Is 0.25 greater or less than 0.03?

The smallest value of the function f(x₁, x₂) = x₁ + x₂ on the disk D with a radius of 1 is -√2, and the largest value is √2. The relative change in the smallest value, expressed in percent, can be calculated if the radius of the disk decreases to 0.99.

a) The problem can be formulated as an optimization problem with constraints. We want to find the smallest and largest values that the function f(x₁, x₂) = x₁ + x₂ achieves on the set D, which is defined as the disk with radius r = 1, i.e., D = {(x₁, x₂) ∈ ℝ² | x₁² + x₂² < 1}.

To find the smallest value, we can minimize the function f subject to the constraint that (x₁, x₂) is within the disk D. Mathematically, this can be written as:

Minimize: f(x₁, x₂) = x₁ + x₂

Subject to: x₁² + x₂² < 1

To find the largest value, we can maximize the function f subject to the same constraint. Mathematically, this can be written as:

Maximize: f(x₁, x₂) = x₁ + x₂

Subject to: x₁² + x₂² < 1

b) To find the points at which the function f achieves the maximum and minimum on the set D, we can analyze the problem. The function f(x₁, x₂) = x₁ + x₂ represents a plane with a slope of 1.

Considering the constraint x₁² + x₂² < 1, we observe that it represents a circle with radius 1 centered at the origin.

Since the function f represents a plane with a slope of 1, the maximum and minimum values occur at the points on the boundary of the disk D where the plane is tangent to the disk. In other words, the maximum and minimum values occur at the points where the plane f(x₁, x₂) = x₁ + x₂ is perpendicular to the boundary of the disk.

Considering the disk D: x₁² + x₂² < 1, we can see that the boundary of the disk is x₁² + x₂² = 1 (the equation of a circle).

At the boundary, the gradient of the function f(x₁, x₂) = x₁ + x₂ is parallel to the normal vector of the boundary circle. The gradient of f is (∂f/∂x₁, ∂f/∂x₂) = (1, 1), which represents the direction of steepest ascent of the function.

Thus, at the points where the plane f(x₁, x₂) = x₁ + x₂ is tangent to the boundary circle, the gradient of f is parallel to the normal vector of the circle. Therefore, the gradient of f at these points is proportional to the vector pointing from the origin to the tangent point.

To find the tangent points, we can use the fact that the tangent line to a circle is perpendicular to the radius at the point of tangency. The radius of the circle D is the vector from the origin to any point (x₁, x₂) on the boundary, which is (x₁, x₂).

So, the tangent points occur when the gradient vector (1, 1) is proportional to the radius vector (x₁, x₂), which means:

1/1 = x₁/1 = x₂/1

Simplifying, we get:

x₁ = x₂

Substituting this back into the equation of the boundary circle, we have:

x₁² + x₂² = 1

x₁² + x₁² = 1

2x₁² = 1

x₁² = 1/2

Taking the positive square root, we get:

x₁ = √(1/2)

Since x₁ = x₂, the corresponding values are:

x₂ = √(1/2)

Thus, the points where the function f achieves the maximum and minimum on the set D are (x₁, x₂) = (√(1/2), √(1/2)) and (x₁, x₂) = (-√(1/2), -√(1/2)).

Plugging these values into the function f(x₁, x₂) = x₁ + x₂, we get:

f(√(1/2), √(1/2)) = √(1/2) + √(1/2) = 2√(1/2) = √2

f(-√(1/2), -√(1/2)) = -√(1/2) - √(1/2) = -2√(1/2) = -√2

Therefore, the largest value of f is √2, and the smallest value of f is -√2.

c) Denoting the smallest value as fin = -√2, we can find the relative change in fin expressed in percent if the radius of the disk decreases to D = {(x₁, x₂) ∈ ℝ² | x₁² + x₂² ≤ 0.99}.

To calculate the relative change, we can use the formula:

Relative Change = (New Value - Old Value) / Old Value * 100

The new value of fin, denoted as fin', can be found by minimizing the function f subject to the constraint x₁² + x₂² ≤ 0.99.

Solving the minimization problem, we find the new smallest value fin' on the set D with a radius of 0.99.

Comparing fin' to fin, we can calculate the relative change:

Relative Change = (fin' - fin) / fin * 100

By solving the new minimization problem, you can find the new smallest value fin' and calculate the relative change using the formula provided.

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a) For these 24 dog sitters, as experience increases, the amount charged tends to increase.

b)  For these 24 dog sitters, there is a positive correlation between experience and amount charged.

c) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately $16.5 per hour.

(a) For these 24 dog sitters, as experience increases, the amount charged tends to increase. This means that the amount charged per hour increases with an increase in years of experience for dog sitters.

(b) For these 24 dog sitters, there is a positive correlation between experience and amount charged. The points on the scatter plot show a generally upward trend, and the line of best fit is also sloping upward.

(c) Using the line of best fit, we would predict that a dog sitter with 5 years of experience would charge approximately $16.5 per hour. This can be determined by locating the point on the X-axis corresponding to 5 years of experience, and then drawing a vertical line to the line of best fit. From there, we can draw a horizontal line to the Y-axis to find the predicted amount charged per hour, which is about $16.5.

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Answer:

[tex]8.71*10^{-3}[/tex]

Step-by-step explanation:

Moving the decimal point 3 spots to the right, we get 8.71, which shows that 0.00871 is equal to 8.71*10^-3 in scientific notation

Answer:

Step-by-step explanation:

0.00871 written in scientific notation is 8.71 x 10^(-3).

hope it helps!

Answer:

221 m

Step-by-step explanation:

Let h = height of dam

tan 33 = h/340

340 tan 33 = h

h = 221 m

The height of the dam to the nearest meter will be 221 m.

The vertical distance between the object's top and bottom is defined as height. It is measured in centimeters, inches, meters, and other units.

From the trigonometry;

[tex]\rm tan \theta = \frac{P}{B} \\\\ \rm tan 33^0 = \frac{h}{340} \\\\ h=221 \ m[/tex]

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Answer:

can you please post with pictures?

because I don't know what degrees the angles are, therefore can't help

Answer:

a)  2.7 sec

b)  2.6 sec

c)  30.3 ft,  1.3 sec

Step-by-step explanation:

graphed the equation and determined answers from the curve

c × c = c²

c × -5 = -5c

8 × c = 8c

8 × -5 = -40

arrange

c² -5c +8c -40

c² +3c -40

answer is B

Answer:

c^2 + 3c - 40

Step-by-step explanation:

seriously dude all you have to do is use an online calculator or something

Answer:

A= 30

B= 24.3

Step-by-step explanation:

If c > 0, then lim_(x->c) (x^c - c^x) / (x - c) = 1 - inc/1+inc.

To show that if c > 0, then lim Xc - Cx / Xx - Cc = 1, we need to make use of L'Hopital's Rule and some algebraic manipulations. Here's how to do it:

Given that c > 0, let f(x) = x^c - c^x and g(x) = x - c.

Then, the given limit is equivalent to: lim_(x->c) f'(x) / g'(x)

By differentiating f(x) and g(x), we get: f'(x) = c * x^(c-1) - c^x * ln(c) and g'(x) = 1.

So, the limit becomes: lim_(x->c) [c * x^(c-1) - c^x * ln(c)] / 1

Now, let's rewrite the numerator: [c * x^(c-1) - c^x * ln(c)] = c * (x^(c-1) - c^(x-1) * ln(c))

Therefore, the limit becomes: lim_(x->c) [c * (x^(c-1) - c^(x-1) * ln(c))] / 1

Now, we can use L'Hopital's Rule to evaluate the limit:

lim_(x->c) [c * (x^(c-1) - c^(x-1) * ln(c))] / 1

= lim_(x->c) [c * ((c-1) * x^(c-2) - (x-1) * c^(x-2) * ln(c))] / 0

= -c * ln(c) * lim_(x->c) (x-1) * c^(x-2) / ((c-1) * x^(c-2))

Now, let's evaluate the limit inside the parentheses:

lim_(x->c) (x-1) * c^(x-2) / ((c-1) * x^(c-2))

= lim_(x->c) (x/c)^(c-2) * (x-1)/(c-1)

We can simplify the above expression as follows:

(x-1)/(c-1)

= (x-c+c-1)/(c-1)

= (x-c)/(c-1) + (c-1)/(c-1)

= (x-c)/(c-1) + 1

Therefore, the limit becomes:

lim_(x->c) -c * ln(c) * [(x-c)/(c-1) + 1]

= lim_(x->c) -c * ln(c) * (x-c)/(c-1) - c * ln(c)

= 1 - inc/1+inc

Hence, we have shown that if c > 0, then lim_(x->c) (x^c - c^x) / (x - c) = 1 - inc/1+inc.

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Show that if c > 0, then lim Xc-Cx/Xx-Cc=1 - Inc/1+ Inc1+ Inc

Answer:

see explanation

Step-by-step explanation:

Since the marked angles are congruent, let them be x

(6)

The sum of the interior angles of a quadrilateral = 360°

Sum the angles and equate to 360

x + x + 120 + 90 = 360

2x + 210 = 360 ( subtract 210 from both sides )

2x = 150 ( divide both sides by 2 )

x = 75

Then ∠ X = ∠ Y = 75°

---------------------------------------------------------------------

(7)

The sum of the interior angles of a hexagon = 720°

Sum the angles and equate to 720

x + x + 108 + 103 + 149 + 90 = 720

2x + 450 = 720 ( subtract 450 from both sides )

2x = 270 ( divide both sides by 2 )

x = 135

Then ∠ X = ∠ Y = 135°

Answer:

Step-by-step explanation:

Sum of the interior angles of a regular polygon:

Quadrilateral has sum of angles:

Sum the given angles and consider x = y as marked congruent:

Hexagon has sum of angles:

Sum the given angles and consider x = y as marked congruent:

Answer:

24

Step-by-step explanation:

6 ÷ 1/4 = 6 × 4 = 24

Answer:

24

Step-by-step explanation:

Answer:

The correct answer in each case is:

Step-by-step explanation:

First, to calculate the surface area or the volume you must have all the measures in the same units, for the exercise we're gonna use centimeters and after we can replace the units in the answer if we need, then:

Now, to obtain the surface area of the triangle we're gonna use the next formula:

And we replace the values in centimeters:

To obtain this same value now in square milimeters, you must know:

Now, you must multiply:

To obtain this value now in [tex]m^{2}[/tex], you must know:

You must divide:

By last, to obtain the volume of the piece of cake, you can use the next formula:

And we replace the surface area in [tex]cm^{2}[/tex] because the answer must be in [tex]cm^{3}[/tex]:

Answer:

x = 33

Step-by-step explanation:

sin 29° = 16/x

x = 33

200x3=600

50x3=150

600+150=750

750

Answer:

1=140  2=9  3=19

Step-by-step explanation:

Answer:

1=140
2=9
3=19

Step-by-step explanation:

The correlation coefficient is:r = -0.332 / (0.683 × 0.566) = -0.994

Correlation Coefficient The correlation coefficient, denoted by r, is a statistical measure of the strength of the relationship between two quantitative variables.

It is always between -1 and 1, where 1 indicates a perfect positive correlation, -1 indicates a perfect negative correlation, and 0 indicates no correlation at all.

In this problem, we have the joint probability function for X1 and X2 as follows:

P(0,0) = 0.06P(2,0) = 0.20P(1,0) = 0.14P(2,1) = 0.30P(1,1) = 0.10P(2,2) = 0.20

The probability of two calls and one order is given as 0.30. Thus:P(2,1) = 0.30

Now, let us find the marginal probabilities of X1 and X2.P(X1 = 0) = P(0, 0) + P(0, 1) + P(0, 2) = 0.06 + 0 + 0 = 0.06P(X1 = 1) = P(1, 0) + P(1, 1) + P(1, 2) = 0.14 + 0.10 + 0 = 0.24P(X1 = 2) = P(2, 0) + P(2, 1) + P(2, 2) = 0.20 + 0.30 + 0.20 = 0.70Similarly,P(X2 = 0) = P(0, 0) + P(1, 0) + P(2, 0) = 0.06 + 0.14 + 0.20 = 0.40P(X2 = 1) = P(0, 1) + P(1, 1) + P(2, 1) = 0 + 0.10 + 0.30 = 0.40P(X2 = 2) = P(0, 2) + P(1, 2) + P(2, 2) = 0 + 0 + 0.20 = 0.20

The expected values of X1 and X2 are:

E(X1) = 0 × 0.06 + 1 × 0.24 + 2 × 0.70 = 1.64

E(X2) = 0 × 0.40 + 1 × 0.40 + 2 × 0.20 = 0.80

Let us now find the expected value of X1X2:

E(X1X2) = 0 × 0.06 + 1 × 0.00 + 2 × 0.30 + 0 × 0.14 + 1 × 0.10 + 2 × 0.20 = 1.14

Thus, the covariance of X1 and X2 is:Cov(X1, X2) = E(X1X2) - E(X1)E(X2) = 1.14 - 1.64(0.80) = -0.332

Finally, the correlation coefficient is given as

r = Cov(X1, X2) / (σ(X1)σ(X2))

Where σ(X1) is the standard deviation of X1 and σ(X2) is the standard deviation of X2.

Let us find the standard deviations of X1 and X2:

Variance of X1:Var(X1) = E(X1^2) - [E(X1)]^2 = 0^2(0.06) + 1^2(0.24) + 2^2(0.70) - (1.64)^2 = 0.4676

Standard deviation of X1:σ(X1) = √Var(X1) = √0.4676 = 0.683

Variance of X2:Var(X2) = E(X2^2) - [E(X2)]^2 = 0^2(0.40) + 1^2(0.40) + 2^2(0.20) - (0.80)^2 = 0.3200

Standard deviation of X2:σ(X2) = √Var(X2) = √0.3200 = 0.566

Thus, the correlation coefficient is:r = -0.332 / (0.683 × 0.566) = -0.994

Therefore, the correlation coefficient is very close to -1.

This means that there is a very strong negative correlation between the number of calls and the number of orders placed. This can be interpreted as follows: As the number of calls increases, the number of orders placed decreases, and vice versa.

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Answer:

56 degree and 68 degree

plz mark me as brainliest

Answer:

62°

Step-by-step explanation:

Since the triangle is isosceles then the 2 base angles are congruent.

Given there is one angle measuring 56° then it must be the vertex angle.

Thus the 2 congruent base angles are

[tex]\frac{180-56}{2}[/tex] = [tex]\frac{124}{2}[/tex] = 62°

Answer:

2/3

5/8

5/9

7/10

etc.

Answer: 2 2/3

Step-by-step explanation:

Answer:

2 2/3

Step-by-step explanation:

Answer:

B. 37.5

Step-by-step explanation:

100/24 = 4.1666666667

4.1666666667*9 = 37.5

Answer:

84

Step-by-step explanation:

To find the range, find the difference between the largest value and the smallest value.

544 - 460 = 84

Let the height of the prism be h and the apothem of the hexagonal base be a, and let the distance between the bases be d. The volume of the prism is given by the formula: V = (1/2) × 6a × h × 2 + 6 × (1/2) × 6 × a × h

[Note: The hexagon has 6 equilateral triangles as its sides. Each triangle has base 6 cm and height a. The volume of the prism is equal to the sum of the volumes of the 12 equal triangular prisms that are formed by the 12 triangular faces of the hexagonal prism]

V = 6ah + 18ah = 24ahGiven that the volume of the prism is 450 cc, we can equate this expression to 450 to obtain:450 = 24ahDividing both sides by 24a, we obtain:450 / 24a = h

The bases of the prism are parallel to each other, and each is a regular hexagon. To obtain the distance between them, we can add twice the apothem to the height of the prism: Distance between bases = 2a + h We can substitute h with the expression we derived earlier: Distance between bases = 2a + 450 / 24aFor the volume of the prism to be positive, the height and the apothem must be positive.

Therefore, the distance between the bases is also positive. We can now use calculus to minimize this expression for the distance between the bases. However, we can also use the arithmetic mean-geometric mean inequality as follows:(2a) + (450 / 24a) ≥ 2 √(2a × 450 / 24a) = √(2 × 450) = 3√50

Therefore, the distance between the bases is at least 3√50 cm. The equality holds when 2a = 450 / 24a. To show that this is indeed the minimum value of the distance between the bases, we need to demonstrate that the value is achievable. We can solve this equation for a to obtain:a² = 75/4a = √(75/4) = (5/2)√3

Therefore, the minimum value of the distance between the bases is 3√50 cm, and it is achieved when a = (5/2)√3.

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Answer:

In explanation

Step-by-step explanation:

ln(x) = 5

log (base e) x = 5

exponential form is e ^ 5 = x

you should be able to generate 2 sentences from here, hope it helps!

Answer:

The percent of the air ticket went up 148%  plus the baggage fee would be 168%. The new total is 420$

Step-by-step explanation:

Answer:

7.2

Step-by-step explanation:

Answer:

I dont think you realize you posted the answer...

Step-by-step explanation:

7.2

Answer:

260 minutes > 4.25 hours

Step-by-step explanation:

1h=60 min.

0.25 h=¼h=¼*60min=15min.

4.25h=4h+0.25h=60*4+15min=240+15=255min

260>255

Answer:

60+60+60+60=240

240/4=4,25

=4,25

Answer:

If both expressions have the same value after substituting and simplifying two different values for the variable, then they are  

✔ equivalent

.

!Step-by-step explanation:

hop3 it helped

The reference angles are in the first, second, third, and fourth is 78°, 102°, 258°, and 282° respectively.

the angle between the terminal side of the angle and the x-axis.

For the first quadrant:

The reference angle = 258 - 180 = 78°

For the second quadrant:

= 180 - 78 = 102°

For the third quadrant:

= 78 + 180

= 258°

For the fourth quadrant:

= 360- 78

= 282°

Thus, the reference angles are in the first, second, third, and fourth is 78°, 102°, 258°, and 282° respectively.

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Answer:

A

Step-by-step explanation:

Answer:

A is correct. Hope it helps.