Answers
Answer:
0
Step-by-step explanation:
since X= -2 and -2 < -1
f(-2)= (-2) +2
=0
Good luck
f( - 2 ) = 0 for the given piecewise function.
To find f( - 2 ) for the given piecewise function, we need to determine which part of the function applies to the input value x = - 2.
Since - 2 is less than or equal to - 1, we use the first part of the function where f(x) = x + 2 for x ≤ - 1.
Now, substitute x = - 2 into the first part of the function:
f( - 2 ) = ( - 2 ) + 2
f( - 2) = 0
So, f( - 2 ) = 0 for the given piecewise function.
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Related Questions
Prove by induction
((x/y)^n+1)<((x/y)^n) n≥1 and 0
Answers
[tex] \: [/tex]
((x/y)^(n+1)) > ((x/y)^n) for all n ≥ 1 and x > y > 0.
To prove this statement by induction, we will use the principle of mathematical induction.
Base case: When n = 1, we have:
((x/y)^(1+1)) = ((x/y)^2) = (x^2)/(y^2)
((x/y)^1) = (x/y)
Since x > y > 0, we have x/y > 1. Therefore, (x^2)/(y^2) > (x/y), which means that the base case is true.
Inductive step: Assume that ((x/y)^(k+1)) > ((x/y)^k) for some arbitrary positive integer k. We want to prove that this implies that ((x/y)^(k+2)) > ((x/y)^(k+1)).
Starting with ((x/y)^(k+2)), we can write:
((x/y)^(k+2)) = ((x/y)^(k+1)) * ((x/y)^1)
Using the induction hypothesis, we know that ((x/y)^(k+1)) > ((x/y)^k), and we also know that x/y > 1. Therefore, we have:
((x/y)^(k+2)) > ((x/y)^k) * (x/y)
Simplifying this expression, we get:
((x/y)^(k+2)) > ((x/y)^k) * (x/y)
((x/y)^(k+2)) > ((x^k)/(y^k)) * (x/y)
((x/y)^(k+2)) > ((x^(k+1))/(y^(k+1)))
Therefore, we have shown that ((x/y)^(k+2)) > ((x/y)^(k+1)) for all positive integers k, which completes the inductive step.
By the principle of mathematical induction, we have proven that ((x/y)^(n+1)) > ((x/y)^n) for all n ≥ 1 and x > y > 0.
What is the perimeter of a rectangle with a base of 9 ft and a height of 10 ft?
Answers
Answer:
P=2(l+w)=2·(9+10)=38ft
Valeria thinks that smoking suppresses a person's appetite so they will weigh less than those who do not smoke. She randomly collected the weights of some smokers and nonsmokers and created the graph shown.
Which statement correctly compares the distributions?
Responses
A Since the range of nonsmokers is 13 lbs more than that of smokers there is much more variability in their weights.Since the range of nonsmokers is 13 lbs more than that of smokers there is much more variability in their weights.
B On average smokers weighed 35 pounds more than nonsmokers.On average smokers weighed 35 pounds more than nonsmokers.
C Almost half of the smokers weighed more than all of the nonsmokers in the sample.Almost half of the smokers weighed more than all of the nonsmokers in the sample.
D On average, nonsmokers weighed 13 lbs less than smokers.On average, nonsmokers weighed 13 lbs less than smokers.
E Even though smokers on average weighed more than nonsmokers the variability in their weights was about the same.
Answers
The correct statement that compares the distributions is:
D On average, nonsmokers weighed 13 lbs less than smokers.
What is the variability?
Variability refers to the amount of spread or dispersion in a set of data. It is a measure of how much the data values in a sample or population differ from each other.
One commonly used measure of variability is the standard deviation, which is the square root of the variance. The variance is the average of the squared differences from the mean.
Looking at the graph, we can see that the center of the distribution of smokers is around 178 lbs, while the center of the distribution of nonsmokers is around 165 lbs. This means that, on average, nonsmokers weigh less than smokers.
Option A is incorrect because the range is not a good measure of variability, and it does not necessarily mean that there is more variability in the weights of nonsmokers.
Option B is incorrect because the graph clearly shows that nonsmokers weigh less on average than smokers.
Option C is incorrect because we cannot make any conclusion about half of the smokers weighing more than all of the nonsmokers from the graph.
Option E is incorrect because the graph shows that the variability in the weights of smokers is greater than that of nonsmokers.
Hence, The correct statement that compares the distributions is:
D On average, nonsmokers weighed 13 lbs less than smokers.
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Consider the line y= -3/2x-8 . Find the equation of the line that is parallel to this line and passes through the point (-2,3). Find the equation of the line and passes through the point (-2,3)
Answers
Answer:
The given line has a slope of -3/2, since it is in the form y = mx + b, where m is the slope. Any line that is parallel to this line will also have a slope of -3/2.
To find the equation of the line that passes through the point (-2,3) and has a slope of -3/2, we can use the point-slope form of the equation of a line:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope. Substituting in the values we have:
y - 3 = (-3/2)(x - (-2))
y - 3 = (-3/2)x - 3
y = (-3/2)x - 3 + 3
y = (-3/2)x
Therefore, the equation of the line that is parallel to y = -3/2x - 8 and passes through the point (-2,3) is y = (-3/2)x.
Note that this line does not have a y-intercept, since it passes through the point (-2,3) and has a slope of -3/2.
Let f be a linear function. If (−3) = 5 and f(5) = −27, find f(x).
Answers
The f(x) of the linear function is:
f(x) = -4x - 7
How to f(x) of a linear function?Since f is a linear function. The general form of a linear function is:
y = mx + b
where y = f(x), m is the slope and b is the y-intercept
Since f (−3) = 5, we have:
x = -3 and y = 5
Substitute into y = mx + b:
y = mx + b
5 = -3m + b ---- (1)
Also, f(5) = −27, we have:
x = 5 and y = -27
Substitute into y = mx + b:
y = mx + b
-27 = 5m + b ---- (2)
Solving (1) and (2) simultaneously by elimination method:
-3m + b = 5
5m + b = -27
-8m = 32
m = 32/(-8)
m = -4
Put m = -4 in (1) and solve for b:
-3x + b = 5
-3(-4) + b = 5
12 + b = 5
b = 5 - 12
b = -7
Put m and b into f(x) = mx + b to get f(x). That is:
f(x) = mx + b
f(x) = -4x - 7
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A small ferry runs every half hour from one side of a large river to the other. The probability distribution for the random variable = money collected (in dollars) on a randomly selected ferry trip is shown here.
Money collected 0 5 10 15 20 25
Probability 0.02 0.05 0.08 0.16 0.27 0.42
Calculate the cumulative probabilities. Do not round.
(≤0) =
(≤5) =
(≤10) =
(≤15) =
(≤20) =
(≤25) =
The median of a discrete random variable is the smallest value for which the cumulative probability equals or exceeds 0.5.
What is the median of ?
Answers
The cumulative probabilities for the given probability distribution were calculated, and the median of the discrete random variable was found to be 20.
To find the median, we need to find the smallest value of the random variable for which the cumulative probability equals or exceeds 0.5.
The cumulative probabilities are:
(≤0) = 0.02
(≤5) = 0.07
(≤10) = 0.15
(≤15) = 0.31
(≤20) = 0.58
(≤25) = 1
The cumulative probability is the sum of the probabilities of all events that have an outcome less than or equal to a given value. For example, the cumulative probability for the event of collecting 5 dollars or less is the sum of the probabilities for collecting 0 dollars and 5 dollars, which is 0.02 + 0.05 = 0.07. Similarly, the cumulative probability for the event of collecting 10 dollars or less is the sum of the probabilities for collecting 0 dollars, 5 dollars, and 10 dollars, which is 0.02 + 0.05 + 0.08 = 0.15. The same process is used to calculate the cumulative probabilities for all other values. The median is the smallest value of the random variable for which the cumulative probability is greater than or equal to 0.5.
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Factor 12m2 + 17m – 5.
Answers
hope this helps
A camel can drink 15 gallons of water in 10 minutes. At this rate, how much water can the camel drink in 11 minutes?
HELP
Answers
Answer: 16.5 gallons of water.
Step-by-step explanation:
If it was me. I would be setting up as a table to keep my work organized.
So first we find how much 1 minute is.
15g : 10m
15/10 : 10m/10
1.5g : 1m
Then I multiply how many minutes there are.
1.5g x 11 : 1m x 1
16.5g : 11m
And there we find the answer of 16.5 gallons.
Happy Solving
Answer:16.5
Step-by-step explanation:
Add.
Your answer should be an expanded polynomial in
standard form.
(−46² + 8b) + (−46³ + 56² – 8b) =
Answers
The polynomial expression (−4b² + 8b) + (−4b³ + 5b² – 8b) when evaluated is −4b³ + b²
Evaluating the polynomial expressionWe can start by combining like terms.
The first set of parentheses has two terms: -4b² and 8b. The second set of parentheses also has three terms: -4b³, 5b², and -8b.
So we can first combine the like terms in the set of parentheses:
(−4b² + 8b) + (−4b³ + 5b² – 8b) = −4b³ + b²
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(5r^2+5r+1)-(-2+2r^2-5r)
Answers
Answer:
3r^2+10r+3
Step-by-step explanation:
The average temperature at the South Pole is - 45" F. The average
temperature on the Equator is 92º F. How much warmer is the average
temperature on the Equator than at the South Pole?
Answers
Answer:
The average temperature on the Equator is 137°F warmer than the average temperature at the South Pole.
76°c
Step-by-step explanation:
a book sold 33,600 copies in its first month of release. suppose this represents 6.7% of the number of copies sold to date. how many copies have been sold to date? answer to the nearest whole number
Answers
6.7% = 33,600/x
To solve for x, we can cross-multiply:
0.067x = 33,600
Dividing both sides by 0.067 gives:
x = 500,000
Therefore, approximately 500,000 copies have been sold to date.
First, 6.7 % can be written in decimal form as 0.067 (6.7 / 100 = 0.067).
Let's use the variable x to represent the number of copies sold to date.
Then we can write and solve the following equation to represent 6.7% of the total sold to date:
0.067 • x = 33600
You can solve this equation by dividing both sides of the equation by 0.067:
0.067 • x = 33600
0.067 0.067
x = 501493
To date, 500000 copies would have been sold rounded to the nearest whole.
Let X1 and X2 denote the proportions of time, out of one working day, that employee A and B, respectively, actually spend performing their assigned tasks. The joint relative frequency behavior of X1 and X2 is modeled by the density function. ( ) ⎩ ⎨ ⎧ + ≤ ≤ ≤ ≤ = 0 ,elsewhere x x ,0 x 1;0 x 1 xf x 1 2 1 2 1 2 , a) Find P( ) X1 ≤ 0.5,X 2 ≥ 0.25 answer 21/64 b) Find P( ) X1 + X 2 ≤ 1
Answers
Answer:
a) To find the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25, we need to integrate the given density function over the region where X1 ≤ 0.5 and X2 ≥ 0.25.
P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫∫(x1,x2) f(x1,x2) dxdy
where the limits of integration are:
0.25 ≤ x2 ≤ 1
0 ≤ x1 ≤ 0.5
Substituting the given density function:
P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 ∫0^0.5 (x1 + x2) dx1 dx2
Evaluating the inner integral:
P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(x1^2/2) + x1x2] |0 to 0.5 dx2
Simplifying the expression:
P(X1 ≤ 0.5, X2 ≥ 0.25) = ∫0.25^1 [(0.125 + 0.25x2)] dx2
Evaluating the upper and lower limits:
P(X1 ≤ 0.5, X2 ≥ 0.25) = [0.125x2 + 0.125x2^2] |0.25 to 1
Substituting the limits:
P(X1 ≤ 0.5, X2 ≥ 0.25) = [(0.125 + 0.125) - (0.03125 + 0.015625)]
Solving for the final answer:
P(X1 ≤ 0.5, X2 ≥ 0.25) = 21/64
Therefore, the probability that X1 is less than or equal to 0.5 and X2 is greater than or equal to 0.25 is 21/64.
b) To find the probability that X1 + X2 is less than or equal to 1, we need to integrate the given density function over the region where X1 + X2 ≤ 1.
P(X1 + X2 ≤ 1) = ∫∫(x1,x2) f(x1,x2) dxdy
where the limits of integration are:
0 ≤ x1 ≤ 1
0 ≤ x2 ≤ 1-x1
Substituting the given density function:
P(X1 + X2 ≤ 1) = ∫0^1 ∫0^(1-x1) (x1 + x2) dx2 dx1
Evaluating the inner integral:
P(X1 + X2 ≤ 1) = ∫0^1 [(x1x2 + 0.5x2^2)] |0 to (1-x1) dx1
Simplifying the expression:
P(X1 + X2 ≤ 1) = ∫0^1 [(x1 - x1^2)/2 + (1-x1)^3/6] dx1
Evaluating the integral:
P(X1 + X2 ≤ 1) = [x1^2/4 - x1^3/6 - (1-x1)^4/24] |0 to 1
Substituting the limits:
P(X1 + X2 ≤ 1) = (1/4 - 1/6 - 1/24) - (0/4 - 0/6 - 1/24)
Solving for the final answer:
P(X1 + X2 ≤ 1) = 1/8
Therefore, the probability that X1 + X2 is less than or equal to 1 is 1/8.
a basement bedroom must have a window with an opening area of at least 5.7 square feet per the international residential code. a rectangular basement window opening is 0.75 meters wide.Among the following heights, in meters, which is the smallest that will qualify the window opening per the code.
Answers
The smallest that will qualify the window opening per the code is 0.71
What is rectangular?
A quadrilateral with four right angles is a rectangle. It can alternatively be described as a parallelogram with a right angle or an equiangular quadrilateral, where equiangular denotes that all of its angles are equal. A square is a rectangle with four equally long sides.
Here, we have
Given: a basement bedroom must have a window with an opening area of at least 5.7 square feet per the international residential code. A rectangular basement window opening 0.75 meters wide.
First, we convert square feet into square meters.
5.7 square feet = 0.53 square meters
Now,
0.53 / 0.75 = 0.71
Hence, the smallest that will qualify the window opening per the code is 0.71
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Polygon JKLMNO and polygon PQRSTU are similar. The area of polygon
JKLMNO is 27. What is the area of PQRSTU?
Answers
Check the picture below.
[tex]\cfrac{3^2}{4^2}=\cfrac{27}{A}\implies \cfrac{9}{16}=\cfrac{27}{A}\implies 9A=432\implies A=\cfrac{432}{9}\implies A=48[/tex]
Find the area of this composite figure: *find the area of each figure, then add those areas together
Answers
Answer:
136 units
Step-by-step explanation:
All sides are equal in a rectangle:
Value of b : 16-8 = 8 units
h = 13-7 = 6 units.
So Area of triangle= bh/2 = 8*6/2 = 24 units
Area of rectangle = lb = 16*7 = 112 units
So Area of figure= 112+24 units = 136 units
< Rewrite the set O by listing its elements. Make sure to use the appropriate set nota O={y|y is an integer and -4≤ y ≤-1}
What is the answer please?
Answers
Answer:
O = { -4,-3,-2,-1,0,-1 }
The set O can be rewritten by listing its elements as:
O = {-4, -3, -2, -1}
Step-by-step Explanation:
This is because O is defined as the set of all integers y such that -4 is less than or equal to y, and y is less than or equal to -1. So, O includes all the integers between -4 and -1, inclusive, which are -4, -3, -2, and -1.
help with math problems.
Answers
Answer:
yes.
Step-by-step explanation:
cause yes.
Find the perimeter and total area
Answers
The perimeter is 27 feet and the area is 35 square feet
Calculating the perimeter and the area
From the question, we have the following parameters that can be used in our computation:
The figure
The perimeter is the sum of tthe side lengths
So, we have
Perimeter = 7.5 + 6 + (6 - 2.5) + 4 + 2.5 + 3.5
Evaluate
Perimeter = 27
The area is calculated as
Area = 6 * 3.5 + 4 * (6 - 2.5)
Evaluate
Area = 35
Hence, teh area is 35 square feet
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HELPP!
When are the values of f(x) positive, and when are they negative?
Answers
The values of function f(x) positive, and negatives are (-infinity, infinity )
What exactly is a function?A function is a procedure or link that connects every element of one non-empty set A to at least one element of another non-empty set B. The phrases "domain" and "co-domain" are used in mathematics to define a function f between two sets, A and B. The constraint F = (a,b)| is satisfied by all values of a and b.
In the case of the question,
f (x) = x²
f (x) will be positive for all x values. As a result of the function:
x² = x × x
That is, when any number or integer is multiplied by itself, the result is positive. (For example, - - = + and + + = +)
As a result, f (x) = x2 will be positive for (,).
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Suppose a jar contains 12 red marbles and 12 blue marbles. If you reach in the jar and pull out 2 marbles at random at the same time, find the probability that both are red.
Answers
As a result, there is a 26% chance that two red marbles will be chosen at random, or around 0.26.
what is probability ?The area of mathematics known as probability is concerned with analysing the results of random events. It represents a probability or likelihood that a specific occurrence will occur. A number in 0 and 1 is used to represent probability, with 0 denoting an event's impossibility and 1 denoting its certainty. In order to produce predictions and guide decision-making, probability is employed in a variety of disciplines, such science, finance, economics, architecture, and statistics.
given
Given that there are 12 red marbles and a total of 24 marbles in the jar, the likelihood of choosing the first red marble is 12/24.
There are 11 red marbles and a total of 23 marbles in the jar after choosing the first red marble.
As a result, the likelihood of choosing a second red marble is 11/23.
We compound the probabilities to determine the likelihood of both outcomes occurring simultaneously (i.e., choosing two red marbles):
P(choosing 2 red marbles) = (12/24) x (11/23) = 0.2609, which is roughly 0.26.
As a result, there is a 26% chance that two red marbles will be chosen at random, or around 0.26.
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If someone has a dog, what is the probability that they also have a cat?
1/6
5/7
5/12
1/3
Answers
Answer:
5/12
Step-by-step explanation:
We are only looking at people who have a dog, which is 12
Now we need to determine if they have a cat
P( cat given that they have a dog)
= number of people with a cat/ they have a dog
= 5/ (5+7)
= 5/12
Given sin x = 4/5 and cos x= 3/5.
What is the ratio for tan x?
Enter your answer in the boxes as a fraction in simplest form.
Answers
Answer:
[tex]tan(x)=\frac{4}{3}[/tex]
Step-by-step explanation:
In the unit circle,
- [tex]cos(a)=\frac{x}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]x[/tex] is the x-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle
- [tex]sin(a)=\frac{y}{r}[/tex] where [tex]a[/tex] is the degree measure, [tex]y[/tex] is the y-coordinate of the triangle, and [tex]r[/tex] is the radius of the circle
Thus, since tangent is equal to sine over cosine, we can simplify our knowledge to: [tex]tan(a)=\frac{sin(a)}{cos(a)}=\frac{y}{x}[/tex]
In this problem, [tex]sin(x)=\frac{4}{5}[/tex]. We can conclude from our previous knowledge that [tex]y=4[/tex] and the radius is 5.
Similarly, [tex]cos(x)=\frac{3}{5}[/tex], which means [tex]x=3[/tex] and the radius is the same, at 5.
Since we know that [tex]x=3[/tex] and [tex]y=4[/tex], we can find the value of [tex]tan(x)[/tex] by using the formula [tex]tan(x)=\frac{y}{x}[/tex] and plug in the numbers.
Therefore, [tex]tan(x)=\frac{4}{3}[/tex].
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matt saves $100 one month, $50 for three months, $150 for four months, and $75 for the rest of the months of that year. how much does he save in one year?
Answers
Step by step
We know our equation can be written
1(100) + 3(50) + 4(150) + 4(75) = $
100 + 150 + 600 + 300 = $1150.
$1150 Saved in one year
The quality control manager at a computer manufacturing company believes that the mean life of a computer is 120 months, with a standard deviation of 10 months. If he is correct, what is the probability that the mean of a sample of 90 computers would be greater than 117.13 months? Round your answer to four decimal places.
Answers
The probability that the mean of a sample of 90 computers would be greater than 117.13 months, if the quality control manager is correct, is approximately 0.9955 or 99.55%.
The sampling distribution of the sample mean follows a normal distribution with a mean of 120 and a standard deviation of 10/sqrt(90) = 1.0541 months (using the formula for the standard deviation of the sample mean).
To find the probability that the mean of a sample of 90 computers would be greater than 117.13 months, we can standardize the sample mean using the formula:
z = (sample mean - population mean) / (standard deviation of sample mean) = (117.13 - 120) / 1.0541 = -2.6089
Using a standard normal distribution table or calculator, we can find that the probability of obtaining a z-score greater than -2.6089 is approximately 0.9955.
Therefore, the probability that the mean of a sample of 90 computers would be greater than 117.13 months, if the quality control manager is correct, is approximately 0.9955 or 99.55%.
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how can 32 div 4 help you solve 320 div 4
Answers
Answer:
you just add a 0 at the end of the answer of what 32 divided by 4 is, so in this case 320 divided by 4 is 80
Step-by-step explanation:
32 divided by 4 is 8.
320 divided by 4 is 80.
To get from 32 to 320 all you need is a 0 at the end, so you can just add the 0 the end of the answer. This means you're going from an 8, to an 80.
OR
Another way you can look at it is 32 multiplied by 10 to get 320. So you need to mutiple your answer by 10 to get the right answer.
32*10=320
8*10=80
Hope this helps!
PLEASE HELP
7x+3y=20 and -4x-6y=11 find the value of 3x-3y
Answers
Answer:
[tex]3x-3y=31[/tex]
Step-by-step explanation:
Adding the equations yields [tex]3x-3y=20+11=31[/tex].
One travel bag is 15 inches long, another is 18 inches long, and a third is 21 inches long. They are all 10 inches deep and 15 inches wide. Which travel bag can hold exactly 3,150 cubic inches? Explain your reasoning. 50 points to whoever answers
Answers
Answer:
The travel bag that can hold exactly 3,150 cubic inches is the bag that is 21 inches long.
Step-by-step explanation:
The travel bags can be modelled as rectangular prisms.
The formula for the volume of a rectangular prism is:
[tex]\boxed{\textsf{Volume} = l \times w\times d}[/tex]
where:
[tex]l[/tex] is the length.[tex]w[/tex] is the width.[tex]d[/tex] is the depth.Given that all three bags have a depth of 10 inches and a width of 15 inches, we can create an equation for the volume of any of the bags by substituting d = 10 and w = 15 into the formula:
[tex]\begin{aligned}\sf Volume &=l \times w \times d\\&= l \times 15\times10\\&=l \times 150\\&=150\;l\end{aligned}[/tex]
To determine which travel bag can hold exactly 3,150 cubic inches, substitute volume = 3150 into the formula and solve for length, l:
[tex]\begin{aligned}\sf Volume &=150\;l\\\\ \implies 3150&=150\;l\\\\\dfrac{3150}{150}&=\dfrac{150\;l}{150}\\\\21&=l\\\\l&=21\;\sf in\end{aligned}[/tex]
Therefore, the travel bag that can hold exactly 3,150 cubic inches is the bag that is 21 inches long.
Solve problem in the picture!
The equation
(x² + y²)² = 4(x² - y²)
defines a lemniscate (a "figure eight" or "oo-shaped curve"). The point P= (√5/8, √3/8) is on this lemniscate. Determine an
equation for the line , which is tangent to the lemniscate at the point P. The figure below, which is drawn to scale, may help to
understand the problem (and may help you to check your answer for "reasonableness").
Bonus Question: [up to 3 points] Let Q = (2,1), and determine an equation for the line which is tangent to the lemniscate at Q.
Answers
1. The equation for the line, which is tangent to the lemniscate at the point P is y = -√3x + (5/4 + √3/8). The equation for the line which is tangent to the lemniscate at Q is y = (-5/3)x + 11/3.
What is derivative of a function?The pace at which a function is changing at a specific point is known as its derivative. It shows the angle at which the tangent line to the curve at that location slopes. A key idea in calculus, the derivative can be utilised to tackle a range of issues, such as curve analysis, rates of change, and optimisation.
The tangent line to the lemniscate at point P, is determined using the derivative of the function.
(x² + y²)² = 4(x² - y²)
Taking the derivative on both sides we have:
2(x² + y²)(2x + 2y(dy/dx)) = 8x - 8y(dy/dx)
dy/dx = (x² + y²)/(y - x)
Substituting P= (√5/8, √3/8) for the x and y we have:
dy/dx = (√5/8)² + (√3/8)²) / (√3/8 - √5/8) = -√3
Thus, the slope of the tangent line at point P is -√3.
Using the point slope form:
y - y1 = m (x - x1)
Substituting the values we have:
y - (√3/8) = -√3(x - √5/8)
y = -√3x + (5/4 + √3/8)
Hence, equation for the line, which is tangent to the lemniscate at the point P is y = -√3x + (5/4 + √3/8).
Bonus question:
The equation of tangent for the lemniscate at point Q = (2,1) is:
dy/dx = (2² + 1²)/(1 - 2) = -5/3
Using the point slope form:
y - 1 = (-5/3)(x - 2)
y = (-5/3)x + 11/3
Hence, equation for the line which is tangent to the lemniscate at Q is y = (-5/3)x + 11/3.
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worth 20 points, pls help!!!
Answers
The probability that a randomly selected light bulb will last between 750 and 900 hours is given as follows:
P = 47.5%.
What does the Empirical Rule state?The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:
The percentage of scores within one standard deviation of the mean of the distribution is of approximately 68%.The percentage of scores within two standard deviations of the mean of the distribution is of approximately 95%.The percentage of scores within three standard deviations of the mean off the distribution is of approximately 99.7%.In the context of this problem, we have that:
750 hours is the mean.900 hours is two hours above the mean.The normal distribution is symmetric, hence the probability of an observation between the mean and two standard deviations above the mean is given as follows:
0.5 x 95 = 0.475 = 47.5%.
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