In the notation "ds," the "d" represents an infinitesimally small increment, while "s" represents the arc length.
In the notation "ds," the "d" represents an infinitesimally small increment or differential. It is used to indicate that we are considering an extremely small part of the whole quantity. In this case, "d" is used to denote an infinitesimally small length along the wire.
The "s" in "ds" represents the arc length. It is the length of the wire segment corresponding to the infinitesimally small increment "d" under consideration. The arc length is the cumulative sum of all these infinitesimally small lengths along the wire.
While it is possible to represent the arc length as just "L" in some contexts, using "ds" helps to explicitly indicate the infinitesimally small nature of the increment. It emphasizes that we are considering a continuous curve and performing calculus operations involving differentials. Thus, using "ds" is more appropriate for these types of problems.
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Among various populations of plants or animals, diseases spread exponentially. Use the function y = 6000(1 - e^-0.154)
to model the spread of the Avian Bird Flu disease among a flock of 6000 chickens on a chicken farm, with t equal to
the number of days since the first case of the disease. How many birds will be infected with the Flu after 6 days?
3560 chickens
C. 1280 chickens
356 chickens
d. 516 chickens
a.
Answer: A. 3560 chickens
Step-by-step explanation:
Plugged in the 6 days into the equation provided
Birds will be infected with the Flu after 6 days will be 3560 chickens
What is exponential?An exponential function is a mathematical function of the following form: f ( x ) = a^x. where x is a variable, and a is a constant called the base of the function.
Given function: y = 6000(1 - [tex]e^{-0.15t}[/tex])
Total birds are =6000
Now, take t= 6 days.
So, y = 6000(1 - [tex]e^{-0.15t}[/tex])
y= 6000(1- [tex]e^{-0.15 * 6}[/tex])
= 6000 (1-0.40656)
= 6000*0.59343
= 3560.58204
= 3560 birds
Hence, 3560 birds get infected in 6 days.
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Write the equation of the line based on the given information explain the answer in slope intercept form
Answer:
y = 2/3 x - 3
Step-by-step explanation:
substitute the given slope and the coordinate into the slope-intercpet form:
y = mx + b to solve for b,
-1 = 2/3(3) + b
=> b = -3
y = 2/3 x - 3
Please help! Will give Brainliest! SHOW ALL WORK
Factor:
2x^(2)+ay-ax^(2)-2y
Answer:
(2 - a) (x² - y)
Step-by-step explanation:
2x² + ay - ax² - 2y
=> 2x² - 2y + ay - ax²
=> 2 (x² - y) + -a (-y + x²)
=> 2 (x² - y) + -a (x² - y)
=> (2 - a) (x² - y)
Mr. Singh needed to understand the trend of his students’ latest test scores. Find the mean and median for each set of data, and then determine which gives a better picture of the scores.
Scores: 89, 99, 80, 5, 90, 0, 100, 95
Answer:
See explanations below
Step-by-step explanation:
Mean is the average of the data
Mean = sum of data/sample size
Sum of data = 89+99+80+5+90+0+100+95
Sum of data = 558
Sample size = 8
Mean = 558/8
Mean = 69.75
Median is the data value in the middle after rearrangement
Rearrange
0 , 5, 80) 89, 90, (95, 99 100
Median = 89+90/2
Median = 89.5
The parameter that gives a better picture of the scores is the Mean since all the given datas are taken into consideration
The mean test score is
✔ 69.75
.
The median test score is
✔ 89.5
.
Which measure of center best represents the test data?
✔ median
I hope this helps!
en que consiste la jerarquía de operaciones?
Answer:
La jerarquía de operaciones es un método para resolver operaciones con múltiples operadores; saber realizarla te servirá para resolver los diversos problemas que te presenten en tu examen CENEVAL EXANI-II. En primer lugar, se deben resolver las operaciones de potencias y raíces.
Step-by-step explanation:
hiiii pppplease help i’ll give brainliest
Answer:
Step-by-step explanation:
For observers right at the north pole and the south pole, there are only two seasons – an almost six-month long winter night followed by an almost six-month long summer day!
A package of 12 pencils costs $0.96. At this rate, how much would 100 pencils cost?
A. $8.35
B. $10.80
C. $9.60
D. $8.00
Answer:
B
Step-by-step explanation:
Please Give brainliest
Answer:
D
Step-by-step explanation:
For a fixed confidence level, when the sample size decreases, the length of the confidence interval for a population mean decreases.
a. True
b. False
The given statement "For a fixed confidence level, when the sample size decreases, the length of the confidence interval for a population mean decreases." is false, length of CI increases in this case.
When the sample size decreases, the length of the confidence interval for a population mean actually increases, not decreases.
The length of a confidence interval is influenced by several factors, including the sample size, the variability of the data, and the desired level of confidence. A confidence interval provides a range of values within which the population mean is estimated to lie.
When the sample size is smaller, there is less information available to estimate the population mean accurately. As a result, the confidence interval needs to be wider to accommodate the increased uncertainty. This wider interval allows for a larger range of potential values for the population mean.
On the other hand, as the sample size increases, there is more data available to estimate the population mean. This increased amount of information leads to a more precise estimate, allowing for a narrower confidence interval.
Therefore, for a fixed confidence level, as the sample size decreases, the length of the confidence interval for a population mean generally increases, not decreases.
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Use the appropriate substitution to find the explicit solution
to the Bernoulli DE dy/dt-y=e^(-2t)*y^2
[tex]y = (e^{(-2t)} + C . e^{(-t)})^{(-1)}[/tex]. is the explicit solution to the given Bernoulli differential equation. we used an appropriate substitution to transform it into a linear differential equation.
The given Bernoulli differential equation is in the form [tex]dy/dt - y = e^{(-2t)} . y^2[/tex]. Bernoulli equations can be transformed into linear equations by using a substitution. In this case, we can let [tex]v = y^{(-1)[/tex]be our substitution.
Differentiating v with respect to t gives dv/dt = -y^(-2) * dy/dt. Substituting this into the original equation, we have:
[tex]-dv/dt - v = e^{(-2t)}.[/tex]
Now, we have transformed the Bernoulli equation into a linear first-order differential equation. We can solve this equation using standard methods. Multiplying through by -1, we get [tex]dv/dt + v = -e^{(-2t)}[/tex].
The integrating factor for this linear equation is given by [tex]e^{ \[ \int_{}^{}1\,dt \] = e^t[/tex]. Multiplying both sides of the equation by e^t, we have:
[tex]e^t. (dv/dt + v) = -e^{(t-2t)[/tex].
Simplifying further, we get [tex]d/dt (e^t.v) = -e^{(-t)[/tex].
Integrating both sides with respect to t, we obtain:
[tex]e^t . v = \[ \int_{}^{} -e^{(-t)} \,dt \][/tex].
Evaluating the integral, we get:
[tex]e^t . v = e^{(-t) }+ C[/tex],
where C is the constant of integration.
Finally, solving for v, we have:
[tex]v = e^{(-2t)} + C .e^{(-t)[/tex].
Since [tex]v = y^{(-1)[/tex], we can take the reciprocal of both sides to find the explicit solution for y:
[tex]y = (e^{(-2t) }+ C .e^{(-t)})^{(-1)}[/tex].
This is the explicit solution to the given Bernoulli differential equation.
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Find the critical t-value that corresponds to 95% confidence. Assume 15 degrees of freedom.
The critical t-value that corresponds to a 95% confidence level with 15 degrees of freedom is approximately 2.131.
To find the critical t-value that corresponds to a 95% confidence level with 15 degrees of freedom, we can use a t-table or a statistical software.
Using a t-table or a statistical software, we look up the critical t-value for a two-tailed test with a confidence level of 95% and 15 degrees of freedom.
For a two-tailed test, we divide the desired confidence level by 2 to account for both tails of the t-distribution. In this case, we divide 95% by 2, which gives us 0.475.
Looking up the critical t-value for a confidence level of 0.475 and 15 degrees of freedom, we find that the critical t-value is approximately 2.131.
Therefore, the critical t-value that corresponds to a 95% confidence level with 15 degrees of freedom is approximately 2.131.
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Paper lanterns are $3.50 each. Write
a direct proportion equation using p
for paper lanterns and c for total
cost.
A. 3.50c = p
B. 3.5pc
C. 3.50 + p = C
D. 3.50p = 0
Answer:
it's probably C
Step-by-step explanation:
️♀️
a box in mrs booths class contains 200 loose white or yellow golf balls
Answer:
Dot Plot
Step-by-step explanation:
hehe guess it right;)
Answer:
I don't understand this at all but I wanted to take a stab at it anyway.
It's not an even number, it's less than 50, it has to be divisible by six or three in order to be 5 more than a perfect square, and it has to be two or three digits that equal 40.
Sounds hard.
Okay first I want to get all the odd numbered equations that could equal 40
1+39 19+21
3+37 17+23
5+35 15+25
7+33
9+31
11+29
13+27
This is taking so much brain power, wow... So none of those so far would work, which makes me think that it would be a single digit equation to make it double digit, but nothing, not even 9+9 equals 40, so at least one of the numbers would have to be double digit but that makes no sense because then the number would have to have three digits, which makes it more than 100!
This is impossible. I think I have to rephrase my answer above in saying that the number has to be more than 100, but that means I have no idea what that first line means...
I'm so frustrated by this imao... Asker of the question, if you've figured out the answer please let me know because I am so confused.
Answer:
I DONT KNOW
Step-by-step explanation:
hehe
a manufacturer is designing a new container for their chocolate-covered almonds. Their original container was a cylinder with the height of 18 cm and a diameter of 14 cm. the new container can be modeled by a rectangular prism with a square base and will contain the same amount of chocolate-covered almonds. if the new container's height is 16 cm, determine and state, to the nearest tenth of a centimeter, the side length of the new container if both containers contain the same amount of almonds
Answer:
answer is 24
Step-by-step explanation:
The side length of the rectangular prism with a square base is 13.2 cm if the original container was a cylinder with the height of 18 cm and a diameter of 14 cm.
What is a cylinder?In geometry, it is defined as the three-dimensional shape having two circular shapes at a distance called the height of the cylinder.
We know the volume of the cylinder is given by:
[tex]\rm V = \pi r^2 h[/tex]
The radius of the cylinder r = 14/2 = 7 cm
The volume of the cylindrical container V = π(7)²(18) = 882π cubic cm
Let's suppose the side length is L:
Then the volume of the rectangular prism is the same as the cylindrical container:
882π = L×L×16 (h = 16 cm)
882π = 16L²
L = 13.15 ≈ 13.2 cm
Thus, the side length of the rectangular prism with a square base is 13.2 cm if the original container was a cylinder with the height of 18 cm and a diameter of 14 cm.
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Which graph represents the function f(x) = 4|x|?
Answer:
A
Step-by-step explanation:
got it correct in edu
The equation of a line in the form y = a number creates this kind of line.
Answer:
a diagonal one??? like its linear uhm okay here i have a picture one sec
Step-by-step explanation:
The population of a small town is decreasing by 2% per year. The population in the year 2017 was 3,850. A. By how many people will be the population decrease from 2027 to 2018? B. What will be the populatiom of the town in 2018?
Answer:
A. 77
B. 3,773
Step-by-step explanation:
According to the problem, computation of given data are as follows,
Population in 2017 = 3,850
Decrease rate = 2% per year
(A) We can calculate number of population decrease from 2017 to 2018 by using following formula,
Number of population decrease = Population in 2017 × Decrease rate
= 3,850 × 2%
= 77
(B). Total population in 2018 = 3,850 - 77
= 3,773
2) given that z is a standard normal random variable, compute the following probabilities. a. p(z ≤ -1.09) = b. p(z ≤ -1.5) = c. p(z ≥ 1.3) = d. p(z ≥ 2.54) = e. p(-0.71 < z ≤ 2.54) =
The probability of z being less than or equal to,-1.09 is approximately 0.1379,-1.5 is approximately 0.0668, 1.3 is approximately 0.0968.2.54 is approximately 0.0057,-0.71 and 2.54 (exclusive) is approximately 0.6467.
To compute these probabilities, we can use the standard normal distribution table or a statistical software. In the standard normal distribution table, we look up the given z-values and find the corresponding probabilities.
For (a), the probability of z being less than or equal to -1.09 is approximately 0.1379.
For (b), the probability of z being less than or equal to -1.5 is approximately 0.0668.
For (c), the probability of z being greater than or equal to 1.3 is approximately 0.0968.
For (d), the probability of z being greater than or equal to 2.54 is approximately 0.0057.
For (e), we subtract the probability of z being less than or equal to -0.71 from the probability of z being less than or equal to 2.54. This gives us approximately 0.6467.
These probabilities provide information about the likelihood of certain values occurring in a standard normal distribution.
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Question is in picture
1 3/4 cups of flour for one batch but he wants to make 1 1/2 batches. How many more cups of flour will he need
Answer:
[tex]2\frac{5}{8} cups = 1\frac{1}{2}\ batches[/tex]
Step-by-step explanation:
Given
[tex]1\frac{3}{4} cups = 1\ batch[/tex]
Required
Number of cups for [tex]1\frac{1}{2}[/tex] batches
[tex]1\frac{3}{4} cups = 1\ batch[/tex]
Multiply both sides by [tex]1\frac{1}{2}[/tex]
[tex]1\frac{1}{2} * 1\frac{3}{4} cups = 1\ batch * 1\frac{1}{2}[/tex]
[tex]1\frac{1}{2} * 1\frac{3}{4} cups = 1\frac{1}{2}\ batches[/tex]
Express fractions as improper fractions
[tex]\frac{3}{2} * \frac{7}{4} cups = 1\frac{1}{2}\ batches[/tex]
[tex]\frac{21}{8} cups = 1\frac{1}{2}\ batches[/tex]
Express fraction as improper fraction
[tex]2\frac{5}{8} cups = 1\frac{1}{2}\ batches[/tex]
Find the general solution of the third order equation, in real form without using Laplace Transform ỹ + 343y = 0
Given equation is y''' + 343y = 0 which is a third-order linear differential equation. To find the general solution of the equation, we can use the characteristic equation of the differential equation as follows;
Let y = erx be the trial solution of the differential equation, where e is the exponential function and r is an unknown constant to be determined. Substituting the trial solution into the differential equation, we have; y''' + 343y = 0 y' = rerx, y'' = rerx, y''' = rerx. Substituting into the differential equation, we have;r³erx + 343erx = 0. Factorizing out erx, we have erx(r³ + 343) = 0For erx ≠ 0;r³ + 343 = 0r³ = -343r = (-343)¹/³ = -7 (r = -7, since we are working with real forms of the equation). Therefore, the general solution of the third order linear differential equation is;
y = c₁e^-7x + c₂e^-7x cos(12.124x) + c₃e^-7x sin(12.124x) where c₁, c₂, and c₃ are arbitrary constants.
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Libby can finish a paper in 2 hours. Ethan can finish the same paper in 4 hours. How
long will it take them to finish a paper if they work together?
Answer:
1.33333 hours
Step-by-step explanation:
Given :
Time taken by Libby = 2 hours
Time taken by Ethan = 4 hours
rate = 1 / time taken
Libby's rate = 1 /2
Ethan's rate = 1/4
Combined rate = Libby's rate + Ethan's rate
Combined rate = 1/2 + 1/4 = 3/4
Time taken = 1 / combined rate
Time taken = 1 / (3/4)
Time taken = 4 /3
Time taken = 1.33333 hours
NO LINKS PLEASE- OR I SWEAR-
Read the question and answer below. Remember that one measure of center is better than another when you have outliers in the data set. *
A. A
B. B
C. C
D. D
Answer:
C
Step-by-step explanation:
Use these equations to find ∂z/∂x and ∂z/∂y for the following.
x8 + y8 + z5 = 6xyz
For the given equation x⁸ + y⁸ + z⁵ = 6xyz, the value of partial derivatives are ∂z/∂x = (6yz - 8x⁷) / (5z⁴ - 6xz) and ∂z/∂y = (6xz) / (8y⁷ - 6xy)
To find the partial derivatives ∂z/∂x and ∂z/∂y for the equation x⁸ + y⁸ + z⁵ = 6xyz, we need to differentiate the equation with respect to x and y while treating y and z as constants.
Taking the partial derivative with respect to x (∂z/∂x), we differentiate each term separately:
8x⁷ + 0 + 5z⁴ (∂z/∂x) = 6yz + 6xz(∂z/∂x)
Simplifying the equation, we get:
8x⁷ + 5z⁴ (∂z/∂x) = 6yz + 6xz(∂z/∂x)
Now, let's take the partial derivative with respect to y (∂z/∂y):
0 + 8y⁷ + 0 (∂z/∂y) = 6xz + 6xy(∂z/∂y)
Simplifying further:
8y⁷ (∂z/∂y) = 6xz + 6xy(∂z/∂y)
To find the values of ∂z/∂x and ∂z/∂y, we need to isolate the partial derivatives:
∂z/∂x = (6yz - 8x⁷) / (5z⁴ - 6xz)
∂z/∂y = (6xz) / (8y⁷ - 6xy)
These equations give the partial derivatives of z with respect to x and y for the given equation x⁸ + y⁸ + z⁵ = 6xyz.
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Given the differential equation: dy/dx +xy = 3x- e+ 2 with the initial condition y(O) = 1, find the values of y corresponding to the values of Xo+0.1 and Xo+0.2 correct to four decim
the approximate values of y at x = X₀ + 0.1 and x = X₀ + 0.2 are:
y ≈ 0.9282 at x = 0.1
y ≈ 0.8281 at x = 0.2
To solve the given differential equation, we will use an appropriate method, such as the Euler's method, to approximate the values of y at specific points.
The Euler's method uses the equation:
y₍ₙ₊₁₎ = yₙ + h * f(xₙ, yₙ),
where:
yₙ is the value of y at xₙ
h is the step size,
f(x, y) is the derivative of y with respect to x (i.e., dy/dx), and
y₍ₙ₊₁₎ is the approximation of y at the next point x₍ₙ₊₁₎ = xₙ + h.
Let's apply Euler's method to find the values of y at x = X₀ + 0.1 and x = X₀ + 0.2, where X₀ is the initial condition x = 0 and y(X₀) = 1.
Given the differential equation:
dy/dx + xy = 3x² - [tex]e^y[/tex] + 2
Rewriting the equation in the form:
dy/dx = -xy + 3x² - [tex]e^y[/tex] + 2
We have the initial condition:
y(0) = 1
Using Euler's method with a step size of h = 0.1:
1. At x = X₀ + 0.1:
y₁ = y₀ + h * [ -x₀ * y₀ + 3 * x₀² - [tex]e^y_0[/tex] + 2 ]
= 1 + 0.1 * [ -(0) * (1) + 3 * (0)² - e¹ + 2 ]
= 1 + 0.1 * (0 - e + 2)
= 1 + 0.1 * (-e + 2)
= 1 + 0.1 * (-2.7183 + 2)
= 1 + 0.1 * (-0.7183)
= 1 - 0.07183
≈ 0.9282
2. At x = X₀ + 0.2:
y₂ = y₁ + h * [ -x₁ * y₁ + 3 * x₁² - [tex]e^y_1[/tex] + 2 ]
= 0.9282 + 0.1 * [ -(0.1) * (0.9282) + 3 * (0.1)₂ - [tex]e^{0.9282}[/tex] + 2 ]
= 0.9282 + 0.1 * (-0.009282 + 0.003 - [tex]e^{0.9282}[/tex] + 2)
≈ 0.8281
Therefore, the approximate values of y at x = X₀ + 0.1 and x = X₀ + 0.2 are:
y ≈ 0.9282 (correct to four decimal places) at x = 0.1
y ≈ 0.8281 (correct to four decimal places) at x = 0.2
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Use cylindrical coordinates. ∭_e x^2 dv .where E … solid thati x2 dV, where E is the solid that lies within the cylinder x^2 + y^2=4, above the plane z = 0, and below the cone z^2 4x^2 + 4y^2
The value of the triple integral ∭_E x² dV over the given solid E is 64π/5.
To evaluate the triple integral ∭_E x^2 dV, where E is the solid that lies within the cylinder x² + y² = 4, above the plane z = 0, and below the cone z² = 4x² + 4y², we can express the integral in terms of cylindrical coordinates.
In cylindrical coordinates, we have:
x = r cos(theta)
y = r sin(theta)
z = z
The limits for the cylindrical coordinates are as follows:
0 ≤ r ≤ 2 (limits for the radius)
0 ≤ theta ≤ 2π (limits for the angle)
0 ≤ z ≤ sqrt(4r²) = 2r (limits for the height, as per the cone equation)
Now let's express the integral in cylindrical coordinates:
∭_E x² dV = ∭_E (r cos(theta))² r dz dr d(theta)
Let's evaluate the integral step by step:
∫[0 to 2π] ∫[0 to 2] ∫[0 to 2r] (r³ cos²(theta)) dz dr d(theta)
We can simplify the innermost integral with respect to z:
∫[0 to 2π] ∫[0 to 2] [r³ cos²(theta)z] |[0 to 2r] dr d(theta)
= ∫[0 to 2π] ∫[0 to 2] (r³ cos²(theta)(2r)) dr d(theta)
= 2 ∫[0 to 2π] ∫[0 to 2] (2r⁴ cos²(theta)) dr d(theta)
Now, let's integrate with respect to r:
2 ∫[0 to 2π] [(1/5) r⁵ cos²(theta)] |[0 to 2] d(theta)
= 2 ∫[0 to 2π] [(1/5)(32 cos²(theta))] d(theta)
= (64/5) ∫[0 to 2π] cos²(theta) d(theta)
To evaluate the remaining integral, we can use the identity cos²(theta) = (1 + cos(2theta))/2:
(64/5) ∫[0 to 2π] [(1 + cos(2theta))/2] d(theta)
= (64/10) ∫[0 to 2π] (1 + cos(2theta)) d(theta)
= (64/10) [(theta + (1/2)sin(2theta))] |[0 to 2π]
= (64/10) [(2π + (1/2)sin(4π)) - (0 + (1/2)sin(0))]
= (64/10) (2π + 0 - 0)
= (64/10) (2π)
= 128π/10
= 64π/5
Therefore, the value of the triple integral ∭_E x² dV over the given solid E is 64π/5.
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Find the perimeter of rectangle with length 6 inches and width 3 inches.
Answer:
18
Step-by-step explanation:
Formula
P = 2 * (L + W)
Solution
L = 6
W = 3
P = 2 (6 + 3)
P = 2 (9)
P = 18
Which graph represents an equation in the form =?
Please help me!!
That's a question about graphs.
In the Cartesian coordinate system, we have two axes: The x-axis (horizontal) and the y-axis (vertical).
When we want to represent a equation in the cartesian coodinate system, we associate a point in the x-axis and other in the y-axis. The Cartesian coordinate is represented by (x, y).
To represent some equation in a graph, we need at least two points to draw a line that cross these 2 points.
For example: To represent the equation [tex]y = x +1[/tex], let's replace the value of x:
When x = 0 → [tex]y = 0 + 1 = 1[/tex]. Therefore, the coordenate is (0, 1).When x = 1 → [tex]y = 1 + 1 = 2[/tex]. Therefore, the coordenate is (1, 2).Finally, if we draw a line tha cross these 2 points, we will have the graph that represents [tex]y = x +1[/tex].
Now you know about it, let's solve our problem?
In that question, we should to find the graph that represents the equation [tex]y = kx[/tex], but we don't know the value of k, so what should we do? It's easy:
k is a constant. That means that the value of k never changes. Now, let's analyze a specific case:
If x = 0, so y = 0 as well. But why? We know that any number multiplied by 0 is equal 0. Therefore:
When If x = 0 → [tex]y = 0 \times k = 0[/tex]. So, this graph cross the coordinate (0,0).With the informations that the problem give to us, these are the unique things that we can find, but that's enough.
When we look at the options, just the second graph have a line that cross the coordenate (0, 0). Therefore, the correct answer is the graph below:
I hope I've helped. :D
Enjoy your studies! \o/
Plz help ASAP and thank you!
Answer:
Plot attached
Step-by-step explanation:
We want to locate √36 and ∛36 on the given number line
Now, √36 = 6.
And ∛36 = 3.3
I have attached a plot number line both values as we are told in the question
The Consumer Reports National Research Center conducted a telephone survey of 2000 adults to learn about the major economic concerns for the future (Consumer Reports, January 2009). The survey results showed that 1760 of the respondents think the future health of Social Security is a major economic concern. a. What is the point estimate of the population proportion of adults who think the future health of Social Security is a major economic concern (to 2 decimals)
Answer:
The point estimate is 0.88
Step-by-step explanation:
Given
[tex]n = 2000[/tex] -- sample size
[tex]x = 1760[/tex] -- adults who think future health of social security is a major economic concerm
Required
Determine the point estimate
This is calculated as:
[tex]\hat p= \frac{x}{n}[/tex]
[tex]\hat p= \frac{1760}{2000}[/tex]
[tex]\hat p = 0.88[/tex]
The point estimate is 0.88