Answer:
The minimum distance of √((195-19√33)/8) occurs at ((-1+√33)/4; (-1+√33)/4; (17-√33)/4) and the maximum distance of √((195+19√33)/8) occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)
Step-by-step explanation:
Here, the two constraints are
g (x, y, z) = x + y + 2z − 8
and
h (x, y, z) = x ² + y² − z.
Any critical point that we find during the Lagrange multiplier process will satisfy both of these constraints, so we actually don’t need to find an explicit equation for the ellipse that is their intersection.
Suppose that (x, y, z) is any point that satisfies both of the constraints (and hence is on the ellipse.)
Then the distance from (x, y, z) to the origin is given by
√((x − 0)² + (y − 0)² + (z − 0)² ).
This expression (and its partial derivatives) would be cumbersome to work with, so we will find the the extrema of the square of the distance. Thus, our objective function is
f(x, y, z) = x ² + y ² + z ²
and
∇f = (2x, 2y, 2z )
λ∇g = (λ, λ, 2λ)
µ∇h = (2µx, 2µy, −µ)
Thus the system we need to solve for (x, y, z) is
2x = λ + 2µx (1)
2y = λ + 2µy (2)
2z = 2λ − µ (3)
x + y + 2z = 8 (4)
x ² + y ² − z = 0 (5)
Subtracting (2) from (1) and factoring gives
2 (x − y) = 2µ (x − y)
so µ = 1 whenever x ≠ y. Substituting µ = 1 into (1) gives us λ = 0 and substituting µ = 1 and λ = 0 into (3) gives us 2z = −1 and thus z = − 1 /2 . Subtituting z = − 1 /2 into (4) and (5) gives us
x + y − 9 = 0
x ² + y ² + 1 /2 = 0
however, x ² + y ² + 1 /2 = 0 has no solution. Thus we must have x = y.
Since we now know x = y, (4) and (5) become
2x + 2z = 8
2x ² − z = 0
so
z = 4 − x
z = 2x²
Combining these together gives us 2x² = 4 − x , so
2x² + x − 4 = 0 which has solutions
x = (-1+√33)/4
and
x = -(1+√33)/4.
Further substitution yeilds the critical points
((-1+√33)/4; (-1+√33)/4; (17-√33)/4) and
(-(1+√33)/4; - (1+√33)/4; (17+√33)/4).
Substituting these into our objective function gives us
f((-1+√33)/4; (-1+√33)/4; (17-√33)/4) = (195-19√33)/8
f(-(1+√33)/4; - (1+√33)/4; (17+√33)/4) = (195+19√33)/8
Thus minimum distance of √((195-19√33)/8) occurs at ((-1+√33)/4; (-1+√33)/4; (17-√33)/4) and the maximum distance of √((195+19√33)/8) occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)
In this question, we have 2 constraints:
The plane g ( x , y , z ) = x + y + 2 z - 8
The paraboloid h ( x , y , z ) = x² + y² - z
We need to apply Lagrange Multipliers to answer it
The solution are:
The nearest point P = ( -9.06/4 , -9.06 /4 , 6.23 )
The farthest point Q ( (7.06) /4 , (7.06) /4 , 10.26)
The Objective Function (F) is the distance between the ellipse and the origin, In this case, we don´t need to know the equation of the ellipse
The Objective Function is
F = √ x² + y² + z² and as this function has the same critical points that
F = x² + y² + z² we will use this one
Then:
δF/δx = 2×x δF/δy = 2×y δF/δz = 2×z
λ ×δg/δx = λ λ δg/δy = λ λ δg/δz = 2× λ
μ× δh/δx = 2× μ×x μ× δh/δy = 2× μ×y μ× δh/δz= - μ
Therefore we get our five equations.
2×x = λ + 2× μ×x (1)
2×y = λ + 2× μ×y (2)
2×z = 2× λ - μ (3)
x + y + 2 z - 8 = 0 (4)
x² + y² - z = 0 (5)
Subtracting equation (2) from equation (1)
2×x - 2×y = 2× μ×x - 2× μ×y
( x - y ) = μ × ( x - y ) then μ = 1 and by substitution in eq. (2)
2×y = λ + 2×y then λ = 0
From eq. (3)
2×z = - 1 z = -1/2
By subtitution in eq. (4) and (5)
x + y - 1 - 8 = 0 ⇒ x + y = 9
x² + y² + 1/2 = 0 this equation has no solution.
If we make x = y
Equation (4) and (5) become
2× x + 2× z = 8
2×x² - z = 0 ⇒ z = 2×x²
2× x + 4×x² = 8 ⇒ 2×x² + x - 8 = 0
Solving for x x₁,₂ = ( -1 ± √ 1 + 64 ) / 4
x₁,₂ = ( -1 ± √65 ) 4
x₁ = (-1 + √65) /4 x₂ = ( -1 - √65) /4 √ 65 = 8.06
x₁ = 1.765 x₂ = - 2.265
And z = 2×x² ⇒ z₁ = 6.23 z₂ =
And critical points are:
P ( x₁ y₁ z₁ ) ( (7.06) /4 , (7.06) /4 , 6.23 )
Q ( x₂ y₂ z₂ ) ( -9.06/4 , -9.06 /4 , 10.26 )
And by simple inspeccion we see That
minimum distance is the point P = ( -9.06/4 , -9.06 /4 , 6.23 )
the point Q ( (7.06) /4 , (7.06) /4 , 10.26) is the farthest point
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how is this simplified? what number do you divide with to get the answer
Answer:
Divide both by 3
Step-by-step explanation:
I figured it out that you divide it by three because the divisible rule for 3 is that you add up all the digitd in the number and if it’s divisible by three than it is a multiple of 3. In this case for the numerator I added 4+4+5+5 = 18 which is divisible by three so the numerator is divisible. Now for the denominator you add 1+9+2+3+0+2+4= 21 which is also divisible by three. So you can divide both by 3.
4 divided by 1/2 as a fraction
4 divided by 1/2 as a fraction is equal to 8/1 or simply 8.
How to simplify the fractionTo divide a number by a fraction, you can multiply the number by the reciprocal of the fraction.
In this case, we have 4 divided by 1/2. The reciprocal of 1/2 is 2/1 (or simply 2).
So, we can rewrite the expression as:
4 * (2/1)
Multiplying these numbers, we get:
8/1
Therefore, 4 divided by 1/2 as a fraction is equal to 8/1 or simply 8.
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2e=2 solve the question
E=
Answer: E=1
Step-by-step explanation:
because 2 times 1 is 2 or any number multiplied by 1 is itself
Answer:
e=1, e= -1
Step-by-step explanation:
Case 1:
2e=2
e=2/2
e=1
Case 2:
2e= -2
e= -2/2
e= -1
Hope this helped!!!
a racetrack is 40 yards long. How many feet is that?
Answer:
120 feet in total
Step-by-step explanation:
Hi there! I'm glad I was able to help you out!
We are given that one racetrack is 40 yards long in length.
We also know that there are three feet in one yard alone. In order to get the answer to this math problem, all we have to do is multiply 40 by 3, where the 40 represents the amount of yards and the 3 represents the amount of feet IN a single yard.
40 × 3 = 120
Therefore, there are 120 feet in total, in terms of the racetrack's length.
I hope I was able to help you understand this a little bit more! :)
There are 120 feet in total, in terms of the racetrack's length.
Here, we have,
a racetrack is 40 yards long.
we know that,
1 yard = 3 feet
We are given that one racetrack is 40 yards long in length.
We also know that there are three feet in one yard alone.
In order to get the answer to this math problem, all we have to do is multiply 40 by 3,
where the 40 represents the amount of yards
and the 3 represents the amount of feet IN a single yard.
40 × 3 = 120
Therefore, there are 120 feet in total, in terms of the racetrack's length.
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If the variance of the water temperature in a lake is 30°, how many days should the researcher select to measure the temperature to estimate the true mean within 4° with 95% confidence? Round the intermediate calculations to two decimal places and round up your final answer to the next whole number
Using the z-distribution, it is found that the researcher should select to measure 8 days.
The margin of error of a z-confidence interval is given by:
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which:
z is the critical value. [tex]\sigma[/tex] is the population standard deviation. n is the sample size.The first step is finding the critical value, which is z with a p-value of [tex]\frac{1 + \alpha}{2}[/tex], in which [tex]\alpha[/tex] is the confidence level.
In this problem, [tex]\alpha = 0.95[/tex], thus, z with a p-value of [tex]\frac{1 + 0.95}{2} = 0.975[/tex], which means that it is z = 1.96.
For this problem, the variance is of 30º, hence [tex]\sigma = \sqrt{30}[/tex].
The number of days is n for which M = 4, hence:
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
[tex]4 = 1.96\frac{\sqrt{30}}{\sqrt{n}}[/tex]
[tex]4\sqrt{n} = 1.96\sqrt{30}[/tex]
[tex]\sqrt{n} = \frac{1.96\sqrt{30}}{4}[/tex]
[tex](\sqrt{n})^2 = \left(\frac{1.96\sqrt{30}}{4}\right)^2[/tex]
[tex]n = 7.2[/tex]
Rounding up, the researcher should select to measure 8 days.
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To estimate the true mean water temperature within 4° with 95% confidence and a variance of 30°, the researcher should measure the temperature on approximately 8 days.
To determine the sample size needed to estimate the true mean water temperature in a lake within a margin of error of 4° with 95% confidence, we can use the formula for sample size in estimating a population mean with a known variance:
n = (Z^2 * σ^2) / E^2
where:
n is the required sample size,
Z is the Z-score corresponding to the desired confidence level (for 95% confidence, Z is approximately 1.96),
σ^2 is the variance of the water temperature (σ^2 = 30),
E is the margin of error (E = 4).
Substituting the values, we get:
n = ((1.96)^2 * 30) / (4^2)
n ≈ (3.8416 * 30) / 16
n ≈ 115.248 / 16
n ≈ 7.203
Rounding up to the nearest whole number, the researcher should select approximately 8 days to measure the water temperature to estimate the true mean within 4° with 95% confidence.
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which equation in standard form has a slope of -1/3 and go through the point (12,-3)?
Answer:
x + 3y = 3
Step-by-step explanation:
The standard form equation ...
ax +by = c
has slope -a/b. Here, we want -a/b = -1/3, and we want a > 0. We can choose ...
a = 1, b = 3
and we can find the constant c using the given point.
x +3y = 12 +3(-3) = 3
The desired standard-form equation is ...
x + 3y = 3
41:02
Machine 1 can complete a task in x hours while an upgraded machine (machine 2) needs 9 fewer hours. The plant manager
knows the two machines will take at least 6 hours, as represented by the inequality:
Which answer choice includes all solutions to the inequality and identifies which interval(s) contain viable completion times of
machine 1?
(0.3] U (9,18]; both contain viable times for machine 1
0 (0.3] U (9,18); only (9,18] contains viable times for machine 1
0 (-0,0) U (3,9) U (18,co); no interval contains viable times for machine 1
0 (-0,0) U[3,9U (18,co); but only (18,co) contains viable times for machine 1
6:36 PM
041 122/2020
O
Type here to search
Answer:the answer is B
Step-by-step explanation:
Answer:
B 0 (0.3] U (9,18); only (9,18] contains viable times for machine 1
Step-by-step explanation:
The graph is 0<x<3 or 9<x<18
Thank you guys so much for the help on my previous question! I’m stuck on one more (this one)
Answer:
C. 324 square inches
Step-by-step explanation:
The area of the post is 240, and the sign is 84 including the triangle. 240 + 84, is 324
What situation could be modeled with the equation 40÷8=5
The required situation could be modeled with the equation 40 ÷ 8 = 5 as "There are eight in each of the 40 groups. Which groupings are there, exactly?"
What is the division operation?In mathematics, divides left-hand operands into right-hand operands in the division operation.
The equation is given in the question
40 ÷ 8 = 5
An equation of this kind may be used to model any situations in which 40 items are divided into 8 or 5 divisions. There are a few possibilities:
"There are eight in each of the 40 groups. Which groupings are there, exactly?"
"There are a total of 40, separated into 8 categories. In how many groups are there?"
Thus, the above situations could be modeled with the equation 40 ÷ 8 = 5.
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2x²(3x²+7x-6)
Can anyone solve it
Answer:
6x∧4 + 14x³ - 12x²
Step-by-step explanation:
2x²(3x² + 7x - 6)
6x∧4 + 14x³ - 12x²
Tell me if I am wrong.
Can I get brainliest
Complete the recursive formula of the geometric sequence 300, 60, 12, 2.4
D(1)=
D(n)= d(n-1)•
Answer:
D(1)=300
D(n)=d(n-1)× 1/5
Step-by-step explanation:
The 1st term is 300 and the common ratio is 1/5. You have to multiply times 1/5 to get to the next number.
Answer:
300
1/5
Step-by-step explanation:
find height of this cylinder
Circle the best answer. The choice between a z-test and a t-test for a population mean depends primarily on: a. the sample size. b. the level of significance. c. whether a one- or two-tailed test is indicated. d. whether the given standard deviation is from the population or the sample. e. a z-test should never be used.
Answer:
Correct option is (d).
Step-by-step explanation:
A hypothesis test for single mean can be used to determine the significance of the claimed value of the population mean μ.
Now there are two test for mean:
z-testt-testA z-test is used when it is provided that the population is normally distributed, the sample is large and the population standard deviation value is provided.
A t-test is used when it is assumed that the population is normally distributed, the sample is large enough and the population standard deviation is not known.
So, in case there is no information about the population standard deviation (σ) but the sample standard deviation is either given or can be calculated from the provided data set, then the hypothesis test for single mean can be performed using the t-test.
Hence, the choice between a z-test and a t-test for a population mean depends primarily on whether the given standard deviation is from the population or the sample.
Thus, the correct option is (d).
Of all the registered automobiles in a city, 8% fail the emissions test. Nine automobiles are selected at random to undergo an emissions test. Round the answers to four decimal places.
a. Find the probability that exactly three of them fail the test. The probability that exactly three of them fail the test is .
b. Find the probability that fewer than three of them fail the test. The probability that fewer than three of them fail the test is .
c. Find the probability that more than two of them fail the test. The probability that more than two of them fail the test is .
d. Would it be unusual for none of them to fail the test?
Answer:
a) The probability that exactly three of them fail the test is 2.61%.
b) The probability that fewer than three of them fail the test is 97.02%.
c) The probability that more than two of them fail the test is 2.98%.
d) It would not be unusual for none of them to fail the test
Step-by-step explanation:
For each automobile, there are only two possible outcomes. Either it fails the test, or it does not. The probability of an automobile faiiling the test is independent of other automobiles. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
8% fail the emissions test.
This means that [tex]p = 0.08[/tex]
Nine automobiles.
This means that [tex]n = 9[/tex]
a. Find the probability that exactly three of them fail the test.
This is [tex]P(X = 3)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 3) = C_{9,3}.(0.08)^{3}.(0.92)^{6} = 0.0261[/tex]
The probability that exactly three of them fail the test is 2.61%.
b. Find the probability that fewer than three of them fail the test.
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{9,0}.(0.08)^{0}.(0.92)^{9} = 0.4722[/tex]
[tex]P(X = 1) = C_{9,1}.(0.08)^{1}.(0.92)^{8} = 0.3695[/tex]
[tex]P(X = 2) = C_{9,2}.(0.08)^{2}.(0.92)^{7} = 0.1285[/tex]
[tex]P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = 0.4722 + 0.3695 + 0.1285 = 0.9702[/tex]
The probability that fewer than three of them fail the test is 97.02%.
c. Find the probability that more than two of them fail the test.
Either fewer than three(two or less) fail, or more than two do. The sum of the probabilities of these events is 100%. So
97.02 + p = 100
p = 2.98
The probability that more than two of them fail the test is 2.98%.
d. Would it be unusual for none of them to fail the test?
More than 2.5 standard deviations from the mean is unusual.
[tex]E(X) = np = 9*0.08 = 0.72[/tex]
[tex]\sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{9*0.08*0.72} = 0.81[/tex]
0 > 0.72 - 2*0.81
So
It would not be unusual for none of them to fail the test
To find the probabilities, we use the binomial probability formula. For each part of the question, we plug in the appropriate values and calculate the probabilities. To determine if it would be unusual for none of them to fail, we calculate the probability of exactly zero failures.
Explanation:To find the probabilities in this question, we can use the binomial probability formula: P(x) = (nCx) * p^x * (1-p)^(n-x), where n is the number of trials (automobiles selected), x is the number of successes (fail the test), and p is the probability of success (8% or 0.08).
a. For exactly three failures, we use x = 3 and n = 9 in the formula. P(3) = (9C3) * 0.08^3 * (1-0.08)^(9-3).
b. For fewer than three failures, we find the probabilities of 0, 1, and 2 failures and sum them up: P(<3) = P(0) + P(1) + P(2).
c. For more than two failures, we find the probabilities of 3, 4, 5, 6, 7, 8, and 9 failures and sum them up: P(>2) = P(3) + P(4) + P(5) + P(6) + P(7) + P(8) + P(9).
d. To determine if it would be unusual for none of them to fail, we calculate P(0) using x = 0 and n = 9 in the formula. If P(0) is very low, it would be considered unusual.
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what is the equation of the line that passes through the point (-2,-2)and has a slope of 2
Answer:
y=2x+2
Step-by-step explanation:
To find the y intercept of the equation you add 4 beacuse the point is 2 under the y intecrept and 2*2 is 4 so -2+4=2 so the y value of the y intercept is 2 so
y=2x+2
Answer:
y=2x+2
Step-by-step explanation:
Since we have a point and a slope, we can use the point slope formula:
y-y1=m(x-x1)
where m is the slope, y1 is the y coordinate of the point, and x1 is the x coordinate of the point
We know the slope is 2, the y coordinate is -2, and the x coordinate is also -2, so we can substitute them in
y-y1=m(x-x1)
y- -2 =2(x - -2)
y+2=2(x+2)
Now we need to solve for/ isolate y
Distribute the 2
y+2=2*x+2*2
y+2=2x+4
Subtract 2 from both sides
y=2x+2
A publisher reports that 42%42% of their readers own a particular make of car. A marketing executive wants to test the claim that the percentage is actually different from the reported percentage. A random sample of 250250 found that 35%35% of the readers owned a particular make of car. Find the value of the test statistic. Round your answer to two decimal places.
Answer:
[tex]z=\frac{0.35 -0.42}{\sqrt{\frac{0.42(1-0.42)}{250}}}=-2.24[/tex]
Step-by-step explanation:
Data given and notation
n=250 represent the random sample taken
[tex]\hat p=0.35[/tex] estimated proportion of readers owned a particular make of car
[tex]p_o=0.42[/tex] is the value that we want to test
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the p value (variable of interest)
Concepts and formulas to use
We need to conduct a hypothesis in order to test the claim that that the percentage is actually different from the reported percentage.:
Null hypothesis:[tex]p=0.42[/tex]
Alternative hypothesis:[tex]p \neq 0.42[/tex]
When we conduct a proportion test we need to use the z statistic, and the is given by:
[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)
The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].
Calculate the statistic
Since we have all the info requires we can replace in formula (1) like this:
[tex]z=\frac{0.35 -0.42}{\sqrt{\frac{0.42(1-0.42)}{250}}}=-2.24[/tex]
−3y−4x=−11
3y−5x=−61
Solve the system of equations.
Answer:
[tex]x = 8\\y=-7[/tex]
Step-by-step explanation:
[tex]-3y-4x=-11\\3y-5x=-61[/tex]
In order to eliminate one of the variables you can add both equations.
[tex]-9x=-72\\x=\frac{-72}{-9} \\x=8[/tex]
Replace in one of the main equations to find y
[tex]3y-5x=-61\\3y-5(8)=-61\\3y-40=-61\\3y=-61+40\\3y=-21\\y=\frac{-21}{3} \\y=-7[/tex]
Replace in one of the main equations to prove that your answers are correct.
[tex]-3y-4x=-11\\-3(-7)-4(8)=-11\\21-32=-11\\-11=-11[/tex]
Kevin has an equal number of dimes, nickel and quarters in his piggy bank. He randomly picks a coin, replaces it, and picks another coin. What is the probability that the sum of the two coins is at least 30cents?
Answer:
5/9
Step-by-step explanation:
2560 divided by 11 equals what
Answer:
232.727
Step-by-step explanation:
Answer:
232.727272727
Step-by-step explanation:
just calculate it
You have been given the task of finding out what proportion of students that enroll in a local university actually complete their degree. You have access to first year enrolment records and you decide to randomly sample 115 of those records. You find that 85 of those sampled went on to complete their degree.
a)Calculate the proportion of sampled students that complete their degree. Give your answer as a decimal to 2 decimal places
Calculate lower bound and upper bound at 95% confidence interval. Give answer decimal to 3 places.
Answer:
The proportion of sampled students that complete their degree is 0.74.
The lower bound for the 95% confidence interval is 0.659 and the upper bound is 0.819.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
For this problem, we have that:
[tex]n = 115, \pi = \frac{85}{115} = 0.739[/tex]
Rounded to two decimal places, the proportion of sampled students that complete their degree is 0.74.
95% confidence level
So [tex]\alpha = 0.05[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.05}{2} = 0.975[/tex], so [tex]Z = 1.96[/tex].
The lower limit of this interval is:
[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.739 - 1.96\sqrt{\frac{0.739*0.261}{115}} = 0.659[/tex]
The upper limit of this interval is:
[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.739 + 1.96\sqrt{\frac{0.739*0.261}{115}} = 0.819[/tex]
The lower bound for the 95% confidence interval is 0.659 and the upper bound is 0.819.
Variables x and y are in direct proportion, and y = -12 when x = -3. Which line in the graph correctly shows the relationship between x and y?
Answer:
Line C
Step-by-step explanation:
I picked this answer because the slope of the line is 4, which is -12/-3.
If this answer is correct, please make me Brainliest!
What is the solution to the equation x^3 = 25
Final answer:
The solution to the equation x^3 = 25 involves taking the cube root of both sides, yielding an approximate solution of x ≈ 2.924, rounded to three decimal places.
Explanation:
The question asks to find the solution to the equation x^3 = 25. This equation requires us to find a value of x such that, when cubed, equals 25. To solve this, we take the cube root of both sides of the equation, which gives us x = √[3]{25}. This cube root can be simplified using a calculator or estimated for a rough solution.
Thus, the solution to the equation is x ≈ 2.924 (rounded to three decimal places). It's important to note that in this case, we are only looking for real numbers as solutions. Complex solutions aren't considered for this specific problem.
The solution to the equation x^3 = 25 is found by taking the cube root of both sides. The approximate numerical solution is x ≈ 2.924.
The solution to the equation x^3 = 25 involves finding a number x that, when cubed, equals 25. We can solve this by taking the cube root of both sides of the equation. This operation will give us the principal root of x. The cube root of 25 is not a whole number, so we will end up with a decimal answer. To solve for x, we proceed as follows:
Write the equation: x^3 = 25.Take the cube root of both sides: x = √325.Calculate the cube root: x ≈ 2.924... (this is an approximation).The actual value of x is a decimal number that, because of the nature of cube roots, cannot be expressed as an exact fraction or simple square root. The approximate numerical solution to the given cubic equation is x ≈ 2.92401773821287.
What is the perimeter of the figure?"
Answer:
14
Step-by-step explanation:
Perimeter= 4× 3.5 in
=14 in
Answer:
14 in
Step-by-step explanation:
[tex]3 \frac{1}{2} = \frac{7}{2} in[/tex]
Perimeter of a square = 4 × sides
[tex] = 4 \times \frac{7}{2} = 14 \: \: in[/tex]
When you divide any number by a fraction less than one, how does the original number change?
Answer:
it depends
Step-by-step explanation:
If you divide the fraction by a number greater than 1 then you will have a smaller fraction. If you divide the fraction by a number equal to 1 then you will have the same fraction. If you divide the fraction by a positive number smaller than 1 then you will have a greater fraction.
) The data below represent the weight losses for people on three different exercise programs. Exercise A 2.5 8.8 7.3 9.8 5.1 Exercise B 5.8 4.9 1.1 7.8 1.2 Exercise C 4.3 6.2 5.8 8.1 7.9 At the 1% significance level, does it appear that a difference exists in the true mean weight loss produced by the three exercise programs
Answer:
See attached files
Step-by-step explanation:
Tatooine is a fictional desert planet that appears in the Star Wars franchise. It is home to many settlers, including humans. Let X = weight of a human inhabitant on Tatooine. You know from prior research that X is approximately Normally distributed with population mean u = 96kg and population standard deviation o = 23.3kg. A. Specify the sampling distribution of statistic X, including stating what (5pts) type of distribution, its sampling mean uy, and its standard deviation Oy, for samples of 9 humans living on Tatooine. B. What is the Probability that a randomly selected group of 9 humans on (10pts) Tatooine would have a mean weight between 95 and 100kg? Include a rough sketch and Probability statement.
Answer:
a) For this case we select a sample size of n =9. And the distribution for the sample mean is given by:
[tex]\bar X \sim N (\mu , \sqrt{\frac{\sigma}{\sqrt{n}}}) [/tex]
With the following parameters:
[tex]\mu_{\bar X}= 96[/tex]
[tex]\sigma_{\bar X} = \frac{23.3}{\sqrt{9}} =7.767[/tex]
b) [tex] P(95< \bar X < 100)[/tex]
And we can use the z score formula given by:
[tex] z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
And if we find the z score for the limit we got:
[tex] z = \frac{95-96}{\frac{23.3}{\sqrt{9}}}= -0.129[/tex]
[tex] z = \frac{100-96}{\frac{23.3}{\sqrt{9}}}= 0.515[/tex]
[tex] P( -0.129 < Z< 0.515) = P(Z<0.515) -P(Z<-0.129) = 0.697-0.449= 0.248[/tex]
The sketch is on the figure attached
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the weights of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(96,23.3)[/tex]
Where [tex]\mu=96[/tex] and [tex]\sigma=23.3[/tex]
Part a
For this case we select a sample size of n =9. And the distribution for the sample mean is given by:
[tex]\bar X \sim N (\mu , \sqrt{\frac{\sigma}{\sqrt{n}}}) [/tex]
With the following parameters:
[tex]\mu_{\bar X}= 96[/tex]
[tex]\sigma_{\bar X} = \frac{23.3}{\sqrt{9}} =7.767[/tex]
Part b
For this case we want this probability:
[tex] P(95< \bar X < 100)[/tex]
And we can use the z score formula given by:
[tex] z = \frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
And if we find the z score for the limit we got:
[tex] z = \frac{95-96}{\frac{23.3}{\sqrt{9}}}= -0.129[/tex]
[tex] z = \frac{100-96}{\frac{23.3}{\sqrt{9}}}= 0.515[/tex]
[tex] P( -0.129 < Z< 0.515) = P(Z<0.515) -P(Z<-0.129) = 0.697-0.449= 0.248[/tex]
The sketch is on the figure attached
What is the circumference of a circle with a radius of 80 inches?(Use 3.14 for pie)
Final answer:
The circumference of a circle with an 80-inch radius is 502.4 inches, calculated using the formula C = 2πr with π approximated as 3.14.
Explanation:
The circumference of a circle with a radius of 80 inches can be calculated using the formula C = 2πr, where π (Pi) is approximately 3.14 and r is the radius. For given radius of 80 inches, the calculation would be:
C = 2 × 3.14 × 80 inches
C = 502.4 inches.
Therefore, the circumference of the circle is 502.4 inches.
A spinner is spun twice. Find the probability of spinning a 5 then spinning an even number.
Answer:
1/16
Step-by-step explanation:
The spinner has 8 equal parts.
The probability of spinning a 5 is: 1/8 (because there is only one 5 to choose and 8 total parts that could be landed on).
The probability of spinning an even number is: 4/8 = 1/2 (because there are 4 even numbers, 2 4 6 8, and there are 8 total parts that could be landed on).
Now, we need to multiply these two probabilities together:
(1/8) * (1/2) = 1/16
Hope this helps!
Answer:
1/16
Step-by-step explanation:
Spinner has 8 sections:
1,2,...8
P(5) = 1/8
P(even) = 4/8 = 1/2
1/8 × 1/2
1/16
A company sells cans of caviar that say they each contain 100g. of product with a standard deviation of 1g. A consumer advocacy group suspects that the company 13 under-filling these cans. The group obtains a simple random sample of 30 cans and measures how much product is In each can. They calculate a sample mean of 99g. They will take further action if this ls significantly lower than the advertised amount. Find the P-value.
Answer:
Step-by-step explanation:
We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean
For the null hypothesis,
µ = 100g
For the alternative hypothesis,
µ < 100g
Due to the <, It means that it is left tailed test.
Since the number of samples is large, the population standard deviation is given, the z test would be used. The formula is
z = (x - µ)/(σ/√n)
Where
x = sample mean
µ = population mean
σ = population standard deviation
n = number of samples
From the information given,
µ = 100g
x = 99 g
σ = 1g
n = 30
z = (99 - 100)/(1/√30) = - 5.48
Looking at the normal distribution table, the probability corresponding to the z score is less than 0.00001
P value < 0.00001
The p-value is approximately 0, which means there is strong evidence to support the consumer advocacy group's suspicion that the company is under-filling the cans of caviar.
Step 1
To determine whether the consumer advocacy group's suspicion is statistically significant, we perform a hypothesis test. We can use a one-sample z-test since we know the population standard deviation.
The null hypothesis [tex](\(H_0\))[/tex] and the alternative hypothesis [tex](\(H_a\))[/tex] are:
- [tex]\(H_0: \mu = 100\)[/tex] (the mean weight is 100 grams)
- [tex]\(H_a: \mu < 100\)[/tex] (the mean weight is less than 100 grams)
Given:
- Population mean [tex](\(\mu\))[/tex] = 100 grams
- Sample mean [tex](\(\bar{x}\))[/tex] = 99 grams
- Population standard deviation [tex](\(\sigma\))[/tex] = 1 gram
- Sample size [tex](\(n\))[/tex] = 30
Step 2
First, calculate the standard error of the mean (SEM):
[tex]\[ \text{SEM} = \frac{\sigma}{\sqrt{n}} = \frac{1}{\sqrt{30}} \approx 0.1826 \][/tex]
Next, calculate the z-score:
[tex]\[ z = \frac{\bar{x} - \mu}{\text{SEM}} = \frac{99 - 100}{0.1826} \approx -5.48 \][/tex]
Now, we find the p-value corresponding to this z-score. Using the standard normal distribution table or a calculator, we find the probability that z is less than -5.48.
The p-value for z = -5.48 is extremely small. For practical purposes, it is close to 0.
Therefore, the p-value is:
[tex]\[ \text{P-value} \approx 0 \][/tex]
Since the p-value is significantly less than the common significance levels (e.g., 0.05, 0.01), we reject the null hypothesis.
Rearrange the equation so b is the independent variable 4a+b=−52
Answer:
b=-4a-52 or -b=4a+52
Step-by-step explanation:
because you can subtract 4a to the other side or you can subtract 52 to the other side then subtract b to get the second equation.
hope this helps :)
The independent variable is the variable you change.
The dependent variable is the variable you measure that depends on the independent variable.
Since b is the independent variable, you need to isolate/get the variable "a" by itself in the equation: [this is because "b" is the variable you change, and "a" is the variable you measure that results/depends on "b"]
4a + b = -52 Subtract b on both sides
4a + b - b = -52 - b
4a = -52 - b Divide 4 on both sides to get "a" by itself
[tex]\frac{4a}{4} =\frac{-52-b}{4}[/tex]
[tex]a=-13-\frac{b}{4}[/tex]