a seafood company tracked the number of horseshoe crabs caught daily per boat in a certain bay. calculate the variance

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.

To learn more about functions visit : https://brainly.com/question/11624077

#SPJ11

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:

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]:

Suppose you have 10 black, 10 white, 10 blue, and 10 brown socks. The minimum number of socks to pick that all 3 brothers wear the same color socks is 4 of the same color.

You are required to determine how many socks to pick blindly since there is no light in the room so all three brothers wear the same color socks. In this question, we have 4 types of socks. Therefore, we can apply Pigeonhole Principle here. Pigeonhole Principle states that If n+1 items are put into n boxes, then at least one box must contain two or more items.

Assuming that a brother needs to wear 3 socks of the same color. Therefore, for all 3 brothers to wear the same color socks, we must pick 4 socks of the same color. Therefore, the minimum number of socks that needs to be picked = 4.  

You can learn more about Pigeonhole Principle at: brainly.com/question/31687163

#SPJ11

Answer:

x^3 + 1 +  (−7) /  x+3

Step-by-step explanation:

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]

To learn more about the height, refer to the link;

https://brainly.com/question/10726356

#SPJ2

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:

x = 33

Step-by-step explanation:

sin 29° = 16/x

x = 33

Answer:

A

Step-by-step explanation:

Answer:

A is correct. Hope it helps.

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:

A vertical line

Answer:

1=140  2=9  3=19

Step-by-step explanation:

Answer:

1=140
2=9
3=19

Step-by-step explanation:

The circle's radius is:

(33.912 : 3.14) : 2 = 5.4 (inches)

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:

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.

To know more about  correlation coefficient visit:

https://brainly.in/question/23382733

#SPJ11

If they make additional payments of $3,000 every four months for the first five years, it will reduce the outstanding balance on the mortgage.

To calculate the values requested, we'll need to perform various calculations based on the given information. Let's go step by step:

Monthly Interest Rate:

The mortgage rate provided is 4.9% compounded semi-annually. To calculate the monthly interest rate, we need to convert the semi-annual rate to a monthly rate. Since there are two compounding periods per year, we divide the annual interest rate by two and then convert it to a decimal:

Monthly interest rate = (4.9% / 2) / 100 = 0.0245

Monthly Payments:

To calculate the monthly payments, we can use the formula for the monthly payment of a mortgage:

Monthly Payment = (Loan Amount - Down Payment) * (Monthly Interest Rate / (1 - (1 + Monthly Interest Rate)^(-Number of Months)))

Loan Amount = $460,000 - (10% * $460,000) = $414,000

Number of Months = 25 years * 12 months/year = 300 months

Substituting these values into the formula, we can calculate the monthly payments:

Monthly Payment = ($414,000) * (0.0245 / (1 - (1 + 0.0245)^(-300)))

Amount Owed after 5 Years:

After five years, the number of months remaining on the mortgage is 25 years - 5 years = 20 years = 240 months. To find out how much they owe after the first five years, we need to calculate the remaining balance on the mortgage:

Remaining Balance = Loan Amount * (1 + Monthly Interest Rate)^Number of Months - Total Payments

In this case, the Loan Amount is $414,000, the Monthly Interest Rate is 0.0245, and the Number of Months is 240. To calculate the Total Payments, we can multiply the Monthly Payment by the number of months paid during the first five years (5 years * 12 months/year = 60 months).

Interest Paid in the First Five Years:

To determine the interest paid in the first five years, we can subtract the remaining balance after five years from the initial loan amount:

Interest Paid = Loan Amount - Remaining Balance

New Semi-Monthly Payments after Renegotiation:

If they renegotiate the mortgage at a new rate of 5.9%, compounded semi-annually, and plan to pay off the remaining balance from part c, we need to recalculate the monthly payments.

We can use a similar formula as in part b, substituting the remaining balance (from part c) as the new loan amount, and the semi-monthly interest rate based on the new mortgage rate.

Payments of $3,000 every Four Months in the First Five Years:

To calculate the new interest paid in the first five years, we need to consider the reduced balance resulting from the additional payments and then subtract this balance from the initial loan amount.

Now, let's perform the calculations and find the values for each part.

Learn more about mortgage

brainly.com/question/31112455

#SPJ11

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.

Learn more about the reference angle here:

brainly.com/question/16210309

#SPJ2

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.

Learn more about prism:

https://brainly.com/question/12649592

#SPJ11

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.

To learn more about manipulations, refer below:

https://brainly.com/question/28701456

#SPJ11

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

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:

2/3

5/8

5/9

7/10

etc.

Answer:

can you please post with pictures?

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

The differentiated 3-form:

d(sin(x₅x₂ + x₃x₄) dx₁ ∧ dx₂∧ dx₃) = cos(x₅x₂ + x₃x₄) * (d(x₅x₂ + x₃x₄) ∧ dx₁ ∧ dx₂∧ dx₃)

To differentiate the 3-form given by sin(x₅x₂ + x₃x₄) dx₁ ∧ dx₂∧ dx₃,

we need to apply the exterior derivative operator (denoted by d) to each component of the 3-form.

First, let's differentiate the component sin(x₅x₂ + x₃x₄):

d(sin(x₅x₂ + x₃x₄)) = cos(x₅x₂ + x₃x₄) * d(x₅x₂ + x₃x₄)

Next, we need to differentiate the differential forms dx₁, dx₂, and dx₃:

d(dx₁) = 0

d(dx₂) = 0

d(dx₃) = 0

Finally, we can assemble the differentiated 3-form:

d(sin(x₅x₂ + x₃x₄) dx₁ ∧ dx₂∧ dx₃) = cos(x₅x₂ + x₃x₄) * (d(x₅x₂ + x₃x₄) ∧ dx₁ ∧ dx₂∧ dx₃)

Learn more about derivative operator here:

https://brainly.com/question/25752367

#SPJ4

Answer:

24

Step-by-step explanation:

6 ÷ 1/4 = 6 × 4 = 24

Answer:

24

Step-by-step explanation:

Answer:

Henry and Valencia are 16 m apart.

Step-by-step explanation:

A sketch to this question shows that their distance apart forms a right angled triangle. So then, we can apply the Pythagoras theorem.

Let the distance between Henry and Valencia be represented by x, so that;

[tex]/hyp/^{2}[/tex] = [tex]/adj 1/^{2}[/tex] + [tex]/adj 2/^{2}[/tex]

[tex]20^{2}[/tex] = [tex]x^{2}[/tex] + [tex]12^{2}[/tex]

400 = [tex]x^{2}[/tex] + 144

[tex]x^{2}[/tex] = 400 - 144

   = 256

x = [tex]\sqrt{256}[/tex]

  = 16

x = 16 m

Therefore, Henry and Valentia are 16 meters apart.

Answer:

Robert

Step-by-step explanation:

If Robert runs 3 miles in 45 minutes, by the end of 90minutes, Robert would have covered 6miles, while Jacob covered 5miles in 90 minutes, therefore, Robert ran faster than Jacob.

Answer: 4a

Step-by-step explanation:

Answer:

4a

Step-by-step explanation:

8 of something minus 4 of something gives you 4 of that thing