The extreme max value is "[tex]f(x,y,z) = 36[/tex]" and min value is "[tex]f(x,y,z) = -36[/tex]".
According to the question,
[tex]f(x, y, z) = 8x + 8y + 4z[/tex]let,
[tex]g(x,y,z) = 4x^2 + 4y^2 + 4z^2 - 36[/tex]By using Lagrange multipliers,
→ [tex]\bigtriangledown f= \lambda \bigtriangledown g[/tex]...(equation 1)
[tex]\bigtriangledown f = <8,8,4>[/tex][tex]\bigtriangledown g = <8x, 8y,8z>[/tex]∴ [tex]<8,8,4>=<8\lambda x, 8 \lambda y, 8 \lambda z >[/tex]
then,
[tex]8 \lambda x = 8[/tex]...(equation 2)[tex]8 \lambda y =8[/tex]...(equation 3)[tex]8 \lambda z = 4[/tex]...(equation 4)→ [tex](x = \frac{1}{\lambda}, y=\frac{1}{\lambda}, z =\frac{1}{2 \lambda} )[/tex]
As we know,
→ [tex]4x^2 +4y^2+4z^2=36[/tex]
By substituting the values, we get
→ [tex]4(\frac{1}{\lambda} )^2+ 4(\frac{1}{\lambda} )^2+4(\frac{1}{2\lambda} )^2 =36[/tex]
→ [tex]\frac{4}{\lambda^2} +\frac{4}{\lambda^2} +\frac{1}{\lambda^2} =36[/tex]
→ [tex]\frac{9}{\lambda^2}=36[/tex]
[tex]\frac{1}{\lambda}= \pm \frac{6}{3}[/tex]
[tex]\frac{1}{\lambda} = \pm 2[/tex]
∴ x = 2, y = 2, z = 1
then,
→ [tex]f(x,y,z) = 36[/tex] (Extreme max value)
and,
∴ x = -2, y = -2, z = -1
then,
→ [tex]f(x,y,z) = -36[/tex] (min value)
Thus the above answer is correct.
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Use the formula to evaluate the series -3+6-12+24-...-a7
Formula:Sn=a1(1-r^n)/1-r
In the formula for a finite series, a1 is the first term, r is the common ratio and n is the number of terms.
What is the area of a circle with a radius of 7 cm? (Use 3.14 for pi and round to the nearest tenth.)
Answer:
[tex]153.9\ cm^{2}[/tex]
Step-by-step explanation:
Hello, thanks for asking this question here at Brainly, I think I can help you with this
the area of circle is given by:
[tex]Area=\pi * radius^{2} \\[/tex]
Step one
Let
[tex]\pi =3.14\\and\\r= 7 cm[/tex]
Step two
put the values into the equation
[tex]Area=\pi * radius^{2} \\Area=3.14* (7 cm)^{2}\\ Area=3.14* 49\ cm^{2} \\Area=153.9\ cm^{2} \\[/tex]
Hence, the area of a circle with radius 7 cm is 153.9 cm2
I hope it helps,do not hesitage in asking me anything about this, have a great day
If 7.02 is subtracted 43.8 the correct answer will be
Kathryn, a college student, works part time at a coffee house. She worked 5 StartFraction 7 Over 20 EndFraction hours on Monday, 4 and one half on Tuesday, 6 and three fourths on Wednesday, 3 and one fourth on Thursday, and 7 on Friday. How many hours did she work altogether? She worked nothing hours altogether.
Answer: I’m just adding this to verify that the other answer was correct
The endpoints of line segment AB are A (9,4) and B (5,-4). Then endpoints of its image after dilation are A' (6,3) and B' (5,-4). Find the scale factor and explain
Answer:To find the image of B, first find the scale factor for the dilation. The scale factor should be greater than 1 because the image of A is farther from the origin than A. Divide the coordinates of the image of A by the coordinates of A: –6/–4 = 3/2 and 9/6 = 3/2, so the scale factor is 3/2. Now, apply the dilation to B by multiplying the coordinates by 3/2 to get ((3/2)(1), (3/2)(4)), or (3/2, 6).
Hope I helped
Joseph needs to find the quotient of 3216 divided by 8. In which place is the first digit of the quotient?
A farmer estimates that he has 9,000 bees producing honey on his farm. The farmer becomes concerned when he realizes the population of bees seems to be decreasing steadily at a rate of 5% per year. If the number of bees in the population after x years is represented by f(x), which statements about the situation are true? Check all that apply. The function f(x) = 9,000(1.05)x represents the situation. The function f(x) = 9,000(0.95)x represents the situation. After 2 years, the farmer can estimate that there will be about 8,120 bees remaining. After 4 years, the farmer can estimate that there will be about 1,800 bees remaining. The domain values, in the context of the situation, are limited to whole numbers. The range values, in the context of the situation, are limited to whole numbers.
Answer:
Step-by-step explanation:
The function f(x) = 9,000(0.95)x represents the situation.
After 2 years, the farmer can estimate that there will be about 8,120 bees remaining.
The range values, in the context of the situation, are limited to whole numbers.
The true representation of the declining bee population is f(x) = 9,000(0.95)^x, accounting for a 5% annual decrease. The function and expected number of bees after 2 years are correct if rounded to approximately 8,120. The domain and range values are limited to whole numbers in this context.
Explanation:The given situation of the farmer noticing a decrease of bees in his population can be represented by a mathematical function that models the decline. This function is f(x) = 9,000(0.95)^x, which correctly represents a 5% annual decrease in the bee population.
After analyzing the situation:
The statement f(x) = 9,000(1.05)^x is false because it suggests a 5% increase rather than a decrease.The correct function, as mentioned before, is f(x) = 9,000(0.95)^x.After 2 years, the population of bees expected can be calculated: f(2) = 9,000(0.95)^2 ≈ 8,122.5, which, when rounded to a whole number due to the context, is about 8,123 bees remaining, making this statement true if it refers to approximately 8,120 bees.After 4 years, using the function, the statement that there will be about 1,800 bees remaining is false; the correct calculation would yield a higher number.The domain values are limited to whole numbers since you cannot have a fraction of a year.The range values are also limited to whole numbers as you cannot have a fraction of a bee.Learn more about Exponential Decay here:https://brainly.com/question/2193799
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When creating a scatterplot, if the points are too close together to see the relationship, how could you adjust your graph?
A.
decrease your scale values
B.
increase your scale values
C.
plot more points
D.
plot fewer points
Speed skiing takes place on a course that is 2/3 mile long. How many feet long is the course? Show work too
1 mile = 5280 feet
5280 * 2/3 =
5280 *2 = 10560
10560/3 = 3520
it is 3520 feet long
Rafael walks 2.5 km every day. How far does he walk in 9 days?
What is the value of m?
13m+3−56m=−15
The value of m in the equation [tex]13m +3 -56m = -15[/tex] is [tex]\frac{18}{43}[/tex]
Given: [tex]13m +3 -56m = -15[/tex]
To find: value of m
To find the value of m we need to move all the terms containing variables on LHS and the constants on RHS. This can be done as follows:
[tex]13m +3 -56m = -15\\13m -56m =-15-3\\[/tex]
Now we can solve for m as follows:
[tex]13m-56m =-15-3\\-43m = -18\\[/tex]
The negative sign on both LHS and RHS will cancel out and it will give us:
[tex]m = \frac{18}{43}[/tex]
Since 18 and 43 do not have any common multiples so the fraction cant be simplified further. So, [tex]m = \frac{18}{43}[/tex]
8/10 divided by 1 5/6
y=0.6x–3;(–2,2) parallel lines
A car company said that its top five models had the following fuel efficiency ratings: 26.3 mpg, 28.95 mpg, 33.6 mpg, 35 mpg, 38.24 mpg. What is the average mpg rating of these five models, recorded to the appropriate degree of precision? A. 32 mpg B. 32.0 mpg C. 32.4 mpg D. 32.42 mpg
The average fuel efficiency rating, calculated by adding the mpg ratings and dividing by five, is 32.418 mpg, rounded to the nearest hundredth as 32.42 mpg. The answer is D. 32.42 mpg.
To find the average fuel efficiency rating of the top five models, we need to calculate the mean mpg rating. To do this, we add up all the mpg ratings and then divide by the number of models:
Add the fuel efficiency ratings together: 26.3 + 28.95 + 33.6 + 35 + 38.24.
The sum is 162.09 mpg.
Divide this sum by the number of cars, which is 5: 162.09 ÷ 5.
The average is 32.418 mpg.
To record the average to the appropriate degree of precision, we consider the place values of the original data provided. The given ratings include mpg values recorded to the nearest hundredth, as in 28.95 mpg and 38.24 mpg. This means the average should also be recorded to the nearest hundredth: 32.42 mpg. Therefore, the correct answer is D. 32.42 mpg.
An electric current, I, in amps, is given by I=cos(wt)+√8sin(wt), where w≠0 is a constant. What are the maximum and minimum values of I
The maximum value of the electric current is [tex]I_{max}=\dfrac{1}{3}[/tex]
the minimum value of the electric current is [tex]I_{min}=-\dfrac{1}{3}[/tex]
The given expression for electric current is [tex]I=\cos(\omega t)+\sqrt{8}\sin(\omega t)[/tex] where [tex]\omega \neq 0[/tex].
Differentiate the given function with respect to [tex]t[/tex]-
[tex]\dfrac{dI}{dt}=\dfrac{d(\cos(\omega t)+\sqrt 8 \sin(\omega t))}{dt}\\\dfrac{dI}{dt}=-\omega \sin(\omega t)+\sqrt 8 \omega \cos(\omega t)[/tex]
Equate the derivative to 0-
[tex]\dfrac{dI}{dt}=0\\-\omega \sin(\omega t)+\sqrt 8 \omega \cos(\omega t)=0\\\sin(\omega t)=\sqrt 8 \cos (\omega t)\\\tan (\omega t)=\sqrt 8\\\omega t=\tan^{-1}\sqrt 8\\\omega t=2n\pi-2\tan^{-1}\sqrt 2}\\or,\\\omega t=2n\pi+2\tan^{-1}\sqrt {1/2}}[/tex]
[tex]2n\pi[/tex] is the period of sine and the cosine trigonometric function.
So, the maximum value of the electric current is
[tex]I_{max}=\cos(2\tan^{-1}\sqrt {1/2})\\I_{max}=\dfrac{1}{3}[/tex]
Also, the minimum value of the electric current is
[tex]I_{min}=\cos(2\tan^{-1}\sqrt {2})\\I_{min}=-\dfrac{1}{3}[/tex]
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A rectangular board is 18 feet by 24 feet. How far from the center of the board should the foci be located in order to cut the largest elliptical tabletop from the board?
a. 7.94 ft
b. 4.06 ft
c. 15 ft
d. 15.9 ft
x = 24
y = 18
Length of Major Axis = a
Length of Minor Axis = b
Center (0, 0):
h = 0
k = 0
a = 1/2 x
a = 1/2 *(24)
a = 12
b = 1/2 y
b =1/2 (18)
b = 9
So, Equation of Ellipse:
General : (x - h)² / a² + (y - k)² / b² = 1
(x - 0)² / (12)² + (y - 0)² / (9)² = 1
x² / 144 + y² / 81 = 1
Focus = c:
_____
c = √a² - b²
_______
c = √(12)² - (9)²
______
c = √144 - 81 = sqrt63 = 7.9372
Focus : ( -7.9372, 0) and ( 7.9372, 0)
The correct answer is: a. 7.94 ft
Step 1
To find the distance from the center of the board to the foci of the ellipse that will result in the largest possible elliptical tabletop, we can use the formula:
[tex]\[ c = \sqrt{a^2 - b^2} \][/tex]
where:
- a is half the length of the major axis (which is half of the longer side of the rectangle)
- b is half the length of the minor axis (which is half of the shorter side of the rectangle)
Step 2
Given that the rectangle is 18 feet by 24 feet, [tex]\( a = \frac{24}{2} = 12 \)[/tex] feet and [tex]\( b = \frac{18}{2} = 9 \)[/tex] feet.
Step 2
Now, we can substitute these values into the formula:
[tex]\[ c = \sqrt{12^2 - 9^2} \]\[ c = \sqrt{144 - 81} \]\[ c = \sqrt{63} \][/tex]
Taking the square root of 63 gives us approximately 7.94 feet.
On a blueprint, a wall that is 45 feet long is 6 inches long. How long should a girder appear on the blueprint if the girder is actually 30 feet long?
Answer:
4 inches
Step-by-step explanation:
1 foot = 12 inches
45 feet --- 6 inches
From this we can conclude that the scale of the Blueprint is:
1 foot in actual = 1/90 foot on blueprint
If the girder is actually 30 feet long, then
30 feet --- (1/90*30*12) inches
30 feet --- 4 inches
a triangle is classified as equilateral triangle when it has
Answer:
Three sides with the same length because that is what an equilateral triangle is.
Step-by-step explanation:
Use the distributive property to create an equivalent expression to 54+18x
Answer:
An equivalent expression to 54+18x, using the distributive property is 18(3+x)
Step-by-step explanation:
In the problem we have the binomial (54+18x), and we are asked to use the distributive property, this means that we can find a factor wich is distributed in both terms of the binomial.
We know that any number, or in this case any polinomial (or binomial), can be multiplied times 1 and will stay the same number, and we only need to write this 1 in a way that could resolve our problem, in particular we can say that
[tex]1=\frac{18}{18}[/tex]
then we can distribute the denominator, so we calculate:
[tex](54+18x)=1(54+18x)=\frac{18}{18}(54+18x)=18(\frac{54}{18}+\frac{18x}{18})=18(3+x)[/tex]
And the equivalent expression to (54+18x) is 18(3+x).
The graph shows a line and two similar triangles.
The equation of the line will be y = 4/3x + 3. The correct option is C.
What is an equation of the line?An equation of the line is defined as a linear equation having a degree of one. The equation of the line contains two variables x and y. And the third parameter is the slope of the line which represents the elevation of the line.
The general form of the equation of the line:-
y = mx + c
m = slope
c = y-intercept
Slope = ( y₂ - y₁ ) / ( x₂ - x₁ )
From the given graph the slope will be calculated as:-
Rise = 4
Run = 3
Slope = Rise / Run = 4 / 3
The y-intercept is 3.
The equation can be written as:-
y = mx + c
y = 4/3x + 3
Therefore, the equation of the line is y = 4/3x +3.
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Pls help me with these three! Thanks!
To solve 49^3x = 343^2x+1, write each side of the equation in terms of base ____
Each side of [tex]49^{3x} = 343^{2x + 1}[/tex], expressed in terms of their bases, is: [tex]7^{2(3x)} = 7^{3(2x + 1)}\\\\[/tex].
Solving Equations by Expressing Equations in Terms of their BasesTo solve an equation, the bases can be made equivalent and cancelled out by creating exponents.
Given:
[tex]49^{3x} = 343^{2x + 1}[/tex]
First, express each side by simplifying their bases and make them equivalent. Let the base be 7.
Thus:
[tex]7^{2(3x)} = 7^{3(2x + 1)}\\\\[/tex]
To solve, cancel the bases to have, 2(3x) = 3(2x + 1) and solve for x.
Therefore, each side of [tex]49^{3x} = 343^{2x + 1}[/tex], expressed in terms of their bases, is: [tex]7^{2(3x)} = 7^{3(2x + 1)}\\\\[/tex].
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Answer: 7 then B on edge
Step-by-step explanation:
Answer: it’s
Step-by-step explanation:
write 60 as the product of two factors
"is it possible for four consecutive even numbers to have a sum that is ten more than the sum of the smallest two numbers"
An artist agrees to paint a landscape painting at $0.30 per square inch. If the painting is 13 inches by 17 inches, how much should the artist charge?
area of painting = 13 * 17 = 221 square inches
221 * 0.30 = 66.30
the price would be $66.30
Where a is the frontal area of the ball (πd2/4), and cd is the drag coefficient (0.47 for a sphere). what is the drag force on the ball?
Show the tens fact you used. Show the difference.
14-6=
10-?=?
Question 1:
14 - 6 = ?
Further ExplanationTo determine the difference between 14 and 6 is the same is 14 – 6. In other words, the different is the result we get if we subtract both values.
Therefore, the difference between 14 and 6 is 8.
14 – 6
= 8.
There is always three parts in any subtraction equation and these include:
The number that is being subtracted from also called minuendThe number that is being subtracted from the minuend, also called subtrahendThe differences, which is the result you get from subtracting the subtrahend from minuendQuestion 2:
10 - ? = ?
Further ExplanationThe given question means you should for a number that when being subtracted from 10 will give the same number.
Therefore, to solve the question, let the missing be X
10 – X = X
If you simplify the equation, then you should have:
10 = X + X
10 = 2X (divide both side by 2)
[tex]\frac{10}{2}[/tex] = [tex]\frac{2x}{2}[/tex]
5 = X
X = 5
Therefore, the answer for question 1 and 2 are 8 and 5 respectively
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Show the tens fact you used write the difference 14 https://brainly.com/question/13620963Show the tens fact you used. Write the difference. https://brainly.com/question/1904682KEYWORDS:
differencetens factsubtractionequationminuend7-4.9t=15+7.6t solve for t
WILL GIVE BRAINEST
use the elimination method to solve the system of equations.
2x+4y=10
3x-4y=5
a.(1,3)
b.(3,4)
c.(5,0)
d.(3,1)
Answer:
(3,1)
Step-by-step explanation:
Hi, to solve this problem:
[tex]2 x + 4y = 10\\ +\\3x-4y=5\\-----\\5x =15\\x = 15/5\\x= 3[/tex]
then, replacing the value of "x" in one equation.
2x (3) + 4 y = 10
6 +4y =10
4y= 10- 6
4 y = 4
y = 4/4
y =1
So, the correct option is d.(3,1).
Feel free to ask for more if it´s necessary or if you did not understand something.
Write 96 39/40 as a decimal