8x÷4=100 What is the value of x in the equation?

the rectangular equation that is equivalent to the given parametric equations is: [tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]

The given parametric equations describe a curve in the xy-plane traced by a particle that moves along a circular path centered at (2,-1) with radius 3. To find the rectangular equation, we can use the following trigonometric identity:

[tex]cos^2[/tex]θ + [tex]sin^2[/tex]θ = 1

Multiplying both sides by [tex]3^2[/tex], we get:

9[tex]cos^2[/tex]θ + [tex]9sin^2[/tex]θ = 9

Rearranging and using the fact that cosθ = (x-2)/3 and sinθ = (y+1)/2, we get:

[tex]9((x-2)/3)^2 + 9((y+1)/2)^2 = 9[/tex]

Simplifying, we get:

[tex](x-2)^2/3^2 + (y+1)^2/2^2 = 1[/tex]

Multiplying both sides by 36, we get:

[tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]

Therefore, the rectangular equation that is equivalent to the given parametric equations is:

[tex]4(x-2)^2 + 9(y+1)^2 = 36[/tex]

This equation represents an ellipse centered at (2,-1) with semi-axes of length 3 and 2 along the x-axis and y-axis, respectively. The parameter θ varies from 0 to 2π, which means the particle completes one full revolution around the ellipse.

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Answer:

Step-by-step explanation:

The total surface area of the given hemispherical scoop is: 30.41 cm²

The formula for the total surface area of a hemisphere is:

TSA = 3πr² square units

Where:

π is a constant whose value is equal to 3.14 approximately.

r is the radius of the hemisphere.

Since the steel is 0.2cm thick and the outside of the scoop has a radius of 2cm, then we can say that:

TSA = 3π(2 + 0.2)²

= 3π(2.2)²

= 30.41 cm²

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So, the dimensions that minimize the production costs are approximately:

Radius (r): 3.824 cm

Height (h): 7.610 cm

To minimize the production cost of the cup-of-soup package, we need to find the dimensions of the cylindrical container that minimize the surface area, as the cost is directly proportional to the surface area of the materials used.

Let r be the radius of the base and h be the height of the cylinder. The volume V of the cylinder is given by:

V = πr^2h

We know that the volume is 350 cubic centimeters:

350 = πr^2h

Now, let's express the height h in terms of the radius r:

h = 350 / (πr^2)

The surface area A of the cylinder consists of the area of the sides, the bottom, and the top:

A = (side area) + (bottom area) + (top area)

The side area of the cylinder is given by 2πrh, the bottom area is given by πr^2, and the top area is given by πr^2. So, the surface area A can be expressed as:

A = 2πrh + πr^2 + πr^2

Now, let's substitute h from the previous equation to express A in terms of r only:

A = 2πr(350 / (πr^2)) + πr^2 + πr^2

A = 700/r + 2πr^2

Next, we'll find the critical points of the function A(r) to find the minimum production cost. To do this, we'll differentiate A(r) with respect to r and set the derivative equal to 0.

dA/dr = -700/r^2 + 4πr

Now, set dA/dr = 0 and solve for r:

0 = -700/r^2 + 4πr

Rearrange the equation to isolate r:

700/r^2 = 4πr

Now, multiply both sides by r^2:

700 = 4πr^3

Divide both sides by 4π:

r^3 = 700 / (4π)

Now, find the cube root of both sides:

r = (700 / (4π))^(1/3)

Now, we can find the height h using the equation we derived earlier:

h = 350 / (πr^2)

Plug in the value of r:

h = 350 / (π(700 / (4π))^(2/3))

These are the dimensions of the cylinder that minimize the production costs: radius r and height h.

To get the final answers for the radius r and height h, we need to calculate the numerical values:

r = (700 / (4π))^(1/3)

r ≈ (700 / (4 × 3.14159265))^(1/3)

r ≈ (700 / 12.56637061)^(1/3)

r ≈ 55.75840735^(1/3)

r ≈ 3.824

Now, let's calculate the height h:

h = 350 / (π(700 / (4π))^(2/3))

h = 350 / (π(3.824^2))

h ≈ 350 / (π × 14.623)

h ≈ 350 / 45.978

h ≈ 7.610

So, the dimensions that minimize the production costs are approximately:

Radius (r): 3.824 cm

Height (h): 7.610 cm

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Answer:

29 blocks

Step-by-step explanation:

You've to count the number of blocks and that's the volume.

Answer:

whooper write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following write an equation of the line in slope-intercept form for each of the following

Step-by-step explanation:

Taking the square root of both sides of the inequality, we get:

||U + V|| ≤ ||U|| + ||V||

The triangle inequality theorem states that, the sum of the length of two sides must be greater than the length of the third side of that triangle.

To prove it, we start by squaring both sides of the inequality:

||U + V||² ≤ (||U|| + ||V||)²

Expanding the right-hand side of the inequality, we get:

||U + V||² ≤ ||U||² + 2||U|| ||V|| + ||V||²

Now, let's use the fact that ||U + V||² = (U + V) · (U + V), where · denotes the dot product:

(U + V) · (U + V) ≤ ||U||² + 2||U|| ||V|| + ||V||²

Now, we have:

||U||² + 2U · V + ||V||² ≤ ||U||² + 2||U|| ||V|| + ||V||²

Therefore, we have:

||U + V||² ≤ ||U||² + 2||U|| ||V|| + ||V||²

≤ ||U||² + 2U · V + ||V||²

= (||U|| + ||V||)²

Taking the square root of both sides of the inequality, we get:

||U + V|| ≤ ||U|| + ||V||

This is exactly what we wanted to prove. Therefore, we have shown that ||U + V|| ≤ ||U|| + ||V||.

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Answer:

Step-by-step explanation:

500/0.4 = 1250

Answer:

581,58 in^3

Step-by-step explanation:

Given:

Cylinder shaped cups

d (diameter) = 4,2 in

r (radius) = 0,5 × 4,2 = 2,1 in

h (height) = 7 in

π = 3.14

.

First, let's find how much juice can he pour into 1 cup:

.

We need to find the base of the cylinder:

.

[tex]a(base) = \pi {r}^{2} = 3.14 \times( {2.1})^{2} = 13.8474[/tex]

.

Now, we can find the volume of one cup:

V = a (base) × h

[tex]v = 13.8474 \times 7 ≈96.93[/tex]

Multiply this number by 6 and we'll get the answer (since there's 6 cups):

96,93 × 6 = 581,58

Let L be the original length of the rectangle and W be the original width of the rectangle. We know that:

(L - 6) = (W + 3) (1) (since the length is decreased by 6cm and the width is increased by 3cm, the result is a square)

The area of the rectangle is LW, and the area of the square is (L - 6)(W + 3). We also know that the area of the square is 27cm^2 smaller than the area of the rectangle. So we can write:

(L - 6)(W + 3) = LW - 27 (2)

Expanding the left side of equation (2), we get:

LW - 6W + 3L - 18 = LW - 27

Simplifying and rearranging, we get:

3L - 6W = 9

Dividing both sides by 3, we get:

L - 2W = 3 (3)

Now we have two equations with two unknowns, equations (1) and (3). We can solve this system of equations by substitution. Rearranging equation (1), we get:

L = W + 9

Substituting this into equation (3), we get:

(W + 9) - 2W = 3

Simplifying, we get:

W = 6

Substituting this value of W into equation (1), we get:

L - 6 = 9

So:

L = 15

Therefore, the area of the rectangle is:

A = LW = 15 x 6 = 90 cm^2.

Answer:

252

Step-by-step explanation:

lol I take rsm too and I just guessed and checked

The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

Probability refers to the likelihood that an occurrence will occur.

In actual life, we frequently have to make predictions about the future. We might or might not be conscious of how an event will turn out.

When this occurs, we proclaim that there is a possibility that the event will occur. In conclusion, probability has a broad range of amazing uses in both business and this rapidly developing area of artificial intelligence.

Simply dividing the favorable number of possibilities by the total number of possible outcomes using the probability formula will yield the chance of an event.

According to our question-

Total number of books that arrived that day = 23 + 41 = 64.

Let E be the event of arriving at a non-fictional book.

The event of arriving at a non-fictional book is n(E) = 23.

Total sample space, n(S) = 64.

The probability of a non-fictional book to arrive is = n(E) / n(S)

= 23 / 64 = 0.359375

= 0.359375 × 100

= 35.9375 ≈ 36.

Hence, The number of non-fiction novels among the following 100 books that are released is anticipated to be 36%.

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Answer:

the missing values are 2, 8, and 3.

Answer:

[tex] \frac{13y ^{2} - 59y - 90 }{(y + 10)(y - 10)} [/tex]

The rate that yields maximum revenue is $9 per month, and the maximum revenue is $1,296,000.

To find the rate that yields maximum revenue, we need to find the price that maximizes the revenue. Let x be the number of $0.50 decreases in the monthly fee, and let y be the number of subscribers who sign up at the new rate. Then we have the following equations:

y = 4800 + 120x (number of subscribers)

p = 24 - 0.5x (price per month)

r = xy(p) = (4800 + 120x)(24 - 0.5x) (revenue)

To find the rate that yields maximum revenue, we need to take the derivative of the revenue function concerning x, set it equal to zero, and solve for x:

r' = 120(24 - x) - (4800 + 120x)(0.5) = 0

x = 60

Therefore, the optimal number of $0.50 decreases is 60, and the corresponding price per month is $24 - 0.5(60) = $9. The number of subscribers at this rate is 4800 + 120(60) = 12000.

Finally, the maximum revenue is given by r = xy(p) = (4800 + 120(60))(24 - 0.5(60)) = $1,296,000.

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Answer:

the answer is A) -2x² + 4x + 30

Step-by-step explanation:

To find the equation of the quadratic function f(x), we can use the standard form of a quadratic function: f(x) = ax^2 + bx + c, where a, b, and c are constants.

We can plug in the values of x and f(x) from the table to get three equations:

a(-1)^2 + b(-1) + c = 24

a(0)^2 + b(0) + c = 30

a(1)^2 + b(1) + c = 32

Simplifying each equation, we get:

a - b + c = 24

c = 30

a + b + c = 32

We can substitute c = 30 into the first and third equations to get:

a - b + 30 = 24

a + b + 30 = 32

Simplifying these equations, we get:

a - b = -6

a + b = 2

Adding these two equations, we get:

2a = -4

Dividing by 2, we get:

a = -2

Substituting a = -2 into one of the equations above, we get:

-2 - b = -6

Solving for b, we get:

b = 4

Therefore, the equation that represents f(x) is:

f(x) = -2x^2 + 4x + 30

So the answer is A) -2x² + 4x + 30

the roots of the polynomial [tex]x^{5}[/tex] - [tex]7x^{3}[/tex] + 10x are 0, ±√5, and ±√2. Thus, the other factors of f(x) are (x+√5), (x-√5), (x+√2), and (x-√2).

What is polynomial?

A polynomial is a mathematical expression that consists of variables and coefficients, which are combined using arithmetic operations such as addition, subtraction, multiplication, and non-negative integer exponents.

If (x+1) is a factor of f(x), then f(-1) = 0. Thus, we can substitute -1 for x in the expression for f(x) and solve for k:

f(-1) = [tex](-1)^{6}[/tex] - 6[tex](-1)^{4}[/tex] + 17[tex](-1)^{2}[/tex] + k = 1 - 6 + 17 + k = 12 + k

Since f(-1) = 0, we have:

12 + k = 0

Solving for k, we find:

k = -12

So, when k = -12, (x+1) is a factor of f(x).

To find another factor of f(x), we can divide f(x) by (x+1) using polynomial long division or synthetic division. The result is:

f(x) = (x+1)([tex]x^{5}[/tex] - [tex]7x^{3}[/tex] + 10x)

So, another factor of f(x) is (x+a), where a is a root of the polynomial x^5 - 7x^3 + 10x. We can find the roots of this polynomial using a numerical method or by factoring it using the rational root theorem. By inspection, we can see that x=0 is a root, so we can factor out x:

x([tex]x^{4}[/tex] - [tex]7x^{2}[/tex] + 10) = 0

The quadratic factor can be factored further as:

x([tex]x^{2}[/tex] - 5)([tex]x^{2}[/tex] - 2) = 0

So, the roots of the polynomial [tex]x^{5}[/tex] - [tex]7x^{3}[/tex] + 10x are 0, ±√5, and ±√2. Thus, the other factors of f(x) are (x+√5), (x-√5), (x+√2), and (x-√2).

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The total area of the trapezoid is approximately 273.664 square yards. The Option A is correct.

The area of a trapezoid is found using the formula, A = ½ (a + b) h, where 'a' and 'b' are the bases (parallel sides) and 'h' is the height (the perpendicular distance between the bases) of the trapezoid.

To find the area of the figure, we need to divide it into two triangles and find the area of each triangle.

The area of the first triangle (with base 21.3 yards and height 12.8 yards) is:

(1/2) * base * height = (1/2) * 21.3 * 12.8 = 136.704 square yards.

The area of the second triangle (with base 6.4 yards and height 12.8 yards) is:

(1/2) * base * height = (1/2) * 6.4 * 12.8 = 41.216 square yards.

The area of the third triangle (with base 14.9 yards and height 12.8 yards) is:

(1/2) * base * height = (1/2) * 14.9 * 12.8 = 95.744 square yards.

Now, the total area of the figure is the sum of the areas of these three triangles:

= 136.704 + 41.216 + 95.744

= 273.664 square yards.

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The value of the expression is 142.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions are used to represent and solve problems in many areas of mathematics, science, engineering, and finance.

To solve this expression, we need to follow the order of operations (PEMDAS) which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Let's start with the parentheses first:

14 x 0.5 x (4 + 2) + exponent 10*2

= 14 x 0.5 x 6 + exponent 10*2

Next, we can simplify the multiplication and division from left to right:

= 7 x 6 + exponent 10*2

= 42 + exponent 10*2

Now we can evaluate the exponent:

= 42 + 100

= 142

Therefore, the value of the expression is 142.

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The desalination plant can produce [tex]9.84*10^6[/tex] gallons of water in four days in scientific notation.

We must divide the daily production rate by the number of days in order to get the total volume of water that a desalination plant can generate in four days:

4. days at a rate of [tex]2.46*10^6[/tex] gallons equals [tex]9.84*10^6[/tex] gallons.

Hence, in four days, the desalination plant can produce [tex]9.84 * 10^6[/tex]gallons of water.

Large or small numbers can be conveniently represented using scientific notation, especially when working with measurements in science and engineering. A number is written in scientific notation as a coefficient multiplied by 10 and raised to a power of some exponent. For example, [tex]2.46*10^6[/tex] denotes 2,460,000, which is 2.46 multiplied by 10 to the power of 6.

The solution in this case is [tex]9.84*10^6[/tex], or 9,840,000, which is 9.84 multiplied by 10 to the power of 6. Large numbers can be written and compared more easily using this format, and scientific notation rules can be used to conduct computations with them.

In conclusion, the desalination plant has a four-day capacity of [tex]9.84 * 10^6[/tex] gallons of water production.

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Answer:

Without knowing the exact length of Ted's pencil, we cannot give an exact decimal representation of its length. However, we do know that Ted's pencil is longer than Jan's pencil, which is 8.5 cm long.

If we assume that Ted's pencil is one centimeter longer than Jan's pencil, then its length would be 9.5 cm. In decimal form, this would be written as 9.5.

If we assume that Ted's pencil is two centimeters longer than Jan's pencil, then its length would be 10.5 cm. In decimal form, this would be written as 10.5.

So, the decimal that could represent the length of Ted's pencil depends on how much longer it is than Jan's pencil.

The probability that a student who attends the first session also attends the second session is 1/3

We can approach this problem by using conditional probability.

Let's use the notation "S1" to represent the event that a student attends the first session, and "S2" to represent the event that a student attends the second session.

Then we can use the formula for conditional probability:

P(S2 | S1) = P(S1 and S2) / P(S1)

We know that P(S1) = 0.6, since 60% of students attend the first session. We also know that P(S1 and S2) = 0.2, since 20% of students attend both sessions.

So we can plug in these values and solve for P(S2 | S1):

P(S2 | S1) = 0.2 / 0.6

P(S2 | S1) = 1/3

Therefore, the probability that a student who attends is 1/3 or approximately 0.333.

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By calculating the z-score, we can conclude that, Team A has better eligibility rates than Team B.

What is z-score?

A z-score (also called a standard score) is a measure of how many standard deviations a given data point is away from the mean of its distribution. It is calculated by subtracting the mean of the distribution from the data point, and then dividing the difference by the standard deviation.

To determine this, we can compare the z-scores for each team's eligibility rates.

Let's assume that Team A's eligibility rate is 875 and Team B's eligibility rate is 825.

The mean eligibility rate for all teams is given as 870, with a standard deviation of 50. Therefore, we can calculate the z-scores for each team's eligibility rate as follows:

z-score for Team A's eligibility rate = (875 - 870) / 50 = 0.1

z-score for Team B's eligibility rate = (825 - 870) / 50 = -0.9

Since the z-score for Team A's eligibility rate is positive and greater than the z-score for Team B's eligibility rate, we can conclude that Team A has better eligibility rates than Team B.

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Answer:

Step-by-step explanation:

Solution:

   [tex](x-2)^2+(y+3)^2=25[/tex]

Equation of general circle:

    [tex](x-a)^2+(y-b)^2=r^2[/tex]      where (a,b) is circle center and r is the radius.

If the roots of a quadratic equation are given, we can write the equation in factored form as [tex](x - r1)(x - r2) = 0[/tex] , where r1 and r2 are the roots.the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

A quadratic equation can be written in the form:

[tex]ax^2 + bx + c = 0[/tex]

where a, b, and c are constants. Since the roots of the equation are 5 and 2, we can write:

[tex](x - 5)(x - 2) = 0[/tex]

Expanding this equation gives:

[tex]x^2 - 7x + 10 = 0[/tex]

To make the leading coefficient of this equation 4, we can multiply both sides by 4/1, which gives:

[tex]4x^2 - 28x + 40 = 0[/tex]

Therefore, the quadratic equation with roots 5 and 2 and leading coefficient 4 is: [tex]4x^2 - 28x + 40 = 0.[/tex]

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The probability that the fire department arrives at the scene in 8 minutes or less is approximately 0.9772 or 97.72%.

What is the standard deviation?

The standard deviation is a measure of the amount of variation or dispersion in a set of data. It measures how much the data deviates from the mean value.

We are given that the times the fire department takes to arrive at the scene of an emergency are normally distributed with a mean of 6 minutes and a standard deviation of 1 minute.

Let X be the random variable representing the time taken by the fire department to arrive at the scene of an emergency. Then, we have:

X ~ N(6, 1)

We want to find the probability that the fire department arrives at the scene in 8 minutes or less. Mathematically, we want to find:

P(X ≤ 8)

To find this probability, we can standardize the random variable X by subtracting the mean and dividing by the standard deviation:

Z = (X - μ) / σ

Substituting the values of μ and σ, we get:

Z = (X - 6) / 1

Z represents the standard normal distribution with a mean of 0 and a standard deviation of 1. Therefore, we can use a standard normal distribution table or calculator to find the probability that Z is less than or equal to the standardized value of 8:

P(Z ≤ (8 - 6) / 1) ≈ P(Z ≤ 2)

Looking up the probability of Z being less than or equal to 2 in a standard normal distribution table, we find that it is approximately 0.9772.

Therefore, the probability that the fire department arrives at the scene in 8 minutes or less is approximately 0.9772 or 97.72%.

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a) The total volume equals the sum of the volumes.

[tex]500 = x + y[/tex]

The total octane amount equals the sum of the octane amounts.

[tex]89(500) = 87x + 92y[/tex]

[tex]44500 = 87x + 92y[/tex]

b)

As x increases, y decreases.

c) Use substitution or elimination to solve the system of equations.

[tex]44500 = 87x + 92(500-x)[/tex]

[tex]44500 = 87x + 46000 - 92x[/tex]

[tex]5x = 1500[/tex]

[tex]x = 300[/tex]

[tex]y = 200[/tex]

The required volumes are 300 gallons of 87 gasoline and 200 gallons of 92 gasoline.

The vector v can be represented as (-2, 4) and has a magnitude of 2√5.

A vector is a quantity that not only describes the magnitude but also describes the movement of an object or the position of an object with respect to another point or object. It is also known as Euclidean vector, geometric vector or spatial vector.

To plot the vector v with an initial point at (0, -1) and a terminal point at (-2, 3), we can draw a line segment from the initial point to the terminal point. The vector v is represented by the direction and magnitude of this line segment.

To analyze the vector v, we can calculate its components and magnitude:

Components: The components of v are the differences between the x-coordinates and y-coordinates of the terminal point and initial point, respectively. We have:

v = (-2 - 0, 3 - (-1)) = (-2, 4)

Magnitude: The magnitude of v is the length of the line segment connecting the initial point and terminal point. We can use the Pythagorean theorem to find the magnitude:

||v|| = √(-2-0)² + (3 - (-1))²) = √20 = 2√5

Therefore, the vector v can be represented as (-2, 4) and has a magnitude of 2√5.

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Answer:

JK is a line segment

Step-by-step explanation:

The sum of the nth roots of unity is zero.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. Complex numbers are used in mathematics and science to represent quantities that have both a real and imaginary part.

The nth roots of unity are given by:

[tex]\omega ^0, \omega^1, \omega^2, ..., \omega^(n-1)[/tex]

where [tex]\omega = e^{(2\pi i/n)[/tex]is a complex number and i is the imaginary unit.

The sum of these roots is:

[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^{(n-1)[/tex]

To simplify this expression, we can use the formula for the sum of a geometric series:

[tex]a + ar + ar^2 + ... + ar^(n-1) = a(1 - r^n)/(1 - r)[/tex]

Let a = 1 and [tex]r = ω.[/tex] Then the sum of the nth roots of unity is:

[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - \omega^n)/(1 - \omega)[/tex]

Substituting [tex]ω = e^(2πi/n)[/tex], we get:

[tex]\omega^n = (e^(2\pi i/n))^n = e^2 \pi i = 1[/tex]

Therefore, the sum of the nth roots of unity is:[tex]\omega^0 + \omega^1 + \omega^2 + ... + \omega^(n-1) = (1 - 1)/(1 - \omega) = 0/(1 - \omega) = 0[/tex]

Hence, the sum of the nth roots of unity is zero.

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