How to prove this using trig identities?
(Tan B + Cot B)^2 = sec^2B + csc^2B

Thanks and please shows steps. I will vote best answer and helpful and do more.

If the standard deviation of a set of data is 6, then the value of variance is 36

The formula for determining variance is variance = √Standard deviation

Variance of a set of data is equal to square of the standard deviation.

If the standard deviation of a set of data is 6 then we get variance by putting the value of standard deviation in the formula

variance = √Standard deviation

Take square root on both sides

Standard deviation² = 6²

Standard deviation= 36

Hence,  standard deviation of a set of data is 6, then the value of variance is 36

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Answer:

$65

Step-by-step explanation:

50 x 0.30

= 15

50 + 15

= 65

Answer:

B: 8/100 = n/75

Step-by-step explanation:

OPTION B is the correct answer.

Part 1

Other answers are possible.

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

Explanation:

The old width was 5, but then it increases to x+5. The old length was 10, but now it's x+10.

The area of any rectangle is length times width.

So the area of the larger rectangle is (x+5)(x+10). Subtract off the old area of 5*10 = 50 and we get (x+5)(x+10) - 50 as the area of the L shape. Set this equal to 126 to finish setting up the equation. This is one possible answer out of many others. This is because we could expand out the (x+5)(x+10) into x^2+10x+5x+50 or simplify that to x^2+15x+50, as two possible options.

==========================================================

Part 2

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

Explanation:

Solve the equation we set up in the previous part

(x+5)(x+10) - 50 = 126

x^2+10x+5x+50-50 = 126

x^2+15x = 126

x^2+15x-126 = 0

Apply the quadratic formula from here

\(x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(15)\pm\sqrt{(15)^2-4(1)(-126)}}{2(1)}\\\\x = \frac{-15\pm\sqrt{729}}{2}\\\\x = \frac{-15\pm27}{2}\\\\x = \frac{-15+27}{2} \ \text{ or } \ x = \frac{-15-27}{2}\\\\x = \frac{12}{2} \ \text{ or } \ x = \frac{-42}{2}\\\\x = 6 \ \text{ or } \ x = -21\\\\\)

Another method you could use is factoring, so you could say:

x^2+15x-126 = 0

(x-6)(x+21) = 0

x-6 = 0 or x+21 = 0

x = 6 or x = -21

The issue with this is that it may take a while to do this trial and error approach.

Whichever method you used, you'll end up with two solutions. One of those solutions doesn't make sense though. We can't have a negative length or distance value, so we ignore x = -21.

The only practical solution is x = 6

If x = 6, then the old height goes from 5 to x+5 = 6+5 = 11

If x = 6, then the old length goes from 10 to x+10 = 6+10 = 16

The new larger rectangle is 11*16 = 176 sq ft, in which we subtract off the 50 sq ft to get 176-50 = 126 sq ft, and this matches with the 126 given to us. Therefore, the answer is confirmed.

Using Poisson distribution ,

a) The probability that two next inspected disk will have no missing pulse is 0.905...

b) The probability that the next inspected disk will have more than one missing pulse is 0.005..

c) The probabihty that neither of the next two inspected disks will contain any missing pulse is 0.819..

Poisson distribution:

It can be defined as a probability distribution resulting from a Poisson experiment. A Poisson experiment is a statistical experiment that divides the experiment into two categories Success or Failure. The Poisson distribution is a restricted process of the binomial distribution.

A Poisson random variable “x” defines the number of successes in the experiment.

Poisson Probability function is represent as given below i.e Mathematically,

P(x, lamda ) = (e ⁻λ (λ ))/x!

where, λ --> an average rate of value

x --> Poission random variable

e ----> base for the logarithm

we have given that,

The quality of computer disk is measured .

averaged missing pluse per disk in a computer disk brand (λ) = 0.1

let x be Poisson random variable which represents the number of missing pulse.

a) we want to find the probability that two next inspected disk will have no missing pulse

P(x= 0 ) = (e⁰·¹ )(0.1)^0/0! = 1/e⁰·¹ ×1/1

=> P(x=0) = 0.905

b) The probability that the next inspected disk will have more than one missing pulse, P(x>1)

P(x>1) = 1 - P(x=0) - P(x=1) = 1 - 0.905 - e⁰·¹(0.1)/1!

= 1 - 0.905 - 0.0905 = 0.0045 ~ 0.005

c) We can assume the disks are independent. the probabihty that neither of the next two inspected disks will contain any missing pulse

P(X = 0 on first disk and x = 0 on second disk) = P(X = 0).P(X = 0) =(0.905)².

= 0.819

Hence, we got all the required probabilities for different cases that is for a) 0.905 b) 0.005 c) 0.81

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The term utilized to make inferences about certain population parameters is a) samples.

In statistical analysis, samples are a subset of a larger population that are selected for observation and analysis. By studying the sample, statisticians can make inferences about the larger population from which it was drawn. This allows for predictions to be made about the entire population, based on information gathered from a smaller subset.

Equations and metrics are important in statistical analysis, but they are not used to make inferences about population parameters. Equations are used to model relationships between variables and to test hypotheses, while metrics are used to quantify the properties of data sets.

Statistics, on the other hand, is the branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of data. Statistics is used to make sense of data and to draw conclusions about the larger population from which the sample was drawn.

In summary, samples are an important tool for statisticians to make inferences about certain population parameters, and statistics is the field that studies these methods of data analysis.

Therefore, the correct answer is a) samples.

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Answer:

4 1/2

Step-by-step explanation:

Jeremy ate 3 slices. Then bill ate half of 3 slices of pizza which is 1 1/2. Add them up and you get 4 1/2.

Answer:

Step-by-step explanation:

4=2(2), 6=2(3) so the lcd is 2(2)3=12

Comparing the Z-scores, we can see that the Z-score for the height (2.61) is slightly larger than the Z-score for the weight (2.45). This indicates that the height value is more extreme than the weight value. The answer is option (a).

To determine which value is more extreme, we can compare the Z-scores for both the height and weight.

The Z-score is calculated using the formula: Z = (X - μ) / σ, where X is the observed value, μ is the mean, and σ is the standard deviation.

For the height value of 76.2 inches, the Z-score is: Z_height = (76.2 - 68.34) / 3.02 ≈ 2.61.

For the weight value of 237.1 lbs, the Z-score is: Z_weight = (237.1 - 172.55) / 26.33 ≈ 2.45.

Therefore, the answer is (a) The height is more extreme than the weight.

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The readability of a double-pan torsion balance is typically very small, usually in the range of a few milligrams (mg). This is because the double-pan torsion balance is a highly sensitive instrument used to measure very small masses.

The readability of a double-pan torsion balance is determined by the sensitivity of the torsion spring which is used to measure the mass on the scale pans. The higher the sensitivity of the torsion spring, the higher the readability of the double-pan torsion balance. The readability of a double-pan torsion balance is typically in the range of 0.1 to 0.5 mg.

This allows the balance to measure very small increments with a high degree of accuracy and precision. The readability of a double-pan torsion balance is critical for accurate measurements of very small masses, such as in the pharmaceutical, chemical and food industries.

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Clark spends $ 12775 on these items which he does not need in a year (if we consider 365 days) where the average spend in a month is $35.

Clark finds that in an average month, he spends $35 on things he really doesn't need and can't afford.

Let us consider the month in consideration here to be of 30- days and ignore any months other number of days.

Thus, calculating the average, say x' , by formula, we get,

x' = (Summation of values of all observations ) / ( Number of observations)

⇒ 35 = Total spend / 30

⇒ Total spend = $ ( 35*30)

⇒ Total spend = $ 1050

Therefore, total spend on a year, that is 12 months (considering all months to be of 30- days ) = $( 1050*12) = $ 12600

But we know a year does not have 360 days. So we calculate the total spend on these 5 days where average month spend is $35 is $175.

Hence the total spend for a year with 365 days is = $( 12600 + 175 ) = $12775

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Answer:

63

Step-by-step explanation:

252/4 is 63

Answer: The train traveled 63 miles per hour

Step-by-step explanation:

252/4=63

Answer:

10

Step-by-step explanation:

Answer:

10 shoes

Step-by-step explanation:

Step 1: State what is known

Giles has 10 socks for every 5 pairs of shoes

He has 20 socks

Step 2: Set up the ratio

Let 'x' represent how much shoes Giles has

10 socks : 5 shoes

20 socks : x shoes

Step 3: Convert ratio into fraction

\(\frac{Socks}{Shoes}\)

10 socks : 5 shoes = \(\frac{10}{5}\)

20 socks : x shoes = \(\frac{20}{x}\)

Step 4: Set ratio equal to each other and solve

\(\frac{10}{5}=\frac{20}{x}\\ 10x = 100\\\frac{10x}{10}=\frac{100}{10}\\x = 10\)

Therefore Giles has 10 shoes when he has 20 socks

The cost of one guitar string is obtained as follows:

$1.3.

The costs are obtained using a system of equations, for which the variables are given as follows:

He purchases 25 of each, for a total of $40, hence:

25x + 25y = 40.

x + y = 1.6.

A string costs $1 more than a guitar pick, hence:

y = x + 1.

x = y - 1.

Hence the cost of a string is obtained as follows:

y - 1 + y = 1.6

2y = 2.6

y = 2.6/2

y = $1.3.

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Answer:

see image

Step-by-step explanation:

We are given the function f(x) = 8x^2 + 3x + 2 and we are asked to find its derivative at x = 1 using the definition of the derivative.

The derivative of a function at a specific point can be found using the definition of the derivative. The definition states that the derivative of a function f(x) at a point x = a is given by the limit as h approaches 0 of (f(a + h) - f(a))/h, provided the limit exists.

In this case, we want to find the derivative of f(x) = 8x^2 + 3x + 2 at x = 1. Using the definition of the derivative, we substitute a = 1 into the limit expression and simplify:

f'(1) = lim(h->0) [f(1 + h) - f(1)]/h

= lim(h->0) [(8(1 + h)^2 + 3(1 + h) + 2) - (8(1)^2 + 3(1) + 2)]/h

= lim(h->0) [(8(1 + 2h + h^2) + 3 + 3h + 2) - (8 + 3 + 2)]/h

= lim(h->0) [(8 + 16h + 8h^2 + 3 + 3h + 2) - 13]/h

= lim(h->0) (8h^2 + 19h)/h

= lim(h->0) 8h + 19

= 19.

Therefore, the derivative of f(x) = 8x^2 + 3x + 2 at x = 1 is f'(1) = 19.

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Answer:

5/4

Step-by-step explanation:

The area of a rectangle is given by

A = l*w

1/2 = 2/5 * w

Multiply each side by 5/2

5/2 * 1/2 = 2/5 *w * 5/2

5/4 = w

Formula we use,

→ A = l × w

Then the width of the rectangle,

→ A = I × w

→ 1/2 = 2/5 × w

Now multiply each side by 5/2,

→ 1/2 × 5/2 = 2/5 × w × 5/2

→ w = 5/4

Hence, the width is 5/4.

The work done on a particle moving uniformly around the unit circle centered at the origin under a constant force field, f(x, y), is zero.

When a particle moves in a closed path, like a circle, the net work done by a conservative force field is always zero. In this case, the force field is constant, which means it does not change as the particle moves along the path. Since the work done by a constant force is given by the formula W = F * d * cos(θ), where F is the force, d is the displacement, and θ is the angle between the force and the displacement vectors, we can see that the cosine of the angle will always be zero when the particle moves along the unit circle centered at the origin. This implies that the work done is zero. Thus, the work done on the particle is zero.

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In this scenario, the outcome of each flip of the coin can be modeled as a Bernoulli random variable Y, where Y takes the value 1 (heads) with a probability of 0.7 and the value 0 (tails) with a probability of 0.3.

A Bernoulli random variable is a discrete random variable that can take only two possible outcomes, typically labeled as success (1) and failure (0), with a fixed probability associated with success. In this case, the outcome of each flip of the coin can be represented by the Bernoulli random variable Y, where Y = 1 represents the event of getting heads and Y = 0 represents the event of getting tails.

Given that the probability of landing on heads is 0.7, we can assign the following probabilities to the outcomes:

P(Y = 1) = 0.7 (probability of getting heads)

P(Y = 0) = 0.3 (probability of getting tails)

Thus, for each individual flip of the coin, we can use the Bernoulli random variable Y to model the outcome, where Y takes the value 1 (heads) with a probability of 0.7 and the value 0 (tails) with a probability of 0.3.

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Answer:

x = 6.5

Step-by-step explanation:

The image below shows that the equation y = 2x - 2 (which has a slope of 2) passes the the points (3, 4) and (6.5, 11); when x = 6.5

Hope this helps! :)

Answer:

\(y = 2x - 2 \\ y = mx + c\)

When the point is (3,4) and slope=2

\(4 = 2 \times 3 + c \\ 4 = 6 + c \\ \boxed{c = - 2}\)

Now, when the point is (x,11) and slope=2

\(11 = 2x - 2 \\ 2x = 13 \\ x =\frac{13}{2} \\ \boxed{x = 6.5 }\)

The required length of the rectangle is 89 m.

Given that,
The area of a rectangular field is 6942 m. If the width of the field is 78 m, what is its length is to be determined.

Perimeter is the measure of the figure on its circumference.

The rectangle is 4 sided geometric shape whose opposites are equal in lengths and all angles are about 90°.

The area of the rectangle = length * width
                                6942  = length * 78
                                length = 89 m

Thus, the required length of the rectangle is 89 m.

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Answer: y = 14.50x + 2.99

Step-by-step explanation:

The equation is y = mx + b

y is the total cost and x is the number of shirt

We know each shirt is $14.50, and she bought x shirts, so 14.50 is the slope

The 2.99 shipping fee is the y-intercept

So our equation is

y = 14.50x + 2.99

Answer:

l = 21

w = 9

P = 60

Step-by-step explanation:

l = 12 + w

A = l*w = 45

substitute l = 12 +w

45 = (12 + w) * w

45 = 12w + w^2

w^2 + 12w - 45 = 0

w^2 + 2*6*w +36 = 45 + 36 (added 36 on both sides 6^2)

(w + 6) ^2 = 45+36 = 81

w + 6 = +-9

As width cannot be negative w = 9

so l = 12 + 9

= 21

Perimeter = P = 2 (l + b)

= 2 ( 21 + 9)

= 2 (30)

= 60

Regular hexagon ABCDEF is inscribed in a circle with center H, the image of segment BC after 120 degree clockwise rotation about point H is the segment joining the points B' and C', which has endpoints (-0.5r\(\sqrt{3\), -0.5r) and (-0.5r, -0.5r).

Since the hexagon is inscribed in a circle with center H, we can conclude that H is also the center of the circle passing through vertices B, C, and D. Therefore, the circle passing through B, C, and D is also a 120 degree clockwise rotation of the circle passing through A, B, and C.

To find the image of segment BC after a 120 degree clockwise rotation about point H, we need to find the coordinates of B and C relative to H, and then apply a 120 degree rotation matrix to these coordinates.

Let the radius of the circle be r, and let the coordinates of H be (0,0). Then the coordinates of B and C are:

B: (r cos(60), r sin(60))

C: (r cos(0), r sin(0)) = (r, 0)

To apply a 120 degree clockwise rotation matrix, we can use the following matrix:

[ cos(-120) -sin(-120) ]

[ sin(-120) cos(-120) ]

Simplifying, we get:

[ cos(120) sin(120) ]

[ -sin(120) cos(120) ]

Applying this matrix to the coordinates of B and C, we get:

B': [ cos(120) sin(120) ][ r cos(60) ] = [ -0.5r \(\sqrt{3}\)]

[ -sin(120) cos(120) ][ r sin(60) ] [ -0.5r ]

C': [ cos(120) sin(120) ][ r ] = [ -0.5r ]

[ -sin(120) cos(120) ][ 0 ] [ -0.5r ]

Therefore, the image of segment BC after a 120 degree clockwise rotation about point H is the segment joining points B' and C', which has endpoints (-0.5r\(\sqrt{3}\), -0.5r) and (-0.5r, -0.5r), respectively.

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Given, the function is f(x,y,z)=5+yxz+g(x,z)Here, we need to find the directional derivative of f at the point (3,0,3) along the direction of the vector (0,4,0) . The directional derivative of f at the point (3,0,3) along the direction of the vector (0,4,0) is 0.

Using the formula of the directional derivative, the directional derivative of f at the point (3,0,3) along the direction of the vector (0,4,0) is given by

(f(x,y,z)) = grad(f(x,y,z)).v

where grad(f(x,y,z)) is the gradient of the function f(x,y,z) and v is the direction vector.

∴ grad(f(x,y,z)) = (fx, fy, fz)

                       = (∂f/∂x, ∂f/∂y, ∂f/∂z)

Hence, fx = ∂f/∂x = 0 + yzg′(x,z)fy

                = ∂f/∂y

                = xz and

fz = ∂f/∂z = yx + g′(x,z)

We need to evaluate the gradient at the point (3,0,3), then

we have:fx(3,0,3) = yzg′(3,3)fy(3,0,3)

                             = 3(0) = 0fz(3,0,3)

                             = 0 + g′(3,3)

                             = g′(3,3)

Therefore, grad(f(x,y,z))(3,0,3) = (0, 0, g′(3,3))Dv(f(x,y,z))(3,0,3)

                                                 = grad(f(x,y,z))(3,0,3)⋅v

where, v = (0,4,0)Thus, Dv(f(x,y,z))(3,0,3) = (0, 0, g′(3,3))⋅(0,4,0)   = 0

The directional derivative of f at the point (3,0,3) along the direction of the vector (0,4,0) is 0.

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Answer:v = 9

Step-by-step explanation:

Simplifying

2(3v + 8) = 70

Reorder the terms:

2(8 + 3v) = 70

(8 * 2 + 3v * 2) = 70

(16 + 6v) = 70

Solving

16 + 6v = 70

Solving for variable 'v'.

Move all terms containing v to the left, all other terms to the right.

Add '-16' to each side of the equation.

16 + -16 + 6v = 70 + -16

Combine like terms: 16 + -16 = 0

0 + 6v = 70 + -16

6v = 70 + -16

Combine like terms: 70 + -16 = 54

6v = 54

Divide each side by '6'.

v = 9

Answer: the answer is 9

Step-by-step explanation:

1. open the bracket by multiplying: 2(3v+8)=70

6v+16=70

2. take the 16 over the equals sign.  note:(the plus sign in front of the 16 changes to a minus sign.)

6v=70-16

6v=54

3. divide both sides by 6

6v/6=54/6

final answer: v=9

Step-by-step explanation:

(a) To find the position of the car when it has zero velocity, we need to find the times at which the velocity, x'(t), is equal to zero. The velocity of the car is given by:

x'(t) = (4.85m/s^2)t^2 - (0.100m/s^6)t^6

Setting x'(t) equal to zero and solving for t, we find:

0 = (4.85m/s^2)t^2 - (0.100m/s^6)t^6

t^2 = (0.100m/s^6)t^6 / (4.85m/s^2)

We can't solve for t analytically, but we can use numerical methods to find approximate values for the two times at which the velocity is equal to zero.

The first instant is approximately t = 0.68 s and the second instant is approximately t = 2.28 s.

Using these values for t, we can find the positions of the car at these instants:

x(0.68s) = 2.31m + (4.85m/s^2)(0.68s)^2 - (0.100m/s^6)(0.68s)^6

x(2.28s) = 2.31m + (4.85m/s^2)(2.28s)^2 - (0.100m/s^6)(2.28s)^6

(b) The acceleration of the car is given by the derivative of its velocity, x'(t):

x''(t) = 2(4.85m/s^2)t - 6(0.100m/s^6)t^5

Using the values of t from part (a), we can find the acceleration of the car at the instants when it has zero velocity:

x''(0.68s) = 2(4.85m/s^2)(0.68s) - 6(0.100m/s^6)(0.68s)^5

x''(2.28s) = 2(4.85m/s^2)(2.28s) - 6(0.100m/s^6)(2.28s)^5