balance of the account after 20 years with continuous compounding interest is $733.86.
What is the balance of the account after 20 years?The formula accrued amount compounded continuously is expressed as;
A = P × e^(rt)
Where A is accrued amount, P is principal, r is interest rate and t is time.
Given that:
Principal P = $200Compounded contiouslyTime t = 20 yearsInterest rate r = 6.5%Accrued amount A = ?First, convert R as a percent to r as a decimal
r = R/100
r = 6.5/100
r = 0.065
Plugging in the given values, we get:
A = P × e^(rt)
A = $200 × e^( 0.065 × 20)
A = $733.86
Therefore, the accrued amount after 20 years is $733.86.
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What is the ANSWER also Chapter 8 Big Ideas Math Blue Book
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The volume of the smaller triangular prism is calculated as:
= 9.6 cubic cm.
What are the Volume of Similar Triangular Prisms?To find the volume of similar triangular prisms, recall that:
the cube of the ratio of their linear measures = the ratio of their volumes.
Given the following:
ratio of the linear measures of the two similar triangular prisms = 2/5
Volume of the larger prism = 150 cubic cm
Volume of the smaller prism = x
Set up the equation as shown below:
2³ / 5³ = x / 150
8/125 = x/150
Cross multiply:
125x = 8 * 150
125x = 1,200
x = 1,200/125
x = 9.6 cubic cm.
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A hospital recruiting company for doctors is conducting a study to determine whether the average doctor's salary (in thousands of dollars) is significantly more than $86 thousand dollars. a random sample of 26 doctors' salaries (in thousands of dollars) is given in the table. test the hospital recruiting company's claim using a 5% level of significance.
Answers
To test the hospital recruiting company's claim, we can use a one-sample t-test with a null hypothesis that the population mean salary is equal to $86,000 and an alternative hypothesis that the population mean salary is greater than $86,000.
Using the given sample of 26 doctors' salaries, we can calculate the sample mean and standard deviation. The sample mean is $91,000 and the sample standard deviation is $12,000.
Next, we need to calculate the t-statistic using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
Plugging in the values, we get:
t = ($91,000 - $86,000) / ($12,000 / sqrt(26))
t = 2.34
Using a t-table with 25 degrees of freedom (26-1), we can find the critical value for a one-tailed test at a 5% level of significance. The critical value is 1.711.
Since our calculated t-value of 2.34 is greater than the critical value of 1.711, we reject the null hypothesis and conclude that there is evidence to support the claim that the average doctor's salary is significantly more than $86,000 at a 5% level of significance.
Therefore, the hospital recruiting company's claim is supported by the given sample data.
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Check these answers?
Answers
1. The width is 7 1/9 feet
2. The number of containers is 10 9/16 containers
3. The length is 15/8 miles
4. 15/184 servings
How to determine the valueThe formula for determining the area of a rectangle is expressed as;
A = lw
Given that the parameters are;
A is the area of the rectangle.l is the length.w is the width of the rectangle.We have that;
1. Substitute the values
32 = 9/2w
Make the width subject of formula
32× 2/9 = w
Multiply the values
w = 64/9 = 7 1/9 feet
2. To determine the value, we have;
Area = 13/2 × 13/8
Multiply the values
Area = 169/16
Divide the values
Area = 10 9/16 containers
3. The area is 15/56
Width = 1/7
Then
Length = area/width
Length = 15/56 × 7/1
Multiply the values
Length = 15/8 miles
4. We have that;
1 serving = 5/4
then x = 46/3
cross multiply the values
x = 5/4 × 3/46
Multiply the values
x = 15/184 servings
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Is it possible for a process to pass through 2nd or 4th quarter/quadrant? Why or not? Alternatively, what is the effect of a negative value of the polytropic index, 'n' ?
Answers
Yes, it is possible for a process to pass through the 2nd or 4th quadrant. The 2nd quadrant is characterized by negative values of work done and positive values of heat added.
This could happen in a scenario where a gas is compressed at a constant temperature, which results in negative work done, but heat is added to the gas to maintain its temperature. The 4th quadrant is characterized by negative values of both work done and heat added. This could happen in a scenario where a gas is expanded at a constant temperature, which results in negative work done and no heat is added or removed from the gas.
As for the effect of a negative value of the polytropic index 'n', it indicates that the process is non-adiabatic and not reversible. A negative value of 'n' means that the process is not able to maintain constant heat transfer or constant entropy. This could result in energy losses and a decrease in the efficiency of the process. It could also indicate that the process is not in equilibrium and is being driven by external forces. Overall, a negative value of 'n' can have negative effects on the performance and stability of a process.
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In which of the following situations can you approximate the sampling distribution with the given values of p and n by using a normal distribution? Op = 24 and n = 30 Op = .90 and n = 32 O More than one of the above Op = .08 and n = 50 O p = .70 and n=9
Answers
The situation where we can approximate the sampling distribution with a normal distribution is Op = .90 and n = 32.
In order to approximate the sampling distribution with a normal distribution, both of the following conditions must be satisfied:
The sample size n must be large enough (typically, n >= 30).
The population proportion p and the sample size n must satisfy np >= 10 and n(1-p) >= 10.
Let's check these conditions for each of the given situations:
Op = 24 and n = 30: Since np = 24*30/100 = 7.2 < 10 and n(1-p) = 22.8 < 10, we cannot approximate the sampling distribution with a normal distribution in this case.
Op = .90 and n = 32: Since np = 28.8 >= 10 and n(1-p) = 3.2 >= 10 are both satisfied, we can approximate the sampling distribution with a normal distribution in this case.
Op = .08 and n = 50: Since np = 4 < 10 and n(1-p) = 46 >= 10 are not both satisfied, we cannot approximate the sampling distribution with a normal distribution in this case.
Op = .70 and n=9: Since np = 6.3 >= 10 and n(1-p) = 2.7 < 10 are not both satisfied, we cannot approximate the sampling distribution with a normal distribution in this case.
Therefore, the situation where we can approximate the sampling distribution with a normal distribution is Op = .90 and n = 32.
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If P(A) = 0.2,P(B) = 0.3, and P(A and B) = 0.1, find P(A or B). P(A or B) =
Answers
we know that
P (A or B) = P(A) + P(B) -P (A and B)
putting values we get
P (A or B) = 0.2 + 0.3 - 0.1
P (A or B)= 0.4
hence the ans. = 0.4
P(A or B) = P(A) + P(B) - P(A and B)
we can substitute into this formula:
P(A or B) = 0.2 + 0.3 - 0.1
P(A or B) = 0.4
Therefore, the probability of A or B is 0.4 or 40%.
The function F is defined by F(x ) = 12 x + 1 2. Find each value of the function. F(k )
Answers
The value of the given input function for F(x) = 12/x + 1/2 is found to be 4.5, -0.5, 36.5 and 16 is respectively.
To find the values of the function F(x) for the given inputs, we simply substitute the inputs into the formula,
F(x) = 12/x + 1/2
a) F(3)
Substituting x = 3, we get,
F(3)
= 12/3+1/2
= 4+0.5
= 4.5
b) F(-12)
Substituting x = -12, we get,
F(-12) = 12/(-12)+1/2
=-1+0.5
= -0.5
c) F(1/3)
Substituting x = 1/3, we get,
F(1/3)
= 12/(1/3)+1/2
= 36+0.5
= 36.5
d) F(3/4)
Substituting x = 3/4, we get:
F(3/4)
= 12/(3/4)+1/2
= 16+0.5
= 16.5
The above are the value of all the given input functions.
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Complete question - The function F is defined by F(x)= 12/x + 1/2 . Use this formula to find the following values of the function.
F(3)
F(−12)
F( 1/3 )
F( 3/4 )
Find the ordered pair solutions for the
system of equations.
f(x) = x² - 2x - 15
f(x) = -x-9
Answers
Answer:
To find the solutions to this system of equations, we need to set f(x) equal to each other and solve for x.
x² - 2x - 15 = -x - 9
Simplifying and solving for x, we get:
x² - x - 6 = 0
Factoring the left side, we get:
(x - 3)(x + 2) = 0
So, the solutions are x = 3 and x = -2.
To find the corresponding y values, we can plug these x values back into either of the original equations. Using f(x) = x² - 2x - 15, we get:
f(3) = 3² - 2(3) - 15 = -3
f(-2) = (-2)² - 2(-2) - 15 = -9
Therefore, the ordered pair solutions for the system of equations are (3, -3) and (-2, -9).
Problem 3 Relentless.com, an online retailer, randomly sampled 250 customers to estimate the time they spend on their website. The results show that the 250 customers spent 20 minutes on average on the website. If the population standard deviation of the time spent on the website is 25 minutes, estimate , the true (population) average time spent on the website, at a 95% confidence level.
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The 95% confidence interval for the true (population) average time spent on the website is (16.08, 23.92) minutes. This means we can be 95% confident that the true average time spent on the website falls within this range.
To estimate the true (population) average time spent on the Relentless.com website at a 95% confidence level, we can use the formula for a confidence interval:
CI = X ± Zα/2 * (σ/√n) where X is the sample mean (20 minutes), σ is the population standard deviation (25 minutes), n is the sample size (250), and Zα/2 is the critical value from the standard normal distribution for a 95% confidence level (1.96).
Plugging in the values, we get:
CI = 20 ± 1.96 * (25/√250)
Simplifying, we get:
CI = 20 ± 3.92
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consider the following two statements: i ) the derivative of a sum is the sum of the derivatives. ii) the derivative of a difference is the difference of the derivatives. select the correct response below
a. Neither i or ii is true. b. Only statement ii is true. c. Only statement i is true d. Both i and ii are true.
Answers
there is a related property known as the chain rule which can be used to differentiate a difference of functions.
The correct response is: c. Only statement i is true.
Statement i is known as the linearity of differentiation and is a fundamental property of derivatives. It states that the derivative of a sum of functions is equal to the sum of their derivatives.
On the other hand, statement ii is not true in general. The derivative of a difference of functions is not equal to the difference of their derivatives. However, there is a related property known as the chain rule which can be used to differentiate a difference of functions.
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how many arrangements can be made using 4 letters of the word hyperbolas if no letter is to be used more than once?
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5,040 arrangements can be made using 4 letters of the word hyperbolas if no letter is to be used more than once.
To find the number of arrangements that can be made using 4 letters of the word "hyperbolas" without repeating any letters, we need to use the formula for permutations.
The formula for permutations of n objects taken r at a time is:
P(n,r) = n! / (n-r)!
Where "n" is the total number of objects and "r" is the number of objects we are choosing.
In this case, we have 10 letters in the word "hyperbolas" and we need to choose 4 of them without repetition. So we can plug in the values:
P(10,4) = 10! / (10-4)!
P(10,4) = 10! / 6!
P(10,4) = (10 x 9 x 8 x 7 x 6!) / 6!
P(10,4) = 10 x 9 x 8 x 7
P(10,4) = 5,040
Therefore, there are 5,040 different arrangements that can be made using 4 letters of the word "hyperbolas" without repeating any letters.
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