I know that x^2 + y^2 in cartesian coordinates is r^2 in polar coordinates.

However, in my problem, I’m given x^2 + z^2. On a graph, it still forms a circle, except on a different plane than just x^2 + y^2.

Is it okay if I simplify x^2 + z^2 in my work to just r^2, or is that wrong?

Answer:

Step-by-step explanation:

here is an solution

Answer: 2. 50,688 litres.

Step-by-step explanation:

The volume of a cylinder can be calculated using the formula:

V = πr²h

where V is the volume, r is the radius, and h is the height.

Substituting the given values, we get:

V = π(2.4²)(2.8)

V = 50.688 m³

To convert this to litres, we can multiply by 1000:

V = 50.688 × 1000

V = 50,688 litres

Therefore, the answer is option 2: 50,688 litres.

The equation of the curve is y = x³ - 2x² + 5x - 3

Since the gradient of every point on the curve C with equation y = f(x), satisfies dy/dx = 3x2 – 4x + k, where k is a constant. The points P (0, -3) and Q (2, 7) both lie on C. To find the equation of the curve c, we first integrate the equation for the gradient.

So, we proceed as follows.

Since dy/dx = 3x² - 4x + k

dy = (3x² - 4x + k)dx

Integrating both sides, we have that

∫dy = ∫(3x² - 4x + k)dx

∫dy = ∫3x²dx - ∫4xdx + ∫kdx

y = 3x³/3 - 4x²/2 + kx + C

y = x³ - 2x² + kx + C

Using the points P(0, -3) and Q(2, 7) we have that

y = x³ - 2x² + kx + C

-3 = 0³ - 2(0)² + k(0) + C

-3 = 0 + 0 + 0 + C

- 3 = 0 + C

C = - 3

So, y = x³ - 2x² + kx + C

y = x³ - 2x² + kx - 3

So, using the points Q(2, 7), we have that

y = x³ - 2x² + kx - 3

7 = 2³ - 2(2)² + k(2) - 3

7 = 8 - 8 + 2k - 3

7 = 0 + 2k - 3

7 = 2k - 3

2k = 7 + 3

2k = 10

k = 10/2

k = 5

So, substituting this into the equation for the curve, we have that

y = x³ - 2x² + 5x - 3

So, the equation of the curve is y = x³ - 2x² + 5x - 3

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Therefore, the 95% confidence interval for the difference in scores is (9.936, 11.864).

In statistics, a confidence interval is a range of values that is likely to contain the true value of a population parameter with a certain degree of confidence, based on a sample of data.

Here,

To find the confidence interval for the difference in scores, we can use the formula:

CI = (x1 - x2) ± t(α/2, df) * SE

where x1 and x2 are the sample means for groups 1 and 2, t(α/2, df) is the critical t-value based on the level of significance α and the degrees of freedom df, and SE is the standard error of the difference between the means, given by:

SE = √((s₁²/n₁) + (s₂²/n₂))

where s₁ and s₂ are the sample standard deviations for groups 1 and 2, and n₁ and n₂ are the sample sizes.

Given:

x1 = 26.4, s₁ = 4.4, n₁ = 200

x2 = 15.5, s₂ = 2.4, n₂ = 200

α = 0.05

df = n₁ + n₂ - 2 = 398 (assuming equal variances)

First, we need to calculate the standard error:

SE = √((4.4²/200) + (2.4²/200))

= 0.492

Next, we need to find the critical t-value:

t(α/2, df) = t(0.025, 398)

= 1.964

Finally, we can calculate the confidence interval:

CI = (26.4 - 15.5) ± 1.964 * 0.492

= 10.9 ± 0.964

= (9.936, 11.864)

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Answer:

Step-by-step explanation:

You want the reciprocal function of f(x) = 5x² +13x -6 and the equations of its vertical asymptotes. You also want the graph of f(x) = ((x +4)(x -5))/(x² +2x-8) along with its domain, range, and intervals of positive, negative, increase, and decrease.

The reciprocal function will be ...

 1/f(x) = 1/(5x² +13x -6)

Its vertical asymptotes will be found where f(x) = 0. Those values of x are nicely found by factoring the function.

  f(x) = 5x² +13x -6 = (5x +15)(5x -2)/5 = (x +3)(5x -2)

The product of factors will be zero when one of the factors is zero.

  x +3 = 0   ⇒   x = -3

  5x -2 = 0   ⇒   x = 2/5

The equations of the vertical asymptotes are x = -3 and x = 2/5.

The function can be simplified by factoring the denominator anc cancelling common factors:

  [tex]f(x)=\dfrac{(x+4)(x-5)}{x^2+2x-8}=\dfrac{(x+4)(x-5)}{(x+4)(x-2)}=\dfrac{x-5}{x-2}\quad x\ne -4[/tex]

The function changes sign where the factors are zero, at x = 2 and x = 5. It is approaches the horizontal asymptote y = 1 for large positive or negative values of x, and is increasing everywhere it is defined. There is a vertical asymptote at x = 2, and a "hole" at x = -4 where the numerator and denominator factors canceled, but the function remains undefined. The function can take on any value except those at the horizontal asymptote and the "hole".

Domain, Increasing

The function is increasing everywhere it is defined, so the "increasing" intervals and the domain are the same:

  (-∞, -4) ∪ (-4, 2) ∪ (2, ∞) . . . .  . domain, intervals of increase

The function has no intervals of decrease.

Negative, Positive, Range

The function is positive everywhere it is defined, except for the interval between sign changes.

  negative: (2, 5)

  positive: (-∞, -4) ∪ (-4, 2) ∪ (5, ∞)

  range: (-∞, 1) ∪ (1, 1.5) ∪ (1.5, ∞)

Answer: 1 just like my answer dawg

Step-by-step explanation:

To make r the subject from the equation v = br + y, first subtract y from both sides to get v - y = br. Then divide both sides by b, to get r = (v - y) / b.

The question requires us to rearrange the given formula

v = br + y

to make

r

the subject. To do this, we have to isolate

r

on one side of the equation. The first step is to subtract

y

from both sides of the equation, getting us

v - y = br

. The next step is to divide both sides of the equation by

b

to completely isolate

r

. Therefore, the formula with

r

as the subject is

r = (v - y) / b

.

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Answer:

Range: [−4,∞)

Step-by-step explanation:

Hope it helps:)

Answer:

Range: [-4, ∞), {y|y ≥ -4}

Step-by-step explanation:

It helps if you graph the function to better understand what the range is.

In interval and set-builder notation, the range is:

[-4, ∞), {y|y ≥ -4}

This is knowing that the Domain is:

(-∞, ∞), {x|x ∈ R}

Answer:

The answer to your problem is, radian measure of angle AOB is π /3 radian

Step-by-step explanation:

How to find the length of an arc:

Arc Length (L) = θ × r

θ being the radian measure of the angle and r is the radius of the circle

From what is give;

L = 2π/9 yards, r = 2 feet = 2/3 yard

Using the formula, L = θ x r

Then concluding to:

2π/9 = θ × 2/3

θ  = (2π × 3) / (9×2) 

θ  = π /3 radian

Thus the answer to your problem is, radian measure of angle AOB is π /3 radian

Answer:

Diameter-3 inches

Circumference- 9.42 inches

Step-by-step explanation:

In order to get this, you must know that radius is 1/2 of diameter or diameter is 2 times the radius.

As such,

D-Diameter

R-Radius

R = 1.5

D = 2R--> FOR ANY CIRCLE

D = 2(1.5)

D = 3 inches

Now to find circumference you must understand the difference of formulas between the area of a circle and circumference of a circle.

Merely saying, the circumference of a circle is the perimeter of a circle and it is much different from finding the perimeter of other shapes.

The circumference formula is...

C - 2лr

As such,

C - 2(3.14)(1.5)

C - 9.42 inches

As such your answers are

Diameter - 3 inches

Circumference - 9.42 inches

Hope this helped!

The probability of introducing a new product is 0.6.

The probability that the new product will be introduced is calculated as follows:

P(new product) = P(new product | first set wins) * P(first set wins) + P(new product | second set wins) * P(second set wins)

The given probabilities from the statement are as follows:

P(first set wins) = 0.6

P(new product | first set wins) = 0.8

P(second set wins) = 0.4

P(new product | second set wins) = 0.3.

Therefore, the required probability will be:

P(new product) = 0.8 * 0.6 + 0.3 * 0.4

P(new product) = 0.48 + 0.12

P(new product) = 0.6

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Answer: 21//2

Step-by-step explanation: 1 1/2 = 3/2

So it takes 3/2 hours to set up 1 comupter workstation.

If we want to find how many hours would it take to set up 7 computer workstation we would have to do 3/2 x 7 and get 21/2 or 20 1/2

The length of ML rounded to nearest tenth is 4.5.

From figure, KML is a right-angled triangle at M, with KM = 8 and KL = 9.2. Let ML = x be the length of the base.

Using Pythagorean theorem, we have:

KL² = KM² + ML²

Substituting the given values ML=x, KM = 8 and KL = 9.2, we have:

9.2² = 8² + x²

Simplifying the right-hand side, we get:

84.64 = 64 + x²

Subtracting 64 from both sides, we get:

20.64 = x²

Taking the square root of both sides, we get:

x ≈ 4.54

x=4.5 (rounded to nearest tenth)

Therefore, the length of ML is approximately 4.5.

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Answer:

The mean length of time is 27 minutes.

Step-by-step explanation:

To find the mean, we have to first calculate x and fx.

(0+12)/2=6

(12+24)/2=18

(36+48)/2=42

(48+60)/2=54

Now we have x, and f

Calculate fx by multiplying:

6×11=66

18×25=450

30×23=690

42×15=630

54×6=324

The mean is the sum of fx/sum of f

(66+450+690+630+324)/(11+25+23+15+6)=27

Hope this helps!

Answer:

x = 20

Step-by-step explanation:

You have pointed out the angles are supplementary, so they add to 180.

Add both angles to equal 180.

7x + x + 20 = 180

Combine like terms.

8x + 20 = 180

Subtract 20 from both sides.

8x = 160

Divide out the 8 to get x.

x = 20

The value of x in chords is 25.857 and this is a isosceles triangle.

In geometry, a chord is a line segment that connects two points on the circumference of a circle. A chord is the longest distance between two points on a circle, and it passes through the center of the circle. The endpoints of a chord are referred to as its "endpoints." A chord that passes through the center of a circle is called the "diameter" of the circle. The length of a chord can be calculated using the Pythagorean theorem, given the radius and the distance between the center and the chord. Chords are used in various geometric calculations involving circles, such as finding the area of a sector or segment of a circle.

The sum of the angles of a triangle is 180 degrees. In a circle, the angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc at any point on the circumference. Using these facts, we can set up the following equation to find x:

(1/2)(14x - 7) + (1/2)(12x + 4) + (1/2)(4x) = 180

Simplifying and solving for x, we get:

7x - 1 = 180

7x = 181

x = 25.857

Therefore, x is approximately equal to 25.857.

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The number of teenagers who gave data for the survey was 52 teenagers .

The number of teenagers who ate red meat at least once a week was 50 teenagers.

The percentage that ate red meat more than twice per week was 77 %.

The distribution can be described as normal.

The number of teenagers in the survey :

= ∑ ( Frequency of number of times meat eaten )

= 2 + 3 + 7 + 9 + 10 + 8 + 7 + 5 + 1

= 52 teenagers

Number of teenagers eating at least once a week :

= ( 52 total - 2 eating zero times )

= 50 teenagers

Percentage eating red meat more than twice :

= ( 52 total - 2 eating zero times - 3 eating once - 7 eating twice ) / 52 x 100 %

= 77 %

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The conditional statement is solved and the following argument is invalid

Given data ,

Let the first statement be A

Now , (∀x)(∃y)(p(x, y) ⇒ r(x)):

This statement asserts that for every x, there exists a y such that if x is a factor of y, then x is prime. This is a valid statement, as it essentially says that there exists a prime factor for every integer x

Let the second statement be B

Now , (∀x)(∀y)(∀z)(¬q(x, y, z)): This statement asserts that for all x, y, and z, z is not the greatest common factor (GCF) of x and y. This is a generic statement and does not provide any specific information about the values of x, y, and z. It is not relevant to the conclusion of the argument.

Let the third statement be C

Now , (∃x)(∀y)(p(x, y) ∨ r(x)): This statement asserts that there exists an x such that for all y, either x is a factor of y or x is prime. This is also a valid statement, as it states that there exists an integer x that is either a factor of y or prime for all y.

According to the conclusion, there must exist an x and a z such that z is the GCF of x and y for any y. However, the premises merely state that x is a factor of y or that x is prime; they make no direct mention of the largest common factor between x and y.

Hence , the conclusion (y)(z)(x)q(x, y, z) is not supported by the argument. A stronger case for this conclusion would require more evidence or premises

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The largest area the farmer can enclose with 100 feet of fencing is a rectangle with sides that measure 25 feet by 40 feet. The area of this rectangle is 1000 square feet. The farmer can use this information to fence off the largest possible area for his farm with the 100 feet of fencing he has available.

Area is the measurement of a two-dimensional figure or space. It is represented by a numerical value and is typically measured in units of square meters, square centimeters, or square feet. Area is often used to measure the size of a piece of land, the area of a room, or the total surface of a 3D object. In mathematics, area is a fundamental concept and is used to calculate various other figures, such as the perimeter and volume.

The largest area the farmer can enclose with 100 feet of fencing is a rectangle with sides that measure 25 feet by 40 feet. This can be determined through the proper and complete calculation of the perimeter and area of the rectangle. To start, the perimeter of a rectangle is equal to the sum of its four sides, so the farmer needs to use all of the fencing to create a rectangle with a perimeter of 100 feet. This means that the two sides of the rectangle, x and y, have to add up to 100 feet when added together. This equation can be expressed as x + y = 100.

Next, the area of a rectangle is equal to the product of its two sides, so the farmer needs to find the two sides that will give the largest product when multiplied together. To do this, the farmer can use the equation from the perimeter, x + y = 100, to solve for x and y. This gives the equation x = 100 – y. Plugging this equation into the area equation gives the equation A = (100 – y)y. To find the maximum area, the farmer needs to take the derivative of this equation and set it to zero. This gives the equation y = 50, meaning that the two sides of the rectangle are both 50 feet.

Therefore, the largest area the farmer can enclose with 100 feet of fencing is a rectangle with sides that measure 25 feet by 40 feet. The area of this rectangle is 1000 square feet. The farmer can use this information to fence off the largest possible area for his farm with the 100 feet of fencing he has available.

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Answer:

33

Step-by-step explanation:

4×4=16

7×3=21

21+16= 37

2×2=4

37-4=33

Answer:

  (c)  3(cos(135°) +i·sin(135°))

Step-by-step explanation:

You want to identify a solution to x^4 = -81.

The principal fourth root of 81∠180° will be ...

  [tex]\sqrt[4]{-81}=\sqrt[4]{81\angle 180^\circ}=\sqrt[4]{81}\angle \dfrac{180^\circ}{4}=3\angle 45^\circ[/tex]

Additional complex roots will be at angles that are multiples of 360°/4 from this one, so ...

  3∠135°, 3∠225°, 3∠315°

The root that is on the list of choices is ...

  3∠135° = 3(cos(135°) +i·sin(135°))

__

Additional comment

The attachment shows the 4th powers of the numbers with the various angles. The fourth power of 3∠135° gives -81.

Let X be the number of employees the company had last year.

According to the problem, the number of employees this year is 225% of the number last year, which can be written as:

135 = 2.25X

To solve for X, we can divide both sides by 2.25:

X = 135 / 2.25

X = 60

Therefore, the company had 60 employees last year.

Answer:

4 units²

Step-by-step explanation:

Calculate some vectors from the point (2,3).

Vector A = <1, 5>

Vector B = <-2, -2>

The area of the triangle is 1/2 * (the magnitude of the cross product of the vectors)

A × B = -2 + 10 = 8

the magnitude of 8 is 8.

Therefore 1/2 * 8 = 4

Answer:

4 units²

Step-by-step explanation:

The null and alternative hypothesis for a physician who check your blood pressure are as follows:

When a physician checks a patient's blood pressure, they may use a diagnostic threshold of 90 mmHg to diagnose hypertension. In statistical terms, the null hypothesis (H0) would be that the patient's blood pressure is less than or equal to 90 mmHg. The alternative hypothesis (H1) would be that the patient's blood pressure is greater than 90 mmHg.

The physician would then use various statistical tests, such as a t-test or a z-test, to determine whether the patient's blood pressure is statistically significantly different from the threshold of 90 mmHg. Based on the results of the test, the physician would either reject the null hypothesis in favor of the alternative hypothesis, indicating that the patient has hypertension, or fail to reject the null hypothesis, indicating that the patient's blood pressure is within the normal range.

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The simplified expression is 3x³ -5x² -13x +10.

We have,

(x-2)(3x² + 4x-5)

Now, simplifying the expression as

(x-2)(3x² + 4x-5)

= x(3x²)+  x (4x) -5x -3(3x²) -2 (4x) +10

= 3x³ + 4x² -5x -9x² -8x +10

= 3x³ -5x² -13x +10

Thus, the simplified expression is 3x³ -5x² -13x +10.

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Using the area formula for rectangle, we can find the height of the rectangle to be 8m.

The quantity of unit squares that can fit inside a rectangle is known as an area. The quantity of space occupied by a flat surface of a specific form is determined by its area. Each shape's area is specified as a "number of" square units and is based on the maximum number of unit squares that can fit inside it (square centimetres, square inches, square feet, etc.).

Area of the rectangle has been given = 52m².

Base of the rectangle has been given as = 6.5m.

Let height be = h

Area = b × h

⇒ 52 = 6.5 × h

⇒ h = 52/6.5

⇒ h = 8m.

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Answer:

Step-by-step explanation:

Given a circle of radius 7 m with central angle AOB = 150°, you want the length of long arc AB and the area of the sector it bounds.

The length of an arc of a circle is ...

  s = rθ

where θ is the central angle in radians.

Here, the long arc in degrees is 360° -150° = 210°. In radians, that is ...

  210° = 210°×π/180° radians = 7π/6 radians

So, the arc length is ...

  s = (7 m)(7π/6) = 49π/6 m . . . . arc length

The area of the sector is half the product of the arc length and the radius; (You can think of it as a kind of triangle.)

  A = 1/2s·r

  A = 1/2(49π/6 m)(7 m) = 343π/12 m² . . . . area

The equation that would be used to find the angle needed to travel 150 meters is 150 = 197.55 sin (2θ).

option D.

The equation that would be used to find the angle needed to travel 150 meters is calculated as follows;

h = (v₀²/4.9) sinθcosθ

where;

Substitute the value of v₀ in the given equation;

h = (44²/4.9) sinθcosθ

h = 395.1 sinθcosθ

In trig identity, 2sinθcosθ = sin (2θ)

h = ¹/₂ [395.1 (2sinθcosθ)]

h = ¹/₂ [395.1 sin (2θ)]

h = 197.55 sin (2θ)

150 = 197.55 sin (2θ)

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The solid figure has a 245.04 total area. The area is the measurement of any two-dimensional region confined by a set of lines, and square footage is a measurement of area.

Therefore,

The equation to a cylinder for the area is A=2πrh+2πr^2

Fill in and solve.

A=2(3.14)(3)(10)+2(3.14)(3)^2

So using PEMDAS you get the answer 245.04

The complete question is:

Find the total area of the solid figure.

60 sq. in.

78 sq. in.

90 sq. in.

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Answer:

the maximum possible area of a Norman window with a perimeter of 76 meters is approximately 43.24 square meters.

Step-by-step explanation:

Let's first write the equation for the perimeter P of a Norman window in terms of its height h and width w:

P = 2h + w + πh

Simplifying this equation, we get:

P = (2 + π)h + w

We are given that the perimeter is 76 meters, so we can set up the equation:

76 = (2 + π)h + w

Solving this equation for w, we get:

w = 76 - (2 + π)h

The area A of a Norman window is the sum of the area of the rectangle and half the area of the circle:

A = hw + (1/2)πh^2

Substituting the expression for w from above, we get:

A = h(76 - (2 + π)h) + (1/2)πh^2

Expanding and simplifying, we get:

A = (38π - 2)h^2 + 76h

This is a quadratic equation in h, with a negative coefficient for the h^2 term. Therefore, the maximum area will occur at the vertex of the parabola, which is located at:

h = -b / 2a = -76 / 2(38π - 2) = 1 / (19π - 1)

Substituting this value of h back into the equation for A, we get:

A = (38π - 2)(1 / (19π - 1))^2 + 76 / (19π - 1)

Simplifying this expression, we get:

A = (76π - 4) / (19π - 1) ≈ 43.24 square meters (rounded to the nearest hundredth)

Therefore, the maximum possible area of a Norman window with a perimeter of 76 meters is approximately 43.24 square meters.