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The function f(x,y,z)equals2 x plus z squared has an absolute maximum value and absolute minimum value subject to the constraint x squared plus 2 y squared plus 3 z squaredequals16. Use Lagrange multipliers to find these values.

Answers

Answer 1

Answer:

Answer:

Absolute maxima an minma both occured at $(25)/(3)$ .

Step-by-step explanation:

Given function is,

$f(x,y,z)=2x+z^2\hfill (1)$

subject to,

$x^2+2y^2+3z^2=16\hfill (2)$

Let $g(x,y,z)=x^2+2y^2=3z^2-16$

To find absolute maxima and absolute minima using Lagranges multipliers method consider $\lambda$ as the multipliers such that,

$\nabla f=\lambda \nabla g$

$\leftrightarrow (2, 0 ,2z )=\lambda (2x, 4y, 6z)$

on compairing both side we get,

$2z=6\lambda z\implies \lambda=(1)/(3)$

$4\labda y=0\implies y=0$

$2=2\lambda x\implies x=(1)/(\lambda)=3$

From (2),

$x^2+2y^2+3z^2=16$

$\implies 9+0+3z^2=16$

$\implies z=\pm\sqrt{(7)/(3)}$

Absolute maxima, at x=3, y=0, $z= \sqrt{(7)/(3)}$ is,

$|f(x,y,z)|_(max)=(2x+z^2)_(3,0,\sqrt{(7)/(3)})=(2*3)+(7)/(3)=(25)/(3)$

Absolute minima, at x=3, y=0, $z= -\sqrt{(7)/(3)}$ is,

$|f(x,y,z)|_(max)=(2x+z^2)_(3,0,-\sqrt{(7)/(3)})=(2*3)+(7)/(3)=(25)/(3)$

Hence the result.

Related Questions

The Labor Bureau wants to estimate, at a 90% confidence level, the proportion of all households that receive welfare. A preliminary sample showed that 17.5% of households in this sample receive welfare. The sample size that would limit the margin of error to be within 0.025 of the population proportion is:_________.
At a given time, the length, L, of the shadow of an object varies directlyas the height of the object, H. If the shadow is 27 ft when the height ofthe object is 12 ft, what is the height of the object if the shadow is 18 ft?
Can you help me simplify it?
I need help please it’s due today I forgot how to do this :(
What is the LCD of 1/2 and 2/5

4. Find the slope and y-intercept.
y= 1/5 x + 4

Answers

Answer:

The slope is $\bold{(1)/(5)}$ and the y-intercept is $\bold{4}$ .

Step-by-step explanation:

We are given an equation in slope-intercept form (y = mx + b).

In this equation, we can define the following variables:

m = slope
b = y-intercept

The slope in an equation and of a line is the rise over run of a line. The numerator of the fraction will be how many y-coordinates the line will ascend or descend (depending on the sign). The denominator will signify how many x-coordinates the line will move to the left or the right.

The y-intercept is the coordinate at which the line crosses the y-axis. The value of x is 0 and the coordinate can be positive or negative.

Therefore, by looking at our equation, we can see that $(1)/(5)$ is the slope, and 4 is the y-intercept.

Solve for x*
(17x - 8)

Answers

For given question, x = 4

What is an equilateral triangle?

"It is a triangle with all three sides of equal length and each angle measures 60°."

What is an equation?

"It is a statement which consists of equal symbol between two mathematical expressions."

From given figure,

we can observe that the all sides of the triangle are equal.

This means, the given triangle is an equilateral triangle.

So, each angle of triangle measures 60°.

So, we get an equation,

⇒ 17x - 8 = 60°

⇒ 17x = 60 + 8

⇒ 17x = 68

⇒ x = 68/17

⇒ x = 4

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Answer:x=4

Step-by-step explanation:

This triangle is an equilateral triangle with all angles equal.

Sum of angles in a triangle=180

17x-8+17x-8+17x-8=180

Collect like terms

17x+17x+17x-8-8-8=180

51x-24=180

51x=180+24

51x=204

Divide both sides by 51

51x/51=204/51

x=4

Statistics can help decide the authorship of literary works. Sonnets by a certain Elizabethan poet are known to contain an average of μ = 8.9 new words (words not used in the poet’s other works). The standard deviation of the number of new words is σ = 2.5. Now a manuscript with six new sonnets has come to light, and scholars are debating whether it is the poet’s work. The new sonnets contain an average of x~ = 10.2 words not used in the poet’s known works. We expect poems by another author to contain more new words, so to see if we have evidence that the new sonnets are not by our poet we test the following hypotheses.H0 : µ = 8.88 vs Ha : µ > 8.88
Give the z test statistic and its P-value. What do you conclude about the authorship of the new poems? (Let a = .05.)
Use 2 decimal places for the z-score and 4 for the p-value.
a. What is z?
b.The p-value is greater than?
c.What is the conclusion? A)The sonnets were written by another poet or b) There is not enough evidence to reject the null.

Answers

Answer:

We conclude that the sonnets were written by by a certain Elizabethan poet.

Step-by-step explanation:

We are given the following in the question:

Population mean, μ = 8.9

Sample mean, $\bar{x}$ =10.2

Sample size, n = 6

Alpha, α = 0.05

Population standard deviation, σ = 2.5

First, we design the null and the alternate hypothesis

$H_(0): \mu = 8.88\nH_A: \mu > 8.88$

We use One-tailed z test to perform this hypothesis.

a) Formula:

$z_(stat) = \displaystyle\frac{\bar{x} - \mu}{(\sigma)/(√(n)) }$

Putting all the values, we have

$z_(stat) = \displaystyle(10.2 - 8.9)/((2.5)/(√(6)) ) = 1.28$

Now, $z_(critical) \text{ at 0.05 level of significance } = 1.64$

b) We calculate the p value with the help of z-table.

P-value = 0.1003

The p-value is greater than the significance level which is 0.05

c) Since the p-value is greater than the significance level, there is not enough evidence to reject the null hypothesis and accept the null hypothesis.

Thus, we conclude that the sonnets were written by by a certain Elizabethan poet.

Final answer:

The z-score is 1.86 and the p-value is 0.0314. As the p-value is less than the level of significance α (0.05), we reject the null hypothesis and conclude that the new sonnets were likely written by another author.

Explanation:

In this statistical testing scenario for authorship of literary works, we need to find out the z-score or z test statistic and then determine the p-value to check if the new sonnets could be the works of the known Elizabethan poet or not.

For calculating the z score, you use the formula z = (x~ - μ) / (σ / √n) = (10.2 - 8.9) / (2.5/ √6) = 1.86 to two decimal places. The p-value is determined from the standard normal distribution table which for a z-score of 1.86 is 0.0314.

Given that α = 0.05, since the p-value is less than α, we reject the null hypothesis H0 (that the works were by the Elizabethan poet). Therefore, we accept the alternative hypothesis Ha (the sonnets were written by another author).

Learn more about Statistical Testing for Authorship here:

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A human brain weighs about 1 kg and contains about 1011 cells. Assuming that each cell is completely filled with water (density = 1 g/mL), calculate the length of one side of such a cell if it were a cube. If the cells were spread out into a thin layer that was a single cell thick, what would be the total surface area (in square meters) for one side of the cell layer?

Answers

Answer:

Step-by-step explanation:

From the information given:

a human brain weighs = 1 kg ; = 1000 grams

Number of cells = 10¹¹ cells

The density of water filled in each cell = 1 g/mL

From above;

the weight of each of the brain cell = total weight of the human brain/the number of cells

the weight of each of the brain cell = 1000/10¹¹

the weight of each of the brain cell = 1 × 10⁻⁸ grams

Now, to calculate the quantity of water in each cell; we have:

= the weight of each brain × density

= $1 * 10^(-8) \ g * (1 \ mL)/(1 \ g)= 1 * 10^(-8) \ mL$

For cube; we know that

1 mL = 1 cm³

Thus:

$1 * 10^(-8) \ mL= 1 * 10^(-8) \ cm^3$

Recall that; the volume of a cube as well = $x^3$

where;

x = length of each sides

∴

$x^3$ = $1 * 10^(-8) \ cm^3$

$x = \sqrt[3]{1 * 10^(-8)}$

x = 0.0022 cm

Thus, the length of each side of the cell = 0.0022 cm

The surface area of a single cell = x²

The surface area of a single cell = (0.0022 cm)²

The surface area of a single cell = 4.84 × 10⁻⁶ cm²

Therefore, the total surface area of is:

$1 * 10^(11) \ cells * (4.84 * 10^(-6) \ cm^2)/(1 \ cell)$

= $4.84 * 10^5 cm^2$

= $5.0* 10^5 cm^2$

= 50 m²

Final answer:

If the human brain's cells were cube-shaped and filled with water, each cell would be roughly 21.5 micrometers on a side. If these cells were spread out into a single-cell-thick layer, the total surface area for one side of the layer would be approximately 4.63 square meters.

Explanation:

To answer your question, the human brain has about 1011 cells, each filled with water. Given the total mass of the brain (about 1 kg) and the number of cells, we can calculate the volume of a single cell. The density of water is 1 g/mL or 1,000 kg/m³, so the volume of all the cells (entire brain) is 1 m³. Therefore, the volume of a single cell must be 1 m³/1011 cells, which is approximately 10-14 m³. For a cubical cell, the side length of the cube (a) would be the cube root of this volume, which is approximately 2.15 x 10-5 m or 21.5 micrometers.

To calculate the total surface area for one side of the cell layer, we multiply the area of a single cell by the total number of cells: (2.15 x 10-5)² m²/cell x 1011 cells = approximately 4.63 m².

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WILL GIVE MEDAL AND FAN! you invest $5,175.00 in a stock plan. it increases 9% the first year and then loses 5% of it's value the second year. what is your gain compared to your original investment?

Answers

Answer:

$183.7125

Step-by-step explanation:

Given,

Original investment, A = $ 5,175.00

In first year,

Thetotal investment = $ 5,175.00

The amount is increased by 9 %,

Thus, the final amount at the end of first year,

$A_1=5,175(1+(9)/(100))^1$

$=5175(1+0.09)$

$=5175(1.09)$

$=\$ 5640.75$

In Second year,

The total investment = $ 5640.75,

The amount is decreased by 5 %,

Thus, the final amount at the end of second year,

$A_2=5640.75(1-(5)/(100))^1$

$=5640.75(1-0.05)$

$=5640.75(0.95)$

$=\$ 5358.7125$

Hence,

$\text{Total Gain}=A_2-A$

$=5358.7125-5175$

$=\$183.7125$

5,175 + 9% = 5640.75

5640.75 - 5% = 5302.305

Original investment = 5,175
Gain = 5,302.305

Subtract them both and get = 127.305

There are 45 new houses being built in a neighborhood. Last month, 1/3 of them were sold. This month, 1/5 of the remaining houses were sold. How many houses are left be sold?

Answers

45 new houses.....last month 1/3 were sold....
1/3(45) = 45/3 = 15 houses sold last month

this month, 1/5 of the remaining houses were sold...
remaining houses left are (45 - 15) = 30
1/5(30) = 30/5 = 6 houses sold this month

houses left : 45 - 15 - 6 = 45 - 21 = 24 houses remain <==

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