The absolute maximum of f(x) on [3,4] is f(3) = 2 and the absolute minimum is f(4) = -7.The extreme value theorem does guarantee the existence of an absolute maximum and minimum for f(x) on this interval.
The extreme value theorem states that if a function is continuous over a closed interval, then it must have an absolute maximum and an absolute minimum on that interval.
In this case, the function f(x) is continuous over the closed interval [3,4], as it is a square root function and the square root of a non-negative number is always defined.
To find the absolute maximum and minimum of f(x) on [3,4], we need to evaluate f(x) at the endpoints and any critical points in the interval. However, since f(x) is a decreasing function on [3,4], the maximum value of f(x) occurs at the left endpoint x=3 and the minimum value occurs at the right endpoint x=4.
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what will be the equilibrium wage rate? group of answer choices $6 $5 $4 $3
The equilibrium wage rate is $6 per hour. This is found by observing the point where the labor demanded and labor supplied curves intersect, which is at a wage rate of $6 and a quantity of labor of 35.
In a competitive labor market, the equilibrium wage is the wage rate at which the quantity of labor demanded equals the quantity of labor supplied. In other words, it is the wage rate that clears the labor market.
To find the equilibrium wage in this problem, we need to compare the labor demanded and the labor supplied at each wage rate. We can see from the table that at a wage rate of $3 per hour, the labor demanded is 50 and the labor supplied is only 20.
This means that the quantity of labor demanded exceeds the quantity of labor supplied, and there will be upward pressure on the wage rate. As the wage rate increases, the quantity of labor demanded decreases and the quantity of labor supplied increases.
From the table, we can see that at a wage rate of $6 per hour, the quantity of labor demanded (35) equals the quantity of labor supplied (50), which is the equilibrium condition. Therefore, the equilibrium wage is $6 per hour.
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--The given question is incomplete, the complete question is given below " Assume that the following data describe labor-market conditions:
Wage rate (per hour) $3 $4 $5 $6 $7 $8 $9 $10
Labor demanded 50 45 40 35 30 25 20 15
Labor supplied 20 30 40 50 60 70 80 90
Find the equilibrium wage.
Instructions: Enter your response as a whole number."--
Cynthia concludes that 8th graders are more likely than 9th graders to spend at most four hours per week studying because more 8th graders than 9th graders responded that they study this long.
Explain whether Cynthia is correct. Include all necessary work to support your answer.
Choosing 8 numbers that match 8 randomly selected balls when the balls are numbered 1 through 33. The probability of the given event is?
Hence, the probability of choosing 8 numbers that match 8 randomly selected balls when the balls are numbered 1 through 33 is given by (1/33)^8 x C(33,8) which is approximately equal to 0.000000206.
The probability of choosing 8 numbers that match 8 randomly selected balls when the balls are numbered 1 through 33 can be calculated using the formula for combination. There are a total of 33 balls and we need to choose 8 out of them. The total number of possible combinations of 8 balls out of 33 is given by C(33,8) which is equal to 69,090,840. Now, if we assume that all the balls are equally likely to be selected, then the probability of selecting a particular ball is 1/33. Therefore, the probability of choosing 8 specific balls out of 33 is (1/33)^8. Hence, the probability of choosing 8 numbers that match 8 randomly selected balls when the balls are numbered 1 through 33 is given by (1/33)^8 x C(33,8) which is approximately equal to 0.000000206. In other words, the probability of this event occurring is very low, which means it is very unlikely to happen.
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the coordinate of point t are (0,6). the midpoing of st is (3,-6). find the corrdinates of points
the coordinates of point S are (6, -18).To find the coordinates of points S and T, we need to use the midpoint formula.
Given that the midpoint of ST is (3, -6) and the coordinate of point T is (0, 6), we can calculate the coordinates of point S.
The midpoint formula states that the coordinates of the midpoint (M) of two points (P and Q) can be found using the following formula:
M = ((Px + Qx) / 2, (Py + Qy) / 2)
Let's substitute the given values into the formula:
(3, -6) = ((0 + Sx) / 2, (6 + Sy) / 2)
Simplifying the equation, we have:
3 = (0 + Sx) / 2 and -6 = (6 + Sy) / 2
Multiplying both sides by 2, we get:
6 = 0 + Sx and -12 = 6 + Sy
Solving for Sx and Sy, we find:
Sx = 6 and Sy = -18
Therefore, thethe coordinates of point S are (6, -18).
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in general, there is more information provided by . a. a confidence interval than a p-value. b. a p-value than a confidence interval. c. a sample statistic than a confidence interval for the corresponding parameter. d. all of the above.
In general, a confidence interval provides more information than a p-value. A confidence interval provides more information than a p-value because it gives us an estimate of the parameter, a measure of uncertainty, and can be derived from a sample statistic.
A confidence interval is a range of values around an estimate of a population parameter that we are fairly certain contains the true value of the parameter. It provides both an estimate of the parameter and a measure of the uncertainty of the estimate. On the other hand, a p-value is a measure of the strength of evidence against a null hypothesis. It tells us the probability of observing a test statistic as extreme as the one we observed, or more extreme, if the null hypothesis were true. However, it does not tell us anything about the magnitude or direction of the effect, or the precision of the estimate.
Furthermore, a confidence interval can be derived from a sample statistic, whereas a p-value cannot. A confidence interval gives us an estimate of the population parameter based on the sample, while a p-value tells us how likely it is to observe a sample statistic as extreme as the one we observed, assuming the null hypothesis is true.
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(x+3)^n+1=(x+3)^n+11
Answer:
No solution to any of the variables or equation
Step-by-step explanation:
Two taps running at the same rate can fill a tank in 45 mins. How long will it take one tap to fill the same tank?
Answer:
90 minutes
Step-by-step explanation:
it will take twice as long therefore 90 minutes
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The data that has a variation, or mean absolute deviation, similar to the data set in the given dot plot is option D.
Why is this so?Deviation describes the extent of the Stata change, do plot in D has the same distribution with the example given to they must have the most similar Mean Absolute deviation.
A data set's mean absolute deviation (MAD) is the average distance between each data value and the mean. A data set's fluctuation can be described using mean absolute deviation. The mean absolute deviation tells us how "spread out" the values in a data collection are.
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answer all pls thx i need it answered
A dot plot can be created by drawing a number line and then adding one dot per value, the first statement is true but the second one is not.
How to draw and interpret a dot plot?The basic steps to draw this type of graph is:
Draw a number line that covers the range of the data, in this case from 18 to 32.Draw a dot for each value, for example, 25 is repeated two times, so two dots should be drawn.Now, let's interpret the information:
The class is an adult class - Yes- All of them are at least 18.Half of the students are between 2 and 3 years old - False- No one is this age.Learn more about dot plots in https://brainly.com/question/22746300
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Suppose a price discriminating monopoly has segregated its market into two submarkets and can prevent resale between the two. Assume that its marginal cost is constant and equal to its average total cost of $8. The firm's demand schedule for the first group is given by the first two columns of the following table. a. Find the firm's total revenue schedule for this submarket, entering the data into the table where indicated. Use these data to determine the marginal revenue schedule in this submarket. b. What output level and price will maximize the firm's profit in this submarket? c. The firm's demand schedule for the second group is given by the first two columns of the following table. d. Find the firm's total and marginal revenue schedules in this second submarket. What output level and price will maximize the firm's profit in this submarket? e. What is this firm's total economic profit?
The price that maximizes profit in this submarket is $10. (e) The firm's total economic profit can be found by subtracting total cost.
The total revenue schedule for the first submarket can be obtained by multiplying the price and quantity demanded in each row. The results are entered in the third column of the table below:
Quantity Price Total Revenue
0 20 0
1 19 19
2 18 36
3 17 51
4 16 64
5 15 75
The marginal revenue schedule can be obtained by calculating the change in total revenue when one more unit is sold. The results are entered in the fourth column of the table below:
Quantity Price Total Revenue Marginal Revenue
0 20 0 -
1 19 19 19
2 18 36 17
3 17 51 15
4 16 64 13
5 15 75 11
b. To maximize profit, the firm should produce where marginal cost equals marginal revenue.
Since marginal cost is constant and equal to $8, the profit-maximizing output level is where marginal revenue equals $8, which is at a quantity of 3. Therefore, the price that maximizes profit in this submarket is $17.
c. The demand schedule for the second submarket is given by the first two columns of the table below:
Quantity Price
0 12
1 11
2 10
3 9
4 8
5 7
d. The total revenue schedule and marginal revenue schedule for the second submarket can be obtained in the same way as in part a. The results are entered in the third and fourth columns of the table below:
Quantity Price Total Revenue Marginal Revenue
0 12 0 -
1 11 11 11
2 10 20 9
3 9 27 7
4 8 32 5
5 7 35 3
To maximize profit, the firm should produce where marginal cost equals marginal revenue.
Since marginal cost is constant and equal to $8, the profit-maximizing output level is where marginal revenue equals $8, which is at a quantity of 2. Therefore, the price that maximizes profit in this submarket is $10. e. The firm's total economic profit can be found by subtracting total cost
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(q20) Is the function f(x) = (x - 9)(x + 9)x even, odd, or neither?
The function f(x) = (x - 9)(x + 9)x is an odd function.
Is the given function even, odd, or neither?Given the function in the question:
f(x) = (x - 9)(x + 9)x
To determine whether the function f(x) = (x - 9)(x + 9)x is even, odd, or neither, we need to check if it satisfies the conditions for even or odd functions.
An even function is symmetric with respect to the y-axis, meaning f(x) = f(-x) for all x in the domain.
An odd function is symmetric with respect to the origin, meaning f(-x) = -f(x) for all x in the domain.
Let's analyze the given function:
f(x) = (x - 9)(x + 9)x
f(x) = x³ - 81x
To check if it's even, we substitute -x for x:
f(x) = x³ - 81x
f(-x) = -x³ + 81x
Since -x³ + 81x is not equal to x³ - 81x, the function is not even.
To check if it's odd, find -f(x):
f(x) = x³ - 81x
-f(x) = -( x³ - 81x )
-f(x) = -x³ + 81x
Since -x³ + 81x equals -x³ + 81x, the function is odd.
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What is the formula for finding the circumstance of a circle
with just the diameter?
A.πx D
B.b x h
C.L×W×H
D.4x1
Answer: A.) π x D
Step-by-step explanation:
The diameter of a circle is 2x the radius
Answer:
A.πx D
Step-by-step explanation: The formula for the circumference of a circle is C=π×d, or it can be written as C=2×π×r.
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Aaron’s major surgery resulted in $25,000 in bills. His insurance has an annual deductible of $5,500, an 80% coinsurance, and an out-of-pocket maximum of $12,000. How much will he be responsible for paying?
Aaron is responsible for paying the sum of $10,700 for his surgery.
How much will Aaron be responsible for paying?Aaron's deductible = $5,500. Thus means he is responsible for paying first $5,500 of his medical bills.
Coinsurance = 80%. Insurance cover 80% of remaining cost and he is responsible for 20%.
Aaron's out-of-pocket maximum is computed as follows
= Surgery cost - Deductible
= $12,000 - $5,500
= $6,500
Amount payable * Coinsurance
= $6,500 x 0.8
= $5,200
So, Aaron's insurance will cover the remaining $19,800 of his $25,000 surgery.
So, he is responsible for paying:
= $5,500 (deductible) + $5,200 (out-of-pocket maximum)
= $10,700
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Which source of bias is most relevant to the following situation: Both members of a couple are asked to indicate if they have remained monogamous in their currently relationship self-interest study voluntary response blas nonrespolse bias or missing data perceived lack of anonymity loaded or leading question
The source of bias most relevant to this situation is "perceived lack of anonymity."
When both members of a couple are asked to indicate if they have remained monogamous in their current relationship, they may be hesitant to answer truthfully due to concerns about their privacy or potential repercussions. They might feel that their responses could be traced back to them, which could lead to negative consequences in their relationship. This fear of not being anonymous can result in participants providing inaccurate or dishonest responses.
In a study asking couples about their monogamy, the most relevant source of bias is perceived lack of anonymity, as it may affect the accuracy and honesty of participants' responses.
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If the conditional proposition p is a contingency, is -p also a contingency? Explain Choose the correct answer below. O A. Yes, because all of the entries in the truth table for-p will be T. B. No, because all of the entries in the truth table for will be F O C. No, because at least one entry in the truth table for p will be T and at least one entry will be F. O D. Yes, because at least one entry in the truth table for -p will be T and at least one entry will be F.
C. If the conditional proposition p is a contingency, it means that there are some truth values of p that make the proposition true and others that make it false. To determine whether -p is also a contingency, we need to consider the truth values of -p for all possible truth values of p.
if we negate a contingency, there will still be some truth values that make the negation true and others that make it false. Therefore, at least one entry in the truth table for p will be T and at least one entry will be F. This means that -p is also a contingency.
if the conditional proposition p is a contingency, then -p is also a contingency because there will be some truth values of p that make the negation true and others that make it false. The correct answer to the question is C.
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if you had to summarize the most important aspect of survey design in one phrase, what would it be? a. complexity is good b. make sure to always include a sample item c. keep it simple d. the more words, the better
The most appropriate phrase that summarizes the most important aspect of survey design is option c. keep it simple.
Survey design requires careful consideration of various factors.
Such as the target population, the research objectives, the types of questions, and the mode of data collection.
However, the fundamental principle of survey design is to keep the survey simple and straightforward.
The survey should be easy to understand, answer, and navigate, ensuring that participants can complete it with ease and accuracy.
When designing a survey, it is essential to prioritize simplicity over complexity.
A survey that is too complicated or difficult to complete can discourage participation, lead to inaccurate responses, and result in biased data.
Therefore, survey designers should aim to create clear, concise, and unambiguous questions that are easy to comprehend and answer.
Survey design should prioritize simplicity, ensuring that the questions are easy to understand, answer, and navigate.
And the survey is straightforward and concise.
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Assume the population has a distribution with a mean of 100 and a standard deviation of 10. For a random sample of size 50, find the following:a. P(99 < X < 102)b. P(X > 97)c. 70th percentile of X
The 70th percentile of X is approximately 100.73.
P(X > 97)=0.9838
P(99 < X < 102)=0.7841
a. Using the Central Limit Theorem, we can assume that the sample mean follows a normal distribution with mean 100 and standard deviation 10/sqrt(50) = 1.41. Therefore,
P(99 < X < 102) = P((99 - 100)/(1.41) < (X - 100)/(1.41) < (102 - 100)/(1.41))
= P(-0.71 < Z < 1.41)
= 0.7841 (using standard normal table or calculator)
b. Using the same reasoning as in part (a),
P(X > 97) = P((X - 100)/(1.41) > (97 - 100)/(1.41))
= P(Z > -2.12)
= 0.9838 (using standard normal table or calculator)
c. To find the 70th percentile of X, we need to find the value x such that P(X < x) = 0.70. Using the same normal distribution with mean 100 and standard deviation 10/sqrt(50) = 1.41, we can find the z-score that corresponds to the 70th percentile:
P(Z < z) = 0.70 => z = 0.52
Then, using the formula (x - 100)/(1.41) = z, we can solve for x:
x = 100 + 0.52(1.41) = 100.73
Therefore, the 70th percentile of X is approximately 100.73.
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Marked price 545 selling price 541 what is the discount offered
Answer: 1% off
Step-by-step explanation:
541 is about 99% of 454, leaving 1% discount
if a review of a product on forest.com has ten words in total, including two negative words and three positive words, what would the sentiment score when conducting a sentiment analysis?
The sentiment score for the review on forest.com would depend on the specific algorithm and weighting system used for the sentiment analysis.
However, a basic calculation could be done by assigning a numerical value to each positive and negative word (e.g. +1 for positive, -1 for negative) and adding them up. In this case, the sentiment score would be +1 (3 positive words) -2 (2 negative words) = -1. However, it's important to note that sentiment analysis is not always straightforward and can be influenced by factors such as context, sarcasm, and language nuances.
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Determine the missing angle measures, Can someone please answer this I need to turn this in Im in a hurry i beg
The measure of angles are;
angle Y = 29 degrees
angle W = 112
angle X = 39
we can write as
29 + angle Z + 112 = 180° (Being linear pair angles)
angle Z = 180 - 112 - 29
angle Z = 39
Therefore,
angle Z = angle X = 39 degrees (by vertically opposite angles)
angle Y = 29 degrees(by vertically opposite angles)
angle W = 112 degrees(by vertically opposite angles)
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suppose ()=100, ()=200, ()=300 (∩)=10, (∩)=15, (∩)=20 (∩∩)=5 (∪∪)= A. 400 B. 600 C. 300 D. NONE
The cardinality of the union is 240, none of the options listed.
How to find union cardinality?
Based on the symbols and values given, it seems like you are referring to a set notation problem. Here is what I understand from the symbols and values given:
()=100 means the cardinality (size) of set A is 100.
()=200 means the cardinality of set B is 200.
()=300 means the cardinality of set C is 300.
(∩)=10 means the intersection of sets A and B has a cardinality of 10.
(∩)=15 means the intersection of sets B and C has a cardinality of 15.
(∩)=20 means the intersection of sets A and C has a cardinality of 20.
(∩∩)=5 means the intersection of sets A, B, and C has a cardinality of 5.
To find the cardinality of the union of all three sets (i.e., (∪∪)), we can use the formula:
|A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |A ∩ C| + |A ∩ B ∩ C|
Substituting the given values, we get:
|A ∪ B ∪ C| = 100 + 200 + 300 - 10 - 15 - 20 + 5 = 240
Therefore, the answer is not one of the options given.
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if a is an m x n matrix, if the equation = has at least two different solutions, and if the equation = is consistent, then the equation = has many solutions. true or false
if the equation Ax=b has at least two different solutions and is consistent, then it has infinitely many solutions.
If the equation Ax=b has at least two different solutions, then there exists more than one vector x that satisfies the equation. This means that the system of equations Ax=b has more than one solution. Since the system is consistent, there exists at least one solution.
If we have at least two solutions to the system Ax=b, say x1 and x2, then any linear combination of x1 and x2 is also a solution. That is, if we take any scalar values c1 and c2, then the vector y=c1x1+c2x2 is also a solution to the equation Ax=b.
Therefore, if the equation Ax=b has at least two different solutions and is consistent, then it has infinitely many solutions.
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why is it possible for condition means to differ at the end of an experiment even if the independent variable had no effect? type ii error has ocurrederror variancerandomizationa null finding has been obtained
Answer:
Answer:- 1) option b) error variance.Have a Nice Best Day : ) Please Give Me Brainliest
The equation can be used to determine T, the temperature in degrees, inside an oven s seconds after the oven is turned on. Which statement relative to this equation is true?
The statement "The initial temperature inside the oven was 78 degrees" is true of the equation (option 3)
Why is the statement true?The equation T = 0.63s + 78 shows that the initial temperature inside the oven is 78 degrees. The temperature inside the oven increases by 0.63 degrees every second. So, after one minute (60 seconds), the temperature inside the oven would be 141 degrees. However, this is not stated in the equation.
Therefore the third statement if true of the equation (The initial temperature inside the oven was 78 degrees" is true of the equation)
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Complete question:
The equation T = 0.63s + 78 can be used to determine the temperature in degrees, inside an oven s seconds after the oven is turned on. Which statement relative to this equation is true?
1. The temperature inside the oven increased by 78 degrees every 63 seconds.
2. The temperature inside the oven in one minute was 141 degrees
3. The initial temperature inside the oven was 78 degrees.
The graph of a quadratic function is shown. What is the equation of this function in f(x) = ax²+bx+c form?
Y
(-2, 2)
-2
Complete the equation of the function: f(x) =
taking a peek at the quadratic graph, we can see it has a vertex at (-2 , 2) and it has a couple of points, hmmm we only need one, say we'll use (-3 , -1)
[tex]~~~~~~\textit{vertical parabola vertex form} \\\\ y=a(x- h)^2+ k\qquad \begin{cases} \stackrel{vertex}{(h,k)}\\\\ \stackrel{a~is~negative}{op ens~\cap}\qquad \stackrel{a~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} h=-2\\ k=2\\ \end{cases}\implies y=a(~~x-(-2)~~)^2 + 2\hspace{4em}\textit{we also know that} \begin{cases} x=-3\\ y=-1 \end{cases}[/tex]
[tex]-1=a(-3-(-2))^2+2\implies -1=a(-1)^2 + 2\implies -3=a \\\\[-0.35em] ~\dotfill\\\\ y=-3(~~x-(-2)~~)^2 + 2\implies y=-3(x+2)^2 +2 \\\\\\ y=-3(x^2+4x+4)+2\implies y=-3x^2-12x-12+2 \\\\\\ ~\hfill {\Large \begin{array}{llll} y=-3x^2-12x-10 \end{array}}~\hfill[/tex]
Check the picture below.
Which situation describes exponential decay? a. The value of a piece of machinery decreases by 19% every year. B. The value of a piece of machinery decreases by $5000 every year
The situation that describes exponential decay is (a) The value of a piece of machinery decreases by 19% every year.
What is the exponential function?
An exponential function is a mathematical function of the form:
f(x) = aˣ
where "a" is a constant called the base, and "x" is a variable. Exponential functions can be defined for any base "a", but the most common base is the mathematical constant "e" (approximately 2.71828), known as the natural exponential function.
Exponential decay is a mathematical term used to describe the process of decay that happens at an exponential rate.
In this case, the value of the machinery decreases by a percentage every year.
Since the percentage decrease is constant over time, this is an example of exponential decay.
On the other hand, option (b) describes a linear decrease in value since the value decreases by a constant amount of $5000 every year.
Hence, the situation that describes exponential decay is (a) The value of a piece of machinery decreases by 19% every year.
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Please solve! Solve for Y.
when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one. Check the picture below.
[tex]\cfrac{24}{y}=\cfrac{y}{30}\implies 720=y^2\implies \sqrt{720}=y\implies 12\sqrt{5}=y\implies 26.83\approx y[/tex]
The midday temperature in Palm
Beach was 86 °F. The
temperature then
changed 2 °F per hour for the next
3 hours. The expression 86 -|-2-3|
represents the current temperature.
What is the current temperature in
degrees Fahrenheit?
After considering all the given data we conclude that the current measured temperature in degree Fahrenheit is 91°F, under the condition that the midday temperature in Palm Beach was 86 °F and the given expression is 86 -|-2-3| which represents current temperature.
To evaluate the temperature let us first consider the expression 86 -|-2-3| which represents the current temperature after 3 hours of temperature change of 2°F per hour from an initial temperature of 86°F.
The given expression can be again simplified as
86 -|-2-3| = 86 -|-5|
= 86 - (-5)
= 91°F
Hence, after evaluating we finally measured the current temperature as 91°F.
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find the length of the curve of x(t)=2t3,y(t)=4t2 2, for t∈[0,1].
The length of the curve for t∈[0,1] is 4/9. To find the length of the curve defined by x(t)=[tex]2t^3[/tex] and y(t)=[tex]4t^2[/tex]+2, we need to use the arc length formula:
L = ∫√[tex](dx/dt)^2 + (dy/dt)^2[/tex] dt from t=0 to t=1.
First, let's find dx/dt and dy/dt:
dx/dt = [tex]6t^2[/tex]
dy/dt = 8t
Now we can plug these into the arc length formula:
L = ∫√[tex](6t^2)^2[/tex]+ [tex](8t)^2[/tex] dt from t=0 to t=1
L = ∫√([tex]36t^4 + 64t^2[/tex]) dt from t=0 to t=1
L = ∫√([tex]4t^2(9t^2 + 16)[/tex]) dt from t=0 to t=1
L = 2∫√([tex]9t^2[/tex] + 16) dt from t=0 to t=1
To integrate this expression, we can use a trig substitution. Let u = (3/4)[tex]t^2[/tex] and du = (3/2)t dt. Then:
L = 2∫√(9(4/3)u + 16) (2/3) du from u=0 to u=3/4
L = (4/3)∫√(12u + 48) du from u=0 to u=3/4
L = (4/3)∫√(12(u + 4)) du from u=0 to u=3/4
L = (4/3)∫2√(u + 4) du from u=0 to u=3/4
L = (8/3)[(2/3)[tex](u + 4)^{(3/2)}[/tex]] from u=0 to u=3/4
L = (8/3)[(2/3)[tex](3/4 + 4)^{(3/2)}[/tex] - [tex](2/3)(4)^{(3/2)}[/tex]]
L = (8/3)[(2/3)[tex](25/4)^{(3/2)}[/tex] - (2/3)(8)]
L = (8/3)[(25√25)/6 - 16/3]
L = (8/3)(25/6 - 16/3)
L = (8/3)(1/6)
L = 4/9
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find an equation of the plane. the plane through the point (9, −9, −6) and parallel to the plane 7x − y − z = 1
Using the normal vector of the parallel plane <7, -1, -1>. By Substituting the given points (9, -9, -6) in the plane equation and got the new plane equation: 7x - y - z = 69.
To find an equation of the plane passing through a given point and parallel to another plane, we need to use the normal vector of the parallel plane and substitute the given point in the equation of the plane.
A plane is defined by a point and a normal vector perpendicular to it. Therefore, to find an equation of the plane passing through the point (9, −9, −6), we need to determine its normal vector.
The given plane 7x − y − z = 1 can be written in the form Ax + By + Cz = D, where A = 7, B = -1, C = -1 and D = 1. The coefficients of x, y, and z in this equation give the components of the normal vector to the plane. Thus, the normal vector is N = (7, -1, -1).
Since the plane we want to find is parallel to the given plane, it has the same normal vector. Therefore, we can write its equation as 7x - y - z = D, where D is a constant. To determine D, we substitute the coordinates of the given point into the equation:
7(9) - (-9) - (-6) = 69
Thus, the equation of the plane passing through the point (9, −9, −6) and parallel to the plane 7x − y − z = 1 is 7x - y - z = 69.
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