The minimum distance between the origin and the plane 3x + y + 2z = 28, using the Lagrange Multipliers is 8.73 units.
In the question, we are asked to use Lagrange multipliers to find the minimum distance between the origin and the plans 3x + y + 2z = 28.
Lagrange multipliers state that if we want to minimize a function, f(x, y, z), subject to a constraint g(x, y, z) = constant, then ∇f = λ∇g.
We want to minimize the distance, and the distance formula says that the distance from the origin to a point (x, y, z), is given as:
D = √(x² + y² + z²), which can also be shown as:
D² = x² + y² + z².
The gradient of this function is: (2x + 2y + 2z), which needs to be equal to the gradient of the constraint equation multiplied by the constant, λ, that is, λ(3, 1, 2), which gives:
2x = 3λ, or, x = (3/2)λ,
2y = λ, or, y = λ/2, and,
2z = 2λ, or, z = λ.
Substituting these in the equation of the plane, 3x + y + 2z = 28, we get:
3(3/2)λ + λ/2 + λ = 28,
or, 6λ = 28,
or, λ = 28/6 = 14/3.
Thus, we get:
x = (3/2)λ = 7,
y = λ/2 = 7/3, and,
z = λ = 14/3.
Substituting these in the distance formula, we get:
D = √(7² + (7/3)² + (14/3)²) = √(49 + 49/9 + 196/9) = √(686/9) = 8.73.
Thus, the minimum distance between the origin and the plane 3x + y + 2z = 28, using the Lagrange Multipliers is 8.73 units.
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There are 25 prime numbers less than 100.
Some of these are two-digit prime numbers.
What fraction of two-digit prime numbers contain the digit 9?
Express this fraction in lowest terms.
Answer:
6/25.
Step-by-step explanation:
They are 19, 29, 59, 79, 89 and 97.
A total of 6.
So the required fraction is 6/25.
What is the slope of the linear equation that has the values shown in the table? Show your work
Slope can be found from two points using the equation [tex]\frac{y_{2}- y_{1}}{x_{2}- x_{1}}[/tex].
Take any two points to substitute in. In this example, I'll use the first two points, (-4, 15), and (-2, 9). Put these into the equation and you'll get
[tex]\frac{(9-15)}{(-2-(-4))}[/tex]. Simplify further to get [tex]\frac{-6}{2}[/tex]. Divide this and you'll get the slope, -3!
hope this helps!
Consider the quadratic function f(x) = x2 – 5x + 12. Which statements are true about the function and its graph? Select three options. The value of f(–10) = 82 The graph of the function is a parabola. The graph of the function opens down. The graph contains the point (20, –8). The graph contains the point (0, 0).
Answer:
numbers
Step-by-step explanation:
numbers
People in a town have a mean hourly wage of $13.69, with a standard deviation of $4.77. The distribution of hourly wages is not assumed to be symmetric. Between what two-hourly wages does Chebyshev's Theorem guarantee that we will find at least 75% of the people?
Using Chebyshev's Theorem, we are guaranteed to find at least 75% of the people earning hourly wages between $4.15 and $23.23.
What does Chebyshev’s Theorem state?When the distribution is not normal, Chebyshev's Theorem is used. It states that:
At least 75% of the measures are within 2 standard deviations of the mean.At least 89% of the measures are within 3 standard deviations of the mean.An in general terms, the percentage of measures within k standard deviations of the mean is given by [tex]100(1 - \frac{1}{k^{2}})[/tex].In this problem, we want the values within 2 standard deviations of the mean, hence the values are:
13.69 - 2 x 4.77 = $4.15.13.69 + 2 x 4.77 = $23.23.More can be learned about Chebyshev's Theorem at https://brainly.com/question/25303620
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11/6 + -2/5 +-13/10?
Answer:
2/15Step-by-step explanation:
11/6 + -2/5 +-13/10?
11/6 + (-2/5) + (-13/10)
43/30 - 13/10
2/15
please give the answer
Step-by-step explanation:
f(a):
So for this you simply plug in "a" as x, and this doesn't really do anything beside replace all values with x, so you just have the equation:
[tex]f(a) = 5a+4[/tex]
2 f(a):
So for this one, you want to represent f(a) using the equation it's equal to (5a + 4), and substitute it in for f(a). In doing so, you get the expression:
2*f(a) -> 2(5a + 4) -> 10a + 8
f(2a):
So this is very similar to the first question, although you will have to do some multiplication. So just plug in 2a as "x" to get the equation:
[tex]f(2a) = 5(2a) + 4 = 10a+4[/tex]
f(a+2):
Basically the same process, you plug in (a+2) as "x" and simplify:
[tex]f(a+2) = 5(a+2) + 4\\f(a+2) = 10a+10+4\\f(a+2) = 10a+14[/tex]
f(a) + f(2):
This is similar to the second question, and you simply want to replace the f(a) with the equation that represents it (5a + 4) and same thing for f(2) = 5(2) + 4
[tex]f(a) + f(2) = (5a+4)+(5(2)+4) \\f(a) + f(2) = (5a+4)+(10+4)\\f(a)+f(2) = (5a+4) + (14)\\f(a) + f(2) = 5a+18[/tex]
Does anyone know the answer? I want to make sure I did it right. Evaluate the expression using the given values type your answer in the open box
=======================================================
Explanation:
Replace x with 4 and y with 5.
Then use the order of operations (PEMDAS) to simplify until you get a single number.
Something like 9x means "9 times x", and a similar situation happens with the 8y term as well.
---------
Here's what the steps look like:
9x + 8y
9*4 + 8*5
36 + 40
76
Therefore, 9x+8y = 76 when x = 4 and y = 5.
The point (4,5) is located on the line 9x+8y = 76
how does a budget help you
Answer:
it helps you control your spending, track your expenses, and save more money.
i need your answer!
[tex](x-1) ^ { 3 } -(x-2) ^ { 2 } =1[/tex]
[tex]x^{3}-3x^{2}+3x-1-\left(x-2\right)^{2}=1 [/tex]
[tex]x^{3}-3x^{2}+3x-1-\left(x^{2}-4x+4\right)=1 [/tex]
[tex]x^{3}-3x^{2}+3x-1-x^{2}+4x-4=1 [/tex]
[tex]x^{3}-4x^{2}+7x-1-4=1 [/tex]
[tex]x^{3}-4x^{2}+7x-5=1 [/tex]
[tex]x^{3}-4x^{2}+7x-5-1=0 [/tex]
[tex]x^{3}-4x^{2}+7x-6=0 [/tex]
[tex]±6,±3,±2,±1 [/tex]
The importance of mendel's work is that it made a connection between observed patterns of ____________ and probability.
Answer:
blank = inheritance
Step-by-step explanation:
Hope that helps
The length of a rectangle is 5 mm longer than the width. If the area of the rectangle is 24 mm^2, what is the width of the rectangle?
Answer:
8
Step-by-step explanation:
This question is asking you to solve for the following system of equations, let x be width and y be height of the rectangle.
[tex]\left \{ {{x=y+5} \atop {x*y=24}} \right.[/tex]
Since the former is already equal to x lets set the second equation to x
[tex]x*y=24[/tex]
[tex]x=24/y[/tex]
Now we have the following system of equations:
[tex]\left \{ {{x=y+5} \atop {x=24/y}} \right.[/tex]
Now that the two equations are equal, we can solve for y:
[tex]y+5=24/y[/tex]
Divide both sides by y
[tex]y^2+5y=24[/tex]
Subtract 24 from the right side setting it to 0:
[tex]y^2+5y-24=0[/tex]
Solve for y using quadratic formula:
[tex]y=\frac{-5\pm \sqrt{5^2-4\cdot \:1\cdot \left(-24\right)}}{2\cdot \:1}[/tex]
[tex]y=3,-8[/tex]
Given the y-values lets plug them into our original equations to find the intersections.
[tex]x=3+5\\x=8[/tex]
Giving intersection vector (8,3)
[tex]x=-8+5\\x=-3[/tex]
Giving intersection vector (-3,-8)
Since both the width and length of the rectangle must be positive lets use the first vector (8,3) as our solution.
The x value being 8 means the width must be 8, y value being 3 the height must be 3, these variable correlate to our original definitions of making the x value equal to the width of the carpet and y value the height of the carpet.
Answer:The width is 3mm
Step-by-step explanation:
Length= 5mm + width
Let the width be represented by W
Area=24mm^2
Since area= length x width
24=(5+ W) x W
24=5W +[tex]W^{2}[/tex]
[tex]W^{2}[/tex] +5W-24=0
The factors are 8 and -3
[tex]W^{2}[/tex]+8w-3w-24=0
Factorise
W(w+8)-3(w+8)=0
(w-3)=0 or (w+8)=0
W=3 0r w= -8
The width cannot be negative.
Hence,W=3mm
Show do i solve for the missing answers
Step 3 of 6 : find the estimated value of y when x=30. round your answer to three decimal places
The estimated value of y when x=30. The answer rounded to three decimal places is 5.
Envisioned cost method, as of any date of willpower, with respect to any paintings of art, the most latest estimate of fee of such paintings of artwork, as determined every so often through the relevant Borrower in accordance with phase 5.12.
To discover the estimated value E(X), or imply μ of a discrete random variable X, in reality,y multiply every value of the random variable by way of its chance and upload the products. The method is given as E ( X ) = μ = ∑ x P ( x ) .
In opportunity theory, the estimated value is a generalization of the weighted common. Informally, the anticipated price is the mathematics suggestion of a big variety of independently decided effects of a random variable.
estimated Y when
b₁x + b
X = 30.
=( 0·302) (30) + (-3-772)
5.288
= 5
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Does x^-2 equal 2/x?
Answer:
No,
Step-by-step explanation:
X^-2 is equal to 1/X^2. so any number or letter to negative power, is the same as 1 divided by that number or letter to the positive power..(power remains with denominator
find the zeros of the following
1.x³+x²-4x-4=0
Answer:
x = -1, x = -2 , x= 2
Step-by-step explanation:
First we factorise this cubic function :
(x+1)(x+2)(x-2) = 0
Using the law a×b×0 = 0
We know that to find the zeros we have to set each bracket equal to 0 :
x+1 = 0 OR x+2 = 0 OR x-2 = 0
x = -1 x = -2 x = 2
Hope this helped and have a good day
How many different integers are there such that the square of the square of the integer is a two-digit integer
The number of different integers there are for the condition described in which case, the square of the square of the integer is a two-digit integer as in the task content is 4 possible integers which includes; -2, -3, 2, 3.
How many integers satisfy the given condition?According to the task content, the square of the square of such integers must be a two-digit number, that is, less than or equal to 99.
The numbers in discuss, x must satisfy;
x⁴ < 99.
The numbers in this regard are therefore, -2, -3, 2, 3.
Ultimately, the integers which satisfy the condition described in which case, the square of the square of the integer is a two-digit integer as in the task content is 4 possible integers which includes; -2, -3, 2, 3.
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greatest common factor of 6,12, and 21
Answer:
3
Step-by-step explanation:
[tex]6=2 \times 3 \\ \\ 12=3 \times 2^2 \\ \\ 21=3 \times 7[/tex]
Mr. Ahmed has 31 students in his class. There are 14 boys and 17 girls.
The ratio
describes the part-to-whole relationship for boys.
The ratio
describes the part-to-whole relationship for girls.
When training models, you would typically place your data into three buckets: Train, Test, and Hold Out. What is the purpose of having hold-out data
Answer:
So, having "hold-out" data allows you to have a nonpartisan set of data (so to speak) for you to refer to later, once you have made use of the "train" and "test" sets. That way you can make a fair assessment as to the difference between the new and old ('hold-out") sets.
Or if you need to repeat the experiment or model again - then you can refer to the "hold out data"
Bob takes a job in which he receives a starting salary of $30000 per year, with raises of $600 per year. Sue takes a job in which she receives a starting salary of $15000 per half year, with raises of $300 every six months. Which person is better paid? Explain.
Using linear functions, it is found that both people are paid the same.
What is a linear function?A linear function is modeled by:
y = mx + b
In which:
m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.b is the y-intercept, which is the value of y when x = 0, and can also be interpreted as the initial value of the function.For Bob, we have that that the intercept is of 30000, with a slope of $600, hence his salary after x years is given by:
B(x) = 30,000 + 600x.
For Sue, we have that that the intercept is of 2 x 15000 = 30000, which is the fixed yearly salary with a slope of 2 x 300 = $600, hence her salary after x years is given by:
S(x) = 30,000 + 600x.
The salaries are equal, that is, both people are paid the same.
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The effectiveness of a blood-pressure drug is being investigated. an experimenter finds that, on average, the reduction in systolic blood pressure is 47.3 for a sample of size 878 and standard deviation 15.9. estimate how much the drug will lower a typical patient's systolic blood pressure (using a 80% confidence level). enter your answer as a tri-linear inequality accurate to one decimal place (because the sample statistics are reported accurate to one decimal place).
Using the z-distribution, the estimate for how much the drug will lower a typical patient's systolic blood pressure is:
[tex]46.6 \leq \mu \leq 48[/tex]
What is a z-distribution confidence interval?The confidence interval is:
[tex]\overline{x} \pm z\frac{\sigma}{\sqrt{n}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.z is the critical value.n is the sample size.[tex]\sigma[/tex] is the standard deviation for the population.In this problem, we have a 80% confidence level, hence[tex]\alpha = 0.8[/tex], z is the value of Z that has a p-value of [tex]\frac{1+0.8}{2} = 0.9[/tex], so the critical value is z = 1.28.
The other parameters are given by:
[tex]\overline{x} = 47.3, \sigma = 15.9, n = 878[/tex]
Then the bounds of the interval are:
[tex]\overline{x} - z\frac{\sigma}{\sqrt{n}} = 47.3 - 1.28\frac{15.9}{\sqrt{878}} = 46.6[/tex]
[tex]\overline{x} + z\frac{\sigma}{\sqrt{n}} = 47.3 + 1.28\frac{15.9}{\sqrt{878}} = 48[/tex]
Hence the interval is:
[tex]46.6 \leq \mu \leq 48[/tex]
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Quick algebra 1 question for 10 points!
Only answer if you know the answer, quick shout-out to tariqareesha2 and MrBrainly, tysm for the help!
In c 11 you cannot use a range-based for loop to modify the contents of an array unless you declare the range variable as a reference variable. true false
This statement is true that the In c 11 you cannot use a range-based for loop to modify the contents of an array unless you declare the range variable as a reference variable.
According to the statement
we have given that the one statement and we have to tell that the those statement is true or false after analyzing it.
So,
In C++ 11 is a programming language which is used to write the coding. And in this language you cannot use a range-based for loop to modify the contents of an array unless you declare the range variable as a reference variable.
Because in the coding it is not possible that the you can make loop without declare the random variable in the loop.
Due to this reasons without declare the variable it is not possible.
So, This statement is true that the In c 11 you cannot use a range-based for loop to modify the contents of an array unless you declare the range variable as a reference variable.
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20 points!! !!
help me
Answer:
The mean absolute deviation (MAD) is 0
Step-by-step explanation:
20 + 16 + 21 + 16 + 22 + 16 + 15
= 126 ÷ 7 = 18
(20 - 18) + (16 - 18) + (21 - 18) + (16 - 18) + (22 - 18) + (16 - 18) + (15 - 18) =
2 +(-2) + 3 + (-2) + 4 + (-2) + (-3) = 0 ÷ 7= 0
The mean absolute deviation (MAD) is 0
Two health clubs offer different pricing plans for their members. Both health clubs charge a one-time sign-up fee and a monthly membership fee. The equation y=60x+30
y=60x+30 represents what Health Club B charges. Health Club A charges a $50 sign-up fee and $27 per month.
Answer is Healthclub A costs $33 less in monthly fees than Healthclub B
The number next to the “x” is your rate.
So Healthclub A charges $33 less than B
A charges 27 monthly fee
B charges 60 monthly fee
60-27=33
please help this is a math question about geometry !!!! please view the attached images to see the question and the answer choices i am going to give 13 points and a brainlist thank you
Answer:
It is the second picture.
Step-by-step explanation:
Find the area of a circle that has a radius of 9 mm. Use 3.14 for π.
254.34 mm²
56.52 mm²
63.59 mm²
28.26 mm²
Answer: 254,34 mm².
Step-by-step explanation:
[tex]S{circle}=\pi R^2\\S{circle}=3,14*9^2\\S{circle}=3,14*81\\S{circle}=254,34\ mm^2.[/tex]
Hi :)
Let's first find the area of this circle
—————————For finding the area of a circle we use
[tex]\longrightarrow\sf{A_{\circ}=\pi r^2}[/tex], where only r (radius) is squared
[tex]\textit{We know that :}\begin{cases} \sf{Radius = 9 mm} \\ \sf{\pi =3.14} \end{cases}}[/tex]
Thus
[tex]\longrightarrow\sf{A_{\circ}=3.14\cdot9^2}[/tex]
[tex]\longrightarrow\sf{A_{\circ}=3.14\times81}[/tex]
[tex]\longrightarrow\sf{\underline{A_{\circ}=254.34\: mm}}[/tex]
[tex]\tt{Learn\:More;Work\;Harder}[/tex]
:)
Given the function h(x) = x² − 10x + 21, determine the average rate of change of the function over the interval -1 ≤ x ≤ 10.
Answer: the average rate of change of the function is 23.
Step-by-step explanation:
[tex]h(x)=x^2-10x+21\ \ \ \ \ -1\leq x\leq 10 \ \ \ \ A(x)=?\\\displaystyle\\A(x)=\frac{\Delta y}{\Delta x}=\frac{h(x-g)-h(x)}{g} \\g=x_{max}-x_{min}\\g=10-(-1)\\g=10+1\\g=11.\ \ \ \ \ \Rightarrow\\A(x)=\frac{(x-11)^2-10*(x-11)+21-(x^2-10x+21)}{11} \\A(x)=\frac{x^2-22x+121-10x+110+21-x^2+10x-21}{11} \\A(x)=\frac{231-22x}{11}\\ A(x)=\frac{11*(21-2x)}{11}\\ A(x)=21-2x.\\x=x_{min}\ \ \ \ \ \Rightarrow\\A(x)=21-2*(-1)\\A(x)=21+2\\A(x)=23.[/tex]
What is the name for the threshold p-value that determines when the null hypothesis is rejected?.
Alpha level. the minimum P-value at which we reject the null hypothesis
What is Null hypothesis ?That two options are identical is the null hypothesis. According to the null hypothesis, the observed difference is solely the result of chance. The probability that the null hypothesis is correct can be determined via statistical testing.
What is alpha level?The "significance level," often known as the alpha level, must be established before performing any statistical test. The likelihood of rejecting the null hypothesis when it is true is what is meant by the term "alpha level" in statistics. The likelihood of making a poor decision is what this phrase means.
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At what rate of interest will the sum of Rs . 5000 give an interest of Rs . 1450 in 1year ?
Answer:
I = (PTR)/100
1450=(5000*1*R)/100
1450*100=5000R
145000/50000=R
29=R
Step-by-step explanation:
We know,
[tex] S.I. = \rm\cfrac{P*R*T}{100}[/tex]
Given,
Principal P = 5000
Rate R = R(Assume)
Time T = 1
Interest = 1450
Plug:
[tex]1450 = \frac{5000 \times r \times 1}{100} [/tex]
Solve:
[tex]1450 \times 100 = 5000 \times r[/tex]
[tex]145000 \div 5000 = r[/tex]
[tex]r \: = \: 29 \%[/tex]
Rate = 29
Therefore,the rate of Interest is 29%.
