(a) g(x) is continuous at c = 0, we need to show that for any ε > 0, there exists a δ > 0 such that |g(x) - g(0)| < ε whenever |x - 0| < δ.
We have g(x) = √3 x, so g(0) = 0. Let ε > 0 be given. Then, for any δ > 0, we have
|g(x) - g(0)| = |√3 x - 0| = √3 |x| < √3 δ
So, to make sure that |g(x) - g(0)| < ε, we can choose δ = ε/√3. Then, whenever |x - 0| < δ, we have |g(x) - g(0)| < ε. Therefore, g(x) is continuous at c = 0.
(b) g(x) is continuous at a point c ≠ 0, we need to show that for any ε > 0, there exists a δ > 0 such that |g(x) - g(c)| < ε whenever |x - c| < δ.
We have g(x) = √3 x, so g(c) = √3 c. Let ε > 0 be given. Then, for any δ > 0, we have
|g(x) - g(c)| = |√3 x - √3 c| = √3 |x - c|
Now, we use the identity a^3 - b^3 = (a - b)(a^2 + ab + b^2). Taking a = x and b = c, we have
a^3 - b^3 = (x^3 - c^3) = (x - c)(x^2 + xc + c^2)
Dividing both sides by (x - c), we get
x^2 + xc + c^2 = (x^3 - c^3)/(x - c)
Taking absolute values and simplifying, we get
|x^2 + xc + c^2| = |x - c||x^2 + xc + c^2|/|x - c| ≤ |x - c|( |x|^2 + |x||c| + |c|^2 )
Since |x - c| < δ, we can choose δ to be the smaller of ε/( |c|^2 + |c||δ| + |δ|^2 ) and 1, so that
|x^2 + xc + c^2| ≤ ε
Therefore, |g(x) - g(c)| = √3 |x - c| < ε/(|c|^2 + |c||δ| + |δ|^2), which shows that g(x) is continuous at c.
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#6 i
Find (a) f(g(x)), (b) g(f(x)), and (c)
f(f(x)).
f(x) = 2x², g(x)=x-1
a. f(g(x)) =
b. g(f(x)) =
C.f(f(x)) =
The solutions are given below,
(a) f(g(x)) = 2x² - 4x + 2
(b) g(f(x)) = 2x² - 1
(c) f(f(x)) = 8x⁴
To find f(g(x)), we substitute g(x) into the function f(x):
f(g(x)) = 2(g(x))²
f(g(x)) = 2(x-1)²
f(g(x)) = 2(x² - 2x + 1)
f(g(x)) = 2x² - 4x + 2
Therefore, f(g(x)) = 2x² - 4x + 2.
b. To find g(f(x)), we substitute f(x) into the function g(x):
g(f(x)) = f(x) - 1
g(f(x)) = 2x² - 1
Therefore, g(f(x)) = 2x² - 1.
c. To find f(f(x)), we substitute f(x) into the function f(x):
f(f(x)) = 2(f(x))²
f(f(x)) = 2(2x²)²
f(f(x)) = 2(4x⁴)
f(f(x)) = 8x⁴
Therefore, f(f(x)) = 8x⁴.
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A town's population
was 345,000 in 1996.
Its population
increased by 3%
each year.
The population after 1.5 years will be 360640.9.
Given that, a town's population was 345,000 in 1996. Its population increased by 3% each year.
The exponential growth =
A = P(1+r)ⁿ
A = final amount, P = initial amount, r = rate and n = time.
A = 345000(1+0.03)ⁿ
A = 345000(1.03)ⁿ
There is a growth factor of 1.03.
For n = 1.5
[tex]A = 345000(1.03)^{1.5[/tex]
A = 360640.9
Hence, the population after 1.5 years will be 360640.9.
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Una muestra de un metal contiene 4. 25 moles de molibdeno y 1. 63 moles de titanio expresa la relación de átomos y molecula
The problem statement is in Spanish and it asks to express the relationship between atoms and molecules for a metal sample containing [tex]4.25 moles[/tex] of molybdenum and [tex]1.63 moles[/tex] of titanium.
However, we can make some assumptions based on the typical behavior of metals. Metals usually exist in a solid state and consist of closely packed atoms arranged in a crystal lattice. Therefore, we can assume that the metal in question is solid, and its atoms are arranged in a regular pattern.
In this case, we can assume that the metal sample contains a mixture of molybdenum and titanium atoms, and the atoms are arranged in a crystal lattice structure. The ratio of moles of molybdenum to moles of titanium in the sample is approximately 2.61:1 (4.25/1.63), which means that there are more molybdenum atoms than titanium atoms in the sample.
Since the metal is solid, we can assume that the atoms are arranged in a crystal lattice, and the ratio of the number of atoms of each element in the crystal lattice is determined by the chemical formula of the compound. Without knowing the chemical formula, we cannot determine the exact ratio of atoms and molecules in the sample.
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two people rented a car from the same agency. the first person drove 1010 miles and paid $232.30 for mileage. the second person drove 765 miles and paid $175.95 for mileage. what is the agency's fee per mile?
To find the agency's fee per mile, we can use the formula: cost/mile = total cost / total miles driven. For the first person, the cost per mile is 232.30/1010 = 0.23 dollars per mile. For the second person, the cost per mile is 175.95/765 = 0.23 dollars per mile. Thus, we can conclude that the agency's fee per mile is 0.23 dollars.
To find the agency's fee per mile, we'll set up a system of equations based on the information provided and then solve for the fee.
Let's use the variables x and y to represent the fee per mile and the fixed cost of renting the car, respectively. The first person's rental can be represented as:
1010x + y = 232.30 (1)
The second person's rental can be represented as:
765x + y = 175.95 (2)
Now, we'll subtract equation (2) from equation (1) to eliminate the y variable:
(1010x + y) - (765x + y) = 232.30 - 175.95
This simplifies to:
245x = 56.35
Next, we'll solve for the fee per mile, x:
x = 56.35 / 245
x ≈ 0.23
Therefore, the agency's fee per mile is approximately $0.23.
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The slope-intercept equation of a line is y = -7x - 2. What is the slope of the
line?
OA. The slope is 2.
OB. The slope is -2.
OC. The slope is 7.
OD. The slope is -7.
SUBMIT
The calculated value of the slope of the line is -7
Calculating the slope of the line?From the question, we have the following parameters that can be used in our computation:
The slope-intercept equation of a line is y = -7x - 2
This means that
y = -7x - 2
A linear equation is represented as
y = mx + c
Where
Slope = m
using the above as a guide, we have the following:
m = -7
This means that the slope of the line is -7
Hence, the slope of the line is -7
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Ava drove her car at a constant rate to the train station. At the train station, she waited for the train to arrive. After she boarded the train, she traveled at a constant rate, faster than she drove her car. She entered the taxi and traveled at a constant speed. This speed was equal to the speed at which she had driven her car earlier. After some time, she arrived at her destination. Which graph represents Ava's travel plans?
1. The rate of change is -1 and the initial value is 1.
2. The graph represents Ava’s travel plans is graph (I).
What is Slope?The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. The net change in y coordinate is Δy, while the net change in the x coordinate is Δx. So the change in y coordinate with respect to the change in x coordinate can be written as,
m = Δy/Δx
where, m is the slope
Note that tan θ = Δy/Δx
We also refer this tan θ to be the slope of the line.
1. We have the coordinates as C(3, -2) and D(-2, 3).
So, the rate of change of linear function is
= 3 - (-2) / (-2 -3)
= 3+ 2 / (-5)
= 5/ (-5)
= -1.
and, the initial values is where the independent variable is zero which is (1, 0).
2. The graph represented for Ava journey is (A).
This, is because the speed of Ava car and speed of taxi is equal which is shown in graph 1 clearly.
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The given question is incomplete, complete question is:
1. A relation is plotted as a linear function on a coordinate plane starting at point C at (3, –2) and ending at point D at (–2, 3). What is the rate of change for the linear function and what is its initial value?
The rate of change is ______ and the initial value is ______.
A. 1 and -1
B. -1 and 1
C. 5 and -2
D. -2 and 5
2. Ava drove her car at a constant rate to the train station. At the train station, she waited for the train to arrive. After she boarded the train, she traveled at a constant rate, faster than she drove her car. She entered the taxi and traveled at a constant speed. This speed was equal to the speed at which she had driven her car earlier. After some time, she arrived at her destination.
Which graph represents Ava’s travel plans? (First 3 graphs are the options to this question.)
the network receives 830 responses, of which 439 indicate that they would like to see the new show in the lineup. the test statistic for this hypothesis would be .
To answer your question, we need to calculate the test statistic for the hypothesis.
Based on the information provided, we have:
- Number of total responses (n) = 830
- Number of positive responses (x) = 439
Assuming you want to test the proportion of positive responses, we can use the formula for the test statistic in a one-sample proportion hypothesis test:
z = (p_hat - p0) / sqrt(p0(1-p0)/n)
where p_hat is the sample proportion, p0 is the null hypothesis proportion, and n is the total number of responses. First, let's calculate p_hat:
p_hat = x/n = 439/830 ≈ 0.529
Now, to determine the test statistic, we need to know the null hypothesis proportion (p0). If you provide that information, I can help you calculate the test statistic (z).
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researchers plan to take another sample of whale and cruise ship encounters in the west arm sub-region of glacier bay. assuming , if the researchers would like to ensure that the standard deviation of the sample proportion is no larger than 0.03, how many encounters would they need to include in their sample? round your answer to the nearest whole number.
The researchers would need to include at least 278 encounters in their sample to ensure that the standard deviation of the sample proportion is no larger than 0.03.
To determine the required sample size, we need to use the formula for the standard deviation of the sample proportion (σp):
[tex]\sigma_p = \sqrt{(p * (1 - p) / n)}[/tex]
where:
p is the estimated proportion (we don't have this information, so we'll use 0.5 as a conservative estimate for maximum variance),
n is the sample size.
Since the researchers want to ensure that the standard deviation of the sample proportion is no larger than 0.03, we can set up the following inequality:
0.03 ≥ √(0.5 * (1 - 0.5) / n)
Squaring both sides of the inequality to eliminate the square root:
0.03² ≥ 0.5 * (1 - 0.5) / n
0.0009 ≥ 0.25 / n
Now, solve for n:
n ≥ 0.25 / 0.0009
n ≥ 277.78
Since the sample size (n) must be a whole number, the researchers would need to include at least 278 encounters in their sample to ensure that the standard deviation of the sample proportion is no larger than 0.03. Rounding up, the required sample size is 278 encounters.
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there are default tab stops every one inch on the horizontal ruler. _________________________
There are default tab stops every one inch on the horizontal ruler is true statement.
In word processing software, there are usually default tab stops set every one inch (or 2.54 cm) on the horizontal ruler. These tab stops are the default positions at which the insertion point will stop when the tab key is pressed.
The purpose of these tab stops is to make it easier to align text in columns or tables. By default, there are left-aligned, centered, right-aligned, decimal-aligned, and bar-aligned tab stops. Users can also add custom tab stops at specific positions on the ruler to suit their formatting needs.
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Use the region in the first quadrant bounded by √x, y=2 and the y - axis to determine the area of the region. Evaluate the integral.
A. 50.265
B. 4/3
C. 16
D. 8
E. 8π
F. 20/3
G. 8/3
E/ -16/3
The integral of the function √x, y=2 and the y is G. 8/3
How to determine the area of the region?You want the area between y=2 and y=√x.
Bounds
The square root curve is only defined for x ≥ 0. It will have a value of 2 or less for m √x ≤ 2
x ≤ 4 . . . . square both sides
So, the integral has bounds of 0 and 4.
Integral
The integral is
[tex]\int\limits^4_0 {[2-xx^{1/2} } \, dx = \int\limits^4_6 {2x-2/3x^{4/3} } \, dx = 8-2/3(\sqrt{4)x^{3} } =8/3[/tex]
Additional comment
You will notice that this is 1/3 of the area of the rectangle that is 4 units wide and 2 units high. That means the area inside a parabola is 2/3 of the area of the enclosing rectangle. This is a useful relation to keep in the back of your mind.
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last year, the revenue for financial services companies had a mean of 90 million dollars with a standard deviation of 22 million. find the percentage of companies with revenue less than 103 million dollars. assume that the distribution is normal. round your answer to the nearest hundredth.
Approximately 72.04% of financial services companies had revenue less than 103 million dollars last year.
To find the percentage of financial services companies with revenue less than 103 million dollars, we first need to standardize the value using the formula z = (x - μ) / σ, where x is the value we want to standardize (103 million), μ is the mean (90 million), and σ is the standard deviation (22 million).
z = (103 - 90) / 22 = 0.59
We then look up the percentage of companies below this z-score in a standard normal distribution table or use a calculator. The percentage of companies with revenue less than 103 million dollars is 72.04%, rounded to the nearest hundredth. Therefore, approximately 72.04% of financial services companies had revenue less than 103 million dollars last year.
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suppose the time a child spends waiting at for the bus as a school bus stop is exponentially distributed with mean 6 minutes. determine the probability that the child must wait at least 8 minutes on the bus on a given morning. a) 0.2636 b) 0.4636 c) 0.5636 d) 0.5272 e) 0.2364 f) none of the above
The probability that the child must wait at least 8 minutes for the bus on a given morning is 0.2636
To determine the probability that a child must wait at least 8 minutes for the bus on a given morning, given that the waiting time is exponentially distributed with a mean of 6 minutes, we'll use the exponential distribution formula:
[tex]P(T > t) = e^{(-t/μ)}[/tex]
where T is the waiting time, t is the specific time we are interested in (8 minutes in this case), μ is the mean waiting time (6 minutes), and e is the base of the natural logarithm (approximately 2.71828).
Step 1: Plug in the values into the formula:
[tex]P(T > 8) = e^{(-8/6)}[/tex]
Step 2: Simplify the exponent:
[tex]P(T > 8) = e^{(-4/3)}[/tex]
Step 3: Calculate the probability using the value of e:
[tex]P(T > 8) ≈ 2.71828^{(-4/3)} ≈ 0.2636[/tex]
Therefore, the probability that the child must wait at least 8 minutes for the bus on a given morning is approximately 0.2636, which corresponds to option (a).
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Whats 4/5 X 3/8?
I am super confused!!
Answer:
12/40
Step-by-step explanation:
to do multiplication with fractions is super simple you just have to multiply the numerators and denominators so the 2 top numbers (4 x 3) and the two bottom numbers (5 x 8) and create your fraction (12/40)
Answer:3/10
Step-by-step explanation: you first multiply the 2 numerator 4*3=12
Then multiply the 2 denominators 5*8=40 now you have 12/40 to simplify you divide both by 4 so you have 3/10
Rolled oats come out in cylindrical containers with a diameter of 5 inches and a height of 9 1/2 inches. these containers are shipped to grocery stores in boxes. each shipping box contains six rolled oats containers. The shipping company is trying to figure out the dimensions of the box for shipping the rolled oats containers so that ill use the least amount of cardboard. They are only considering boxes that are rectangular prisms so they are easy to stack.
a. What is the surface area of the box needed to ship these containers to the grocery store that uses the least amount of cardboard?
b. What is the volume of the box?
Answer:
a. The surface area of the box needed to ship the containers with the least amount of cardboard is about 340.26 square inches.
b. The volume of the box needed to ship the containers is about 923.08 cubic inches.
(Hope this helps)
Step-by-step explanation:
a. To find the surface area of the box, we need to add up the areas of all the sides of the box. Think of wrapping the box in wrapping paper. The area of the paper that covers the box is the surface area.
b. To find the volume of the box, we need to measure how much space is inside the box. Think of the box as a swimming pool. The amount of water that can fit inside the pool is its volume.
SOMEBODY HELP ME PLEASE
use a table of integrals with forms involving eu to find the indefinite integral. (use c for the constant of integration.) ∫ (1 / 1+e^12x) dx
The indefinite integral of (1 / 1+e^12x) is (1/12) ln|1+e^12x| + C, where C is the constant of integration.
To find the indefinite integral of (1 / 1+e^12x), we can use a table of integrals with forms involving eu. The form that matches our integral is ∫(1 / 1+e^u) du, where u=12x.
We can substitute u=12x and du/dx=12 to get ∫(1 / 1+e^12x) dx = (1/12) ∫(1 / 1+e^u) du.
Using the table of integrals, the integral of (1 / 1+e^u) du is ln|1+e^u| + C, where C is the constant of integration.
Substituting back in u=12x and multiplying by 1/12, we get the final answer: ∫(1 / 1+e^12x) dx = (1/12) ln|1+e^12x| + C.
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please do part a and b thank youUse the Mean Value Theorem to show that if x > 0, then sinr S.
The Mean Value Theorem is a crucial theorem of calculus that reveals a relationship between the gradient of a curve and the values of its associated function at the endpoint.
What is the Mean Value Theorem?Specifically, it states that provided f(x) is steady on the enclosed interval [a, b], and differentiable on (a, b), then there must exist a point c within the range of (a, b) such that
f(b) - f(a) = f'(c) * (b - a)
which translates to there being an individual c inside the parameterized region (a, b), such that the inclined angle of the tangent line to the graph at c is equal to the general incline of the graph between a and b.
The Mean Value Theorem possesses a plethora of utilities in mathematical analysis and calculus alike.
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Q7. Find the a) area of the region bounded by spiral r = 20 for 0 Sost, and b) the length of the same spiral (r = 20 for 0 So sn). (5 points each) re20 (2x)
The area of the region bounded by the spiral is 200π square units, and the length of the same spiral is 20π units.
a) To find the area of the region bounded by the spiral r = 20 for 0 ≤ θ ≤ π, we can use the polar coordinate system formula for area: Area = (1/2) ∫(r^2 dθ) from 0 to π
Given r = 20,
Area = (1/2) ∫((20)^2 dθ) from 0 to π
Area = (1/2) * 400 ∫(dθ) from 0 to π
Area = 200 [θ] from 0 to π
Area = 200(π - 0) = 200π square units.
b) To find the length of the same spiral (r = 20 for 0 ≤ θ ≤ π), we can use the formula for arc length in polar coordinates:
Arc Length = ∫(√(r^2 + (dr/dθ)^2) dθ) from 0 to π
Given r = 20 (a constant), dr/dθ = 0.
Arc Length = ∫(√((20)^2 + (0)^2) dθ) from 0 to π
Arc Length = 20 ∫(dθ) from 0 to π
Arc Length = 20[θ] from 0 to π
Arc Length = 20(π - 0) = 20π units.
So, the area of the region bounded by the spiral is 200π square units, and the length of the same spiral is 20π units.
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6. Hsu Mei did a study on reaction times of teenage drivers and used a box plot to display the data. If her reaction time is 0. 50 seconds, how does she compare to the reaction time of other teenage drivers? Explain
Answer:
There can be no answer, as you did not provide the box plot to compare the data.
semi circles i need help i dont get it please!
The radius of the semi circle is 6 inches and the circumference of the semi circle is 18.84 inches.
How to find the radius and circumference of a semi circle?A semi circle is half of a circle. The radius and circumference of the semi circle can be found as follows:
Therefore, radius is half of the diameter of a circle.
Hence,
radius of the semi circle = 12 / 2
radius of the semi circle = 6 inches
Therefore, let's find the circumference of the semi circle.
circumference of the semi circle = πr
circumference of the semi circle = 3.14 × 6
circumference of the semi circle = 18.84 inches
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What is the expected value for the binomial distribution below? Successes 0, 1, 2, 3, 4, 5 probability 243/3125, 162/625, 216/625, 48/625, 32/3125
The expected value for this binomial distribution is 0.5.
To find the expected value for the binomial distribution, we can use the formula:
E(X) = np
where:
X is the random variable representing the number of successes
n is the total number of trials
p is the probability of success in each trial
In this case, the binomial distribution has the following probabilities for the number of successes:
P(X=0) = 243/3125
P(X=1) = 162/625
P(X=2) = 216/625
P(X=3) = 48/625
P(X=4) = 32/3125
The total number of trials is the sum of the probabilities:
n = (243/3125) + (162/625) + (216/625) + (48/625) + (32/3125) = 1
The probability of success in each trial is the sum of the probabilities for X=1, X=2, X=3, and X=4:
p = (162/625) + (216/625) + (48/625) + (32/3125) = 0.5
Now we can use the formula to find the expected value:
E(X) = np = 1 * 0.5 = 0.5.
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Answer:
0.5616
Step-by-step explanation:
The expected value for a binomial distribution can be calculated using the formula E(X) = np, where n is the number of trials and p is the probability of success in each trial.
To calculate the expected value for the given binomial distribution, we need to multiply each number of successes by its corresponding probability and then sum them up.
0 successes: (0)(243/3125)
1 success: (1)(162/625)
2 successes: (2)(216/625)
3 successes: (3)(48/625)
4 successes: (4)(32/3125)
5 successes: (5)(1/3125)
Now, let's calculate each of these values:
0 successes: 0
1 success: 162/625
2 successes: 432/625
3 successes: 144/625
4 successes: 128/3125
5 successes: 5/3125
To find the expected value, we need to sum up these values:
0 + 162/625 + 432/625 + 144/625 + 128/3125 + 5/3125 = 0.5616
Therefore, the expected value for the given binomial distribution is approximately 0.5616.
do you dislike waiting in line? supermarket chain kroger has used computer simulation and information technology to reduce the average waiting time for customers at stores. using a new system called quevision, which allows kroger to better predict when shoppers will be checking out, the company was able to decrease average customer waiting time to just seconds (informationweek website). assume that waiting times at kroger are exponentially distributed. a. which of the probability density functions of waiting time is applicable at kroger? a. for b. for c. for d. for b b. what is the probability that a customer will have to wait between and seconds (to 4 decimals)? 0.5433 c. what is the probability that a customer will have to wait more than minutes (to 4 decimals)?
a. This is a very low probability, indicating that the new system implemented by Kroger is effective in reducing waiting times.
b. The probability that a customer will have to wait between 2 and 4 seconds is approximately 0.5433.
c. The probability that a customer will have to wait more than 5 minutes (300 seconds) is approximately 0.000006, or 0.0006%.
a. The probability density function of waiting time applicable at Kroger is the exponential distribution function.
b. The probability of a customer having to wait between 2 and 4 seconds can be calculated as follows:
Let λ be the rate parameter of the exponential distribution, which represents the average number of customers served per second. Since the waiting times are exponentially distributed, the probability density function of the waiting time t is given by:
[tex]f(t) = \lambda \times e^{(-\lambda\times t)}[/tex]
We want to find the probability that a customer will have to wait between 2 and 4 seconds. This can be calculated as the difference between the cumulative distribution functions (CDF) evaluated at 4 seconds and 2 seconds:
P(2 < t < 4) = F(4) - F(2)
where F(t) is the CDF of the exponential distribution:
[tex]F(t) = 1 - e^{(-\lambda \times t)}[/tex]
Substituting the value of λ (which we need to estimate), we can solve for the probability:
[tex]P(2 < t < 4) = (1 - e^{(-\lambda4)}) - (1 - e^{(-\lambda2)})\\= e^{(-\lambda2)} - e^{(-\lambda4)}[/tex]
To estimate λ, we can use the information given in the problem that the average waiting time is "just seconds". Let's assume that this means an average waiting time of 2 seconds. Then, the rate parameter λ can be estimated as:
λ = 1 / 2
Substituting this value in the equation above, we get:
[tex]P(2 < t < 4) = e^{(-1)} - e^{(-2)[/tex]
≈ 0.5433
c. The probability of a customer having to wait more than 5 minutes (i.e., 300 seconds) can be calculated as follows:
P(t > 300) = 1 - F(300)
where F(t) is the CDF of the exponential distribution as given above. Substituting the value of λ estimated earlier, we get:
[tex]P(t > 300) = 1 - (1 - e^{(-\lambda300)})\\= e^(-\lambda300)[/tex]
Substituting the value of λ, we get:
[tex]P(t > 300) = e^{(-150)}[/tex]
≈ 0.000006
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using the theorem, the curvature is given by (t) = |r'(t) × r''(t)| |r'(t)|3 . we first find the first and second derivatives. for r(t) = 3t i 7 sin(t) j 7 cos(t) k, we have
The first derivative is [tex]r'(t) = 3i + 7 cos(t) j - 7 sin(t) k[/tex], and the second derivative is [tex]r"(t) = -7 sin(t) j - 7 cos(t) k[/tex], then the curvature of the curve is [tex]k(0) = |-21i - 49j - 21k| / |3i + 7j|^3 = 7 / 58[/tex].
The theorem you mentioned is a formula for calculating the curvature of a curve at any point along the curve. In order to use the formula, we need to have a parametric equation for the curve, which is given to us as [tex]r(t) = 3ti + 7 sin(t) j + 7 cos(t) k[/tex].
To find the first and second derivatives of r(t), we simply differentiate each component with respect to t. The first derivative is [tex]r'(t) = 3i + 7 cos(t) j - 7 sin(t) k[/tex], and the second derivative is [tex]r"(t) = -7 sin(t) j - 7 cos(t) k[/tex].
Now we can use the formula for curvature to find the curvature of the curve at any point. For example, at t = 0, we have [tex]r'(0) = 3i + 7j, r"(0) = -7k[/tex], and plugging these values into the formula gives us[tex]k(0) = |-21i - 49j - 21k| / |3i + 7j|^3 = 7 / 58[/tex].
In general, the curvature measures how sharply a curve is turning at each point along the curve. A high curvature indicates a sharp turn, while a low curvature indicates a more gradual turn.
In this case, we can see that the curvature is relatively small, which means the curve is not turning very sharply at this particular point.
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Complete Question:
Using the theorem, the curvature is given by
[tex]k(t) = \frac{|r'(t) \times r"(t)|}{|r'(t)|^3}[/tex]
we first find the first and second derivatives. for r(t) = 3ti + 7 sin(t) j 7 cos(t)k, we have
Based on the mortality table below, what is the probability that a 28-year-old will be alive in 1 year?
A. 99,987
B. 99.87%
C. 99.99%
D. 0.13%
Answer: B. 99.87%
Step-by-step explanation:
What is the length??
[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ c^2=a^2+o^2\implies c=\sqrt{a^2 + o^2} \end{array} \qquad \begin{cases} c=hypotenuse\\ a=\stackrel{adjacent}{2}\\ o=\stackrel{opposite}{8} \end{cases} \\\\\\ c=\sqrt{ 2^2 + 8^2}\implies c=\sqrt{ 4 + 64 } \implies c=\sqrt{ 68 }\implies c\approx 8.2[/tex]
Answer:
8.3
Step-by-step explanation:
The equation to find the length of the hypotenuse from the legs is a² + b² = c². This means that you will need to square the lengths of the legs and then add them together to get the length of the hypotenuse squared. The lengths of the legs of this triangle are 2 and 8. 2 squared is 4, and 8 squared is 64. Add 64 + 4 to get 68. Now we know that 68 equals the hypotenuse squared. The square root of 68, 8.246, is the measurement of the hypotenuse. Now you just need to round up the number to the nearest tenth to get 8.3.
Find the maxima and minima, and where they are reached, of the function In f(x,y) = x² + y² + xy
{(x,y): x² + y² ≤ 1}
(I)Local. (ii) Absolutes. (iii) Identify the critical points inside the disk (not on the border) if any. Say if they are extreme '? what type?'o saddle points,'o we cannot tell using ___
i. The local maxima and minima are 3 and 2
ii. The absolute maximum of f(x,y) over the region is 3/2 at (1/√2, 1/√2), and the absolute minimum is -1/2, which is attained at (-1/√2, -1/√2).
iii. There are no other critical points inside the disk, so we cannot tell whether they are extreme or saddle points.
i. To find the maxima and minima of the function f(x,y) = x² + y² + xy over the region {(x,y): x² + y² ≤ 1}, we first find the critical points by setting the partial derivatives equal to zero:
f(x) = 2x + y = 0
fy = 2y + x = 0
Solving these equations simultaneously gives the critical point (-1/3, 2/3). We now need to check if this is a local maximum, local minimum or a saddle point. To do this, we use the second partial derivative test.
f(xx) = 2, f(xy) = 1, fyy = 2
The determinant of the Hessian matrix is Δ = f(xx)f(yy_ - (fxy)² = 2(2) - (1)² = 3, which is positive, and f(xx) = 2, which is positive. Therefore, the critical point is a local minimum.
ii. To find the absolute maximum and minimum, we need to consider the boundary of the region. Let g(x,y) = x² + y² be the equation of the circle with radius 1 centered at the origin. We can parameterize this curve as x = cos(t) and y = sin(t), where 0 ≤ t ≤ 2π.
Substituting this into the function f(x,y), we get:
h(t) = f(cos(t), sin(t)) = cos²(t) + sin²(t) + cos(t)sin(t) = 1 + (1/2)sin(2t)
We now find the critical points of h(t) by setting dh/dt = 0:
dh/dt = cos(2t) = 0
This gives t = π/4 and 5π/4.
Substituting these values into h(t), we get:
h(π/4) = 3/2
h(5π/4) = -1/2
Therefore, the absolute maximum of f(x,y) over the region is 3/2, which is attained at (1/√2, 1/√2), and the absolute minimum is -1/2, which is attained at (-1/√2, -1/√2).
iii. There are no other critical points inside the disk, so we cannot tell whether they are extreme or saddle points.
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how many different samples of size 2 can be selected from a population of size 10? multiple choice 45 10
The problem asks for the number of different samples of size 2 that can be selected from a population of size 10. To solve this problem, we can use the formula for the number of combinations of n objects taken r at a time, which is given by nCr = n!/(r!(n-r)!), where n is the size of the population and r is the size of the sample.
In this case, we have n=10 and r=2, so the number of different samples of size 2 that can be selected from a population of size 10 is given by 10C2 = 10!/(2!(10-2)!) = 45. Therefore, there are 45 different samples of size 2 that can be selected from a population of size 10.
Another way to think about this problem is to consider that when selecting a sample of size 2 from a population of size 10, we can choose the first element from any of the 10 objects in the population, and then choose the second element from the remaining 9 objects in the population (since we can't choose the same object twice).
Therefore, the total number of different samples of size 2 that can be selected is 10 x 9 = 90. However, since the order in which we choose the elements of the sample doesn't matter, we need to divide by 2 (the number of ways to arrange 2 elements), giving us a total of 45 different samples of size 2.
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Complete question:
How many different samples of size 2 can be selected from a population of size 10?
The use of CDs has been declining at a rate of 19% every year. At this
rate, if there are 109,750 CDs in Buffalo this year, how many are there
likely to be in 19 years?
Answer:
see below
Step-by-step explanation:
109750 * (1-19%)^19 =109750*0.018248 = 2002
a stack of 12 cards has 4 aces, 4 kings, and 4 queens. what is the probability of picking 3 queens from the stack?
To find the probability of picking 3 queens from the stack, we need to first find the total number of ways to pick 3 cards from the stack of 12. This is represented by the combination formula:
nCr = n! / (r! * (n-r)!)
where n is the total number of cards in the stack (12) and r is the number of cards we want to pick (3).
nCr = 12! / (3! * (12-3)!) = 220
So, there are 220 possible ways to pick 3 cards from the stack.
Now, we need to find the number of ways to pick 3 queens from the stack. Since there are 4 queens in the stack, we can use the combination formula again:
nCr = n! / (r! * (n-r)!)
where n is the number of queens in the stack (4) and r is the number of queens we want to pick (3).
nCr = 4! / (3! * (4-3)!) = 4
So, there are 4 possible ways to pick 3 queens from the stack.
Finally, we can find the probability of picking 3 queens by dividing the number of ways to pick 3 queens by the total number of ways to pick 3 cards:
P(3 queens) = 4 / 220 = 0.018 or approximately 1.8%.
To answer your question, let's calculate the probability of picking 3 queens from the stack of 12 cards containing 4 aces, 4 kings, and 4 queens.
The total number of ways to pick 3 cards from the stack of 12 cards is represented by the combination formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of cards and k is the number of cards chosen. In this case, n=12 and k=3.
C(12, 3) = 12! / (3!(12-3)!) = 12! / (3!9!) = (12 × 11 × 10) / (3 × 2 × 1) = 220
calculate the number of ways to pick 3 queens from the 4 queens available:
C(4, 3) = 4! / (3!(4-3)!) = 4! / (3!1!) = (4 × 3 × 2) / (3 × 2 × 1) = 4
Finally, divide the number of ways to pick 3 queens by the total number of ways to pick 3 cards to find the probability:
Probability = (Number of ways to pick 3 queens) / (Total number of ways to pick 3 cards) = 4 / 220 = 1/55 ≈ 0.0182
So, the probability of picking 3 queens from the stack is approximately 0.0182 or 1/55.
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Review for Exam 2 - MATH 2421 1. Find the absolute extrema of the functions on the given intervals: (a) f(x) = 7x2 + 1 on (-1,2] - (b) f(x) = 2x3 – 6x on [0, 3] (c) f(x) = ya on (–27, 27] 6.x2 (d)
(a) The absolute minimum occurs at x = -1, and the absolute maximum occurs at x = 2.
(b) the absolute minimum occurs at x = 0, and the absolute maximum occurs at x = 3
(c) The absolute minimum occurs at x = -27 and x = 27.
How to find the absolute extrema of [tex]f(x) = 7x^2 + 1[/tex] on the interval (-1,2]?
(a) To find the absolute extrema of [tex]f(x) = 7x^2 + 1[/tex] on the interval (-1,2], we need to check the critical points and the endpoints of the interval.
Critical points: We find f'(x) = 14x, so the critical point is x = 0.
Endpoints: f(-1) = 8 and f(2) = 29.
Thus, the absolute minimum occurs at x = -1 with a value of 8, and the absolute maximum occurs at x = 2 with a value of 29.
How to find the absolute extrema of [tex]f(x) = 2x^3 - 6x[/tex] on the interval [0, 3]?(b) To find the absolute extrema of [tex]f(x) = 2x^3 - 6x[/tex] on the interval [0,3], we need to check the critical points and the endpoints of the interval.
Critical points: We find [tex]f'(x) = 6x^2 - 6x = 6x(x - 1),[/tex] so the critical points are x = 0 and x = 1.
Endpoints: f(0) = 0 and f(3) = 45.
Thus, the absolute minimum occurs at x = 0 with a value of 0, and the absolute maximum occurs at x = 3 with a value of 45.
How to find the absolute extrema of f(x) = ya on the interval (–27, 27] 6.x2?(c) To find the absolute extrema of [tex]f(x) = y^{(1/3)}[/tex] on the interval (-27,27], we need to rewrite f(x) in terms of x and then check the critical points and the endpoints of the interval.
[tex]f(x) = (x^2)^{(1/3)} = |x|^{(2/3)}[/tex]
Critical points: We find [tex]f'(x) = (2/3)|x|^{(-1/3)}\sgn(x)[/tex], where [tex]\sgn(x)[/tex] is the sign function.
The critical points are where f'(x) is undefined or equal to zero. Since f'(x) is undefined at x = 0 and not equal to zero anywhere else on the interval, there are no critical points.
Endpoints: f(-27) = 3 and f(27) = 3.
Thus, the absolute minimum occurs at x = -27 and x = 27 with a value of 3.
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