Find the area of the triangle below.

Minimizing curve C is a helix, described by the equation :

C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)

To express the length L(p) of curve C in terms of p, we can use the formula for the length of a curve in cylindrical coordinates. In cylindrical coordinates, the arc length element ds can be given by:

ds² = dr² + r² dt² + dz²

Since dr = 0 (as r = a is constant along the curve C), and dt = -a sin(t) dt (from the parametrization), we have:

ds² = a² sin²(t) dt² + dz²

Integrating ds over the curve C from t = 0 to t = tmax, we get:

L(p) = ∫[0,tmax] √(a² sin²(t) + p'(t)²) dt

where p'(t) denotes the derivative of p(t) with respect to t.

To find the differential equation for the function p(t) that minimizes L(p), we can apply the Euler-Lagrange equation of the calculus of variations. The Euler-Lagrange equation is given by:

d/dt (dL/dp') - dL/dp = 0

Differentiating L(p) with respect to p' and p, we have:

dL/dp' = 0 (since p does not appear explicitly in L(p))

dL/dp = d/dt (dL/dp') = d/dt (a² sin²(t) p'(t) / √(a² sin²(t) + p'(t)²))

Using the chain rule, we can simplify the expression:

dL/dp = (a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2)

Setting the Euler-Lagrange equation equal to zero, we get:

(a² sin²(t) p''(t) - a² sin(t) cos(t) p'(t)²) / (a² sin²(t) + p'(t)²)^(3/2) = 0

Simplifying further, we have:

p''(t) - (sin(t) cos(t) / sin²(t)) p'(t)² = 0

This is the differential equation that the function p(t) must satisfy to minimize L(p).

To solve this differential equation, we can make the substitution u = p'(t). Then the equation becomes:

du/dt - (sin(t) cos(t) / sin²(t)) u² = 0

This is a separable first-order ordinary differential equation. By solving it, we can obtain the solution for u = p'(t). Integrating both sides and solving for p(t), we get:

p(t) = C exp(-cot(t)) + h

where C is a constant determined by the initial condition p(0) = 0, and h is the value of p at t = tmax.

Therefore, the minimizing curve C is a helix, described by the equation :

C(t) = (a cos(t), a sin(t), C exp(-cot(t)) + h)

where C is a constant determined by the initial condition, and h is the value of p at t = tmax.

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Answer:

3.3/2 = 1.65² x 3.14 = 8.54865

Answer:

7/10

Step-by-step explanation:

Answer:

The area of the rectangle is 24.

Step-by-step explanation:

The given points:

a) (-2, 2)

b) (-2, -4)

c) (2, -4)

To complete the rectangle the other point must be (2, 2), so the rectangle formed has the following dimensions:

x: distance in the x-direction (from -2 to 2):

[tex]x = 2 - (-2) = 4[/tex]

y: distance in the y-direction (from -4 to 2):

[tex]y = 2 - (-4) = 6[/tex]

The area of the rectangle is:

[tex] A = x*y = 4*6 = 24 [/tex]

Therefore, the area of the rectangle is 24.

I hope it helps you!      

the solution to the equation log(a) + log(z - 6) = 2 is z = 100/a + 6.

We can simplify the equation using logarithmic properties. The sum of logarithms is equal to the logarithm of the product, so we can rewrite the equation as log(a(z - 6)) = 2.

Next, we can convert the equation to exponential form. In exponential form, the base of the logarithm becomes the base of the exponent and the logarithm value becomes the exponent. Therefore, we have a(z - 6) = 10^2, which simplifies to a(z - 6) = 100.

To solve for z, we need to isolate it. Divide both sides of the equation by a: (z - 6) = 100/a.

Finally, add 6 to both sides to solve for z: z = 100/a + 6.

So, the solution to the equation log(a) + log(z - 6) = 2 is z = 100/a + 6.

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Answer:

9 - 166.42m²

10 - 308.8cm²

Step-by-step explanation:

The first figure shown is a triangular prism

We can find the surface area using this formula

[tex]SA=bh+L(S_1+S_2+h)[/tex]

where

B = base length

H = height

L = length

S1 = base length

S2 = slant height ( base's hypotenuse )

The triangular prism has the following dimensions

Base Length = 4m

Height = 5.7m

Length = 8.6m

S1 = 4m

S2 = 7m

Having found the needed dimensions we plug them into the formula

SA = ( 4 * 5.7 ) + 8.6 ( 4 + 7 + 5.7 )

4 * 5.7 = 22.8

4 + 7 + 5.7 = 16.7

8.6 * 16.7 = 143.62

22.8 + 143.62 = 166.42

Hence the surface area of the triangular prism is 166.42m²

The second figure shown is a pyramid

The surface area of a pyramid can be found using this formula

[tex]SA = A+\frac{1}{2} ps[/tex]

Where

A = Area of base

p = perimeter of base

s = slant height

The base of the pyramid is a square so we can easily find the area of the base by multiplying the base length by itself

So A = 8 * 8

8 * 8 = 64

So the area of the base (A) is equal to 64 cm^2

The perimeter of the base can easily be found by multiplying the base length by 4

So p = 4 * 8

4 * 8 = 32 so p = 32

The slant height is already given (15.3 cm)

Now that we have found everything needed we plug in the values into the formula

SA = 64 + 1/2 32 * 15.3

1/2 * 32 = 16

16 * 15.3 = 244.8

244.8 + 64 = 308.8

Hence the surface area of the pyramid is 308.8cm²

Answer:

Diameter: 26cm

Circumference: 26π / 81.68cm

Area: 169π / 531cm²

Step-by-step explanation:

Diameter is radius x 2, so 13 x 2 = 26

Circumference is diameter x π, so 26 x π = 26π / 81.68cm

Area is π x r², so π x 13² = 169π / 531cm²

Using the corrected linearized form a = c * A * B * log(x) - A * B * log(A) + A * B * log(B), solve for unknowns A, B, c, and x.

To solve the equation a = c * A * B * log(x) - A * B * log(A) + A * B * log(B) for unknowns A, B, c, and x, we need additional information or constraints.

Without specific values or relationships among the variables, it is not possible to provide a numerical solution. However, if you have specific values for any of the variables or if there are constraints or relationships among them, we can apply appropriate mathematical techniques, such as substitution or optimization methods, to find the values of A, B, c, and x that satisfy the equation.

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Answer:

2.25kg

Step-by-step explanation:

500g+850g+900g=2250

2250/1000=2.25kg

Answer:

The last one is symmetrical.

The first one and the last one are correct.

The correct probabilities are 0.1017 and 0.2053.

Given:

P(c\NV)=0.139, P(C|V)= 1- 0.95 = 0.05

P(V) = 0.418

P(NV) = 1- 0.418 = 0.582.

(1). The probability of that person getting Covid? P CP(C) = P(C|NV)        P(NV)+P(C|V) P(V)

0.139*0.582+0.05*0.418

= 0.1017.

(2).  The probability that he or she was vaccinated with P fizer P(V|C).

   [tex]P(V|C) = \frac{P(V|C)P(V)}{P(C|NV)P(NV)+P(CV)P(V)}[/tex]

                        [tex]\frac{0.05\times0.418}{0.139\times0.582+0.05\times0.418} = 0.2053[/tex]

3). P(NV|C) = 1 - P(V|C) = 0.7946.

(4). The chances of Covid is decreased.

(5). A second paragraph using statistics to support someone choosing 0.1017 = 10% got Covid and 0.139 = 13% not vaccinate.

Therefore, the probability of that person getting Covid is 0.1017 and the probability that he or she was vaccinated with Pfizer is 0.2053.

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To approximate the stationary matrix S for the transition matrix P, we need to compute powers of the transition matrix P until it reaches a stable matrix.

Starting with P = [0.37 0.63; 0.19 0.81], we can compute powers of P as follows:

P^2 = P * P

= [0.37 0.63; 0.19 0.81] * [0.37 0.63; 0.19 0.81]

= [0.2746 0.7254; 0.1538 0.8462]

P^3 = P * P^2

= [0.37 0.63; 0.19 0.81] * [0.2746 0.7254; 0.1538 0.8462]

= [0.2421 0.7579; 0.1873 0.8127]

P^4 = P * P^3

= [0.37 0.63; 0.19 0.81] * [0.2421 0.7579; 0.1873 0.8127]

= [0.2222 0.7778; 0.1941 0.8059]

Continuing this process, we find:

P^5 = [0.2149 0.7851; 0.1957 0.8043]

P^6 = [0.2124 0.7876; 0.1961 0.8039]

P^7 = [0.2117 0.7883; 0.1961 0.8039]

As we can see, the matrix P^7 is very close to the stationary matrix S. Therefore, we can approximate the stationary matrix S as:

S ≈ [0.2117 0.7883; 0.1961 0.8039]

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Answer:

b,c,e

Step-by-step explanation:

Answer:

5/8

Step-by-step explanation:

10 * 2^-4 = 10 * 1/16 = 10/16 = 5/8

Answer:

d

Step-by-step explanation:

Answer:

a. 7% b. 1.4 inches

Step-by-step explanation:

(a)What is the probability that the diameter of a randomly selected tree will be at least 10 inches? Will exceed 10 inches.

Since our mean, x = 8.8 and standard deviation, σ = 2.8, and we want our maximum value to be 10 inches,

So, x + ε = 10 where ε = error between mean and maximum value

So,8.8 + ε = 10

ε = 10 - 8.8 = 1.2

Since σ = 2.8, ε/σ = 1.2/2.8 = 0.43

ε/σ × 100 % = 0.43 × 100 = 43%

Since 50% of our values are in the range x - 3σ to x and 43% of our values are in the range x to x + ε = x + 0.43σ, the probability of finding a value less than 10 inches is thus 50 % + 43% = 93%.

So, the probability of finding a value greater than 10 inches is thus 100 % -   93 % = 7 %.

(b) What is the value of c so that the interval (8.8-c, 8.8+c) contains 98% of all diameter values.

Since 98% of the values range from 8.8-c to 8.8+c, then half of the interval is from 8,8 - c to 8.8 or 8.8 to 8.8 + c. So, the number that range in this half interval is 98%/2 = 49%

So, c/σ × 100 % = 49%  

c/σ = 0.49

c = 0.49σ = 0.49 × 2.8 = 1.372 ≅ 1.37 inches = 1.4 inches to 1 d.p

Rounding the cutoff score to the nearest whole number, the cutoff score that will make a student eligible for the reading program is approximately 108.

To find the cutoff score that will make a student eligible for the reading program, we need to determine the score below which the bottom 14% of students fall.

Since the variable is normally distributed and we know the average score and standard deviation, we can use the Z-score formula to find the cutoff score.

The Z-score formula is:

[tex]\[Z = \frac{X - \mu}{\sigma}\][/tex]

Where:

Z is the Z-score,

X is the raw score,

[tex]\mu[/tex] (mu) is the mean, and

[tex]\sigma[/tex] (Sigma) is the standard deviation.

We want to find the Z-score that corresponds to the bottom 14% of students, which means the area to the left of the Z-score is 0.14.

Using a standard normal distribution table or calculator, we can find the Z-score that corresponds to an area of 0.14, which is approximately -1.08.

Now we can rearrange the Z-score formula to solve for X, the cutoff score:

[tex]\[X = Z \cdot \sigma + \mu\][/tex]

Substituting the values we have:

[tex]\[X = -1.08 \cdot 15 + 124.5\][/tex]

Calculating the expression:

[tex]\[X = -16.2 + 124.5\]\\X = 108.3[/tex]

Rounding the cutoff score to the nearest whole number, the cutoff score that will make a student eligible for the reading program is approximately 108.

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Answer:

it would be $1800

Step-by-step explanation:

if each hoop costs $600, and they buy 3, 600 x 3 = 1800

Answer:

1800

Step-by-step explanation:

600 * 3 = 1800

Answer:

yuh, did you bestie

Step-by-step explanation:

≤))≥

_| \_

Answer:

-1 and I hope everyone reading this has a great day!

For A and B to be mutually exclusive, the value of x must be 0. For A and B to be independent, the value of x can be any value between 0 and 0.3, inclusive.

Two events A and B are said to be mutually exclusive if they cannot occur at the same time, meaning that the intersection of A and B is an empty set. In probability terms, if A and B are mutually exclusive, then P(A and B) = 0.

Given that P(A) = 0.4 and P(A or B) = 0.7, we can use the formula for the probability of the union of two events: P(A or B) = P(A) + P(B) - P(A and B). Since we want to find the values of x for which A and B are mutually exclusive, we set P(A and B) = 0:

0.7 = 0.4 + x - 0

0.7 = 0.4 + x

x = 0.3

Therefore, for A and B to be mutually exclusive, the value of x must be 0. For any other value of x, A and B will have a non-empty intersection and therefore will not be mutually exclusive.

On the other hand, two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event. In probability terms, if A and B are independent, then P(A and B) = P(A) * P(B).

Since P(A) = 0.4 and P(B) = x, we can set up the equation:

P(A) * P(B) = 0.4 * x

For A and B to be independent, this equation must hold for any value of x. Therefore, A and B are independent for any value of x between 0 and 0.3, inclusive.

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The dimension of w is 1.

To find a basis for the subspace W = {(0, x, y, z) : x - 6y + 9z = 0} of R4, we can first find a set of vectors that span W, and then apply the Gram-Schmidt process to obtain an orthonormal basis.

Let's find a set of vectors that span W. Since the first component is always zero, we can ignore it and focus on the last three components. We need to find vectors (x, y, z) that satisfy the equation x - 6y + 9z = 0. One way to do this is to set y = s and z = t, and then solve for x in terms of s and t:

x = 6s - 9t

So any vector in W can be written as (6s - 9t, s, t, 0) = s(6,1,0,0) + t(-9,0,1,0). Therefore, {(0,6,1,0), (0,-9,0,1)} is a set of two vectors that span W.

To obtain an orthonormal basis, we can apply the Gram-Schmidt process. Let u1 = (0,6,1,0) and u2 = (0,-9,0,1). We can normalize u1 to obtain:

v1 = u1/||u1|| = (0,6,1,0)/[tex]\sqrt{37}[/tex]

Next, we can project u2 onto v1 and subtract the projection from u2 to obtain a vector orthogonal to v1:

proj_v1(u2) = (u2.v1/||v1||^2) v1 = (-6/[tex]\sqrt{37}[/tex]), -1/[tex]\sqrt{37}[/tex], 0, 0)

w2 = u2 - proj_v1(u2) = (0,-9,0,1) - (-6/[tex]\sqrt{37}[/tex]), -1/[tex]\sqrt{37}[/tex], 0, 0) = (6/[tex]\sqrt{37}[/tex]), -9/[tex]\sqrt{37}[/tex], 0, 1)

Finally, we can normalize w2 to obtain:

v2 = w2/||w2|| = (6/[tex]\sqrt{37}[/tex], -9/[tex]\sqrt{37}[/tex], 0, 1)/[tex]\sqrt{118}[/tex]

Therefore, a basis for W is {(0,6,1,0)/[tex]\sqrt{37}[/tex], (6/[tex]\sqrt{37}[/tex]), -9/[tex]\sqrt{37}[/tex], 0, 1)/[tex]\sqrt{118}[/tex]}.

For the subspace w = {(:a+2c = 0 and b – d = 0} of [tex]M_{2*2}[/tex], we can think of the matrices as column vectors in R4, and apply the same approach as before. Each matrix in w has the form:

| a b |

| c d |

We can write this as a column vector in R4 as (a, c, b, d). The condition a+2c = 0 and b-d = 0 can be written as the linear system:

| 1 0 2 0 | | a | | 0 |

| 0 0 0 1 | | c | = | 0 |

| 0 1 0 0 | | b | | 0 |

| 0 0 0 1 | | d | | 0 |

The augmented matrix of this system is:

| 1 0 2 0 0 |

| 0 1 0 0 0 |

| 0 0 0 1 0 |

The rank of this matrix is 3, which means the dimension of the solution space is 4 - 3 = 1. Therefore, the dimension of w is 1.

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answer: he increased 40 pound

Step-by-step explanation: if it gives negative option please select that

Answer:

The Answer is B

Step-by-step explanation:

The equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k) are the coordinates of the centre and r is the radius

Here diameter = 12, thus radius = 12 ÷ 2 = 6 and (h, k) = (2.5, - 3.5), thus

(x - 2.5)² + (y - (- 3.5))² = 6², that is

(x - 2.5)² + (y + 3.5)² = 36 ← equation of circle

Answer:

caca

Step-by-step explanation:

Answer:

-6

Step-by-step explanation:

To find the slope, you do y₂ - y₁ / x₂ - x₁

y₂ - y₁ / x₂ - x₁

= -35 - 11 / 5 - 1

= -24 / 4

= -6

The slope is -6

Answer:

-6

Step-by-step explanation:

Hi,

To find the slope when given a table, just pick two points, subtract the y values, and then divide them by the x values after you subtract them as well. Here's what I mean...

Let's use 1, -11 and 5, -35

So...

-35 - (-11)

This is the change in y. -35 - (-11) is the same thing as -35 + 11 (subtracting  negative switches to adding it)

You get -24

Now, the change in x.

5 - 1 = 4

So, -24/4 and you get the slope of : -6

I hope this helps :)

Answer:

1776.25

Step-by-step explanation:

35x7=245. 245x7.25=1776.25

Answer:

35 x 24 = 840, 840 hours

840 x 7.25 = $6090 for 35 days

Step-by-step explanation:

answer:

50/50

step-by-step explanation:

hi there!

flip a coin a couple of times, most likely get the same number of heads then you will of tails, since a coin has 2 sides and only two equal sides the probability to flip that very coin and it landing on heads is 50 percent same as landing it on tails, we know its 50 percent because 100 percent is the full amount ( no matter how much the coin is worth ) and dividing that by 2 does indeed equal 50 or 50 percent.

i believe that is one of the reasons why before a lot of sport games start off with a coin toss to chose which team plays first because it is a 50/50 chance for each team, making it fair toss

i hope this helps you! i hope you have a good rest of your day! :)

The solution to the system of equations by substitution is x = 3 and y = -8.

To solve the system of equations by substitution:

Solve one equation for one variable in terms of the other variable. Let's solve the first equation for x:

4x - y = 20

4x = y + 20

x = (y + 20)/4

Substitute this expression for x into the second equation:

-2x - 2y = 10

-2((y + 20)/4) - 2y = 10

(y + 20)/2 - 2y = 10

(y + 20) - 4y = 20

-y - 20 - 4y = 20

-5y = 40

y = -8

Substitute the value of y back into the first equation to find x:

4x - (-8) = 20

4x + 8 = 20

4x = 20 - 8

4x = 12

x = 12/4

x = 3

Therefore, the solution to the system of equations is x = 3 and y = -8.

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