Given the points A(-1, 2) and B(7, 14), find the coordinates of the point P on directed line segment stack A B with bar on top that partitions stack A B with bar on top in the ratio 1:3. Show your work.

The constant of proportionality of 5 tells us that the volume of the cylinder is five times greater than it would be if the radius were 1 cm.

The volume of a cylinder is given by the formula V = πr²h, where r is the radius of the base and h is the height. If h = 5 and V is directly proportional to r² with a constant of proportionality of 5, then we can write the following equation:

V = 5r²

This equation tells us that the volume of the cylinder is 5 times the square of the radius. For example, if the radius is 1 cm, then the volume of the cylinder is 5 cm³. If the radius is 2 cm, then the volume of the cylinder is 20 cm³.

The fact that the volume of the cylinder is directly proportional to the square of the radius means that the volume increases more rapidly as the radius increases. For example, when the radius is doubled from 1 cm to 2 cm, the volume increases from 5 cm³ to 20 cm³, which is a fourfold increase.

For example, if the radius is 2 cm, then the volume of the cylinder is 20 cm³, which is five times greater than the volume of a cylinder with a radius of 1 cm and a height of 5 cm.

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Answer:

irrational

Step-by-step explanation:

because an irrational number goes on forever so no matter what number is added to it it will always be irrational.

Answer:

Rational

Step-by-step explanation:

Answer:

Your answer is -5n=1.75

Step-by-step explanation:

So a quadratic equation in factored form is generally given in the form: \(a(x-b)(x+c)\). Sometimes x may have a coefficient but I'll just create a simple quadratic equation. In this form it's easy to see that the solutions to the equation are b and -c, because if you input them in as x, you'll get one of the factors equal to 0, and it'll make the y-value zero, thus it's a solution. So b and c can literally be anything. But for this example I'll chose 3 and 6 to keep it simple. Inputting these as b and c you'll get: \(a(x-3)(x+6)\). Now the value of a will determine the stretch/compression, but to keep it really simple I'll just say that a=1. So now all you have to do is expand the two factors. When you expand them you'll get it in the form: \((x-b)(x+c) = x^2+cx-bx-bc\) by using the foil method. So if you expand the two factors in this example you'll get: \(x^2+6x-3x-18\) which simplifies to \(x^2+3x-18\). So cool now we have our equation. Now it's time to solve it, you can either factor it (since you literally just had it in factored form), or use the quadratic formula I'll show both methods

Factoring:

   When you factor an equation in the form: \(ax^2+bx+c\). You usually multiply the coefficients a and c. Then you'll find factors of ac that add up to b. In this example ac is really just c because a is 1...

   So given the equation: \(x^2+3x-18\) You want to look for factors of -18 that add up to 3. And whenever you have a negative number as AC you really want to look for factors that have a distance of b which in this case is 3. For example I would say that 9 and 3 have a distance of 6. And then after that depending on whether b is negative or positive. I can choose which factor will be negative and which will be positive. If b was 6 well then 9 would have to be positive and 3 would be negative, but if b was -6 then 9 would have to be negative and 3 would have to be positive.

 So in this example look for factors that have "distance" of 3. Well 6 and 3 have a "distance" of 6 so I know those 2 will be the factors. Since 3 is positive that means 6 will be positive, and 3 will be negative. so they add up to 3 and multiply to get -18

   AC factors that add up to 3 and multiply to -18: 6 and -3

   Now write it in factored form using the two factors:

   \((x-3)(x+6)\)

  Now set up each factor equal to 0 and solve for x:

   \(x-3 = 0\\x=3\)

   \(x+6=0\\x=-6\)

   This gives you the two solutions: x=3 and x=-6

Quadratic Formula:

   The quadratic formula is usually used when a equation can't be easily factored. The quadratic formula gives both zeroes in the equation: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). Notice the plus or minus sign next to the square root? That's where the the two solutions will come from, or possibly one solution if it's at the vertex. or maybe even no real solutions if the quadratic never intersects the x-axis.

  Ok so let's identify  a, b, and c. We're given the equation: \(x^2+3x-18\). so a=1, b=3, and c=-18. Plugging these values into the equation you get:

   \(x=\frac{-(3)\pm\sqrt{(3)^2-4(1)(-18)}}{2(1)}\\\)

   Simplifying it a bit gets you the equation:

   \(x=\frac{-3\pm\sqrt{81}}{2}\)

   Simplifying the radical gives you:

   \(x=\frac{-3\pm9}{2}\)

  Use the positive sign:

   \(x=\frac{-3+9}{2}\\x=\frac{6}{2}\\x=3\)

   Use the negative sign:

   \(x=\frac{-3-9}{2}\\x=\frac{-12}{2}\\x=-6\)

  This gives you the two solutions x=3, and x=-6
   

Answer:

The answer is 1400

Step-by-step explanation:

Answer: Let the width of the rectangle be w and the length be l. We are given that the area of the rectangle is 45 cm^2, so we can write the equation:

w * l = 45

To find the lengths of the sides, we need to solve for w and l. We can divide both sides of the equation by any non-zero number to find an equivalent expression. For example, we can divide both sides by 9:

(w/9) * (l/9) = 45/9

Simplifying, we get:

w/9 * l/9 = 5

So, w/9 = 5/l and w = 5l/9.

Substituting the expression for w into the original equation, we get:

(5l/9) * l = 45

Expanding and simplifying, we get:

5l^2/9 = 45

Multiplying both sides by 9, we get:

5l^2 = 405

Dividing both sides by 5, we get:

l^2 = 81

Taking the square root of both sides, we get:

l = 9

So, the length of the rectangle is 9 cm. To find the width, we can substitute the value of l back into the expression we derived earlier:

w = 5l/9

w = 5 * 9 / 9

w = 5

So, the width of the rectangle is 5 cm.

The lengths of the sides of the rectangle, both exactly and approximately, are 9 cm and 5 cm.

Step-by-step explanation:

We can conclude that there are no nontrivial clean pulsations for the given left-mounted beam with a simple support on the right.

To find the equation of clean pulsations for a left-mounted beam with a simple support on the right, we can use the differential equation that describes the deflection of the beam. Assuming the beam is subject to a distributed load and has certain boundary conditions, the equation governing the deflection can be written as:

d^2y/dx^2 + (chx)^2 * y = 0

Where:

y(x) is the deflection of the beam at position x,

d^2y/dx^2 is the second derivative of y with respect to x,

ch(x) is the hyperbolic cosine function.

To solve this differential equation, we can assume a solution in the form of y(x) = A * cosh(kx) + B * sinh(kx), where A and B are constants, and k is a constant to be determined.

Substituting this assumed solution into the differential equation, we get:

k^2 * (A * cosh(kx) + B * sinh(kx)) + (chx)^2 * (A * cosh(kx) + B * sinh(kx)) = 0

Simplifying the equation and applying the given identity (chx)^2 - (shx)^2 = 1, we have:

(A + A * chx^2) * cosh(kx) + (B + B * chx^2) * sinh(kx) = 0

For this equation to hold for all values of x, the coefficients of cosh(kx) and sinh(kx) must be zero. Therefore, we get the following equations:

A + A * chx^2 = 0

B + B * chx^2 = 0

Simplifying these equations, we have:

A * (1 + chx^2) = 0

B * (1 + chx^2) = 0

Since we are looking for nontrivial solutions (A and B not equal to zero), the expressions in parentheses must be zero:

1 + chx^2 = 0

Using the identity (sinx)^2 + (cosx)^2 = 1, we can rewrite this equation as:

1 + (1 - (sinx)^2) = 0

Simplifying further, we get:

2 - (sinx)^2 = 0

Solving for (sinx)^2, we find:

(sin x)^2 = 2

Since the square of the sine function cannot be negative, there are no real solutions to this equation. Therefore, we can conclude that there are no nontrivial clean pulsations for the given left-mounted beam with a simple support on the right.

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Answer:

1

Step-by-step explanation:

32/4=8

no remainder so it is i^4

i^4=1

The probability would be 7/12.

Here, we have,

Consider A is the event that there is snowing in first three days and B is the event that there is snowing in next four days.

According to the question,

P(A) = 1/3

P(B)= 1/4

Thus, the probability that it snows at least once during the first week of January

= snow in first three days or snow in next four days

= P(A∪B)

=P(A) + P(B) - P(A∩B)

 ( ∵ A and B are independent ⇒P(A∩B) = 0  )

=1/3 + 1/4

=7/12

Hence, The probability would be 7/12.

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\(y = 5x +25~~~~~~~~~~~~~~...(i)\\\\y = 3x +35~~~~~~~~~~~~~~...(ii)\\\\ \text{Substitute (x,y) = (5,50) in both equations.} \\\\y = 5(5) +25 = 25 +25 = 50\\\\y =3(5) +35 = 15 +35 = 50\\\\\text{Hence, (5,50) is a solution to the linear system of equations.}\)

(f ◦ g)(x) = -2x^2 + 11.

What is Composition of two functions ?

The new function obtained by performing f first and then g is the combination of two functions g and f.

The composition of two functions f and g, denoted by (f ◦ g)(x), is defined as the function that maps x to f(g(x)).

Given the functions f(x) = 2x + 3 and g(x) = -x^2 + 5, the composition of these two functions is:

(f ◦ g)(x) = f(g(x)) = f(-x^2 + 5) = 2(-x^2 + 5) + 3 = -2x^2 + 11

So, (f ◦ g)(x) = -2x^2 + 11.

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Answer:

$189

Step-by-step explanation:

Calculate a unit rate (dollars per hour) then times by 9 hours. OR use a proportion. You will get the same answer.

dollars/hour

= $252/12hours

= $21/1hour

= 21 $/hr

21 × 9 = 189

Yoko will make $189 for 9 hours.

OR use a proportion.

$/hour = $/hour

252/12 = x/9

cross-multiply

12x = 252(9)

12x = 2268

divide by 12

x = 2268/12

x = 189

same answer!

Answer:

D

Step-by-step explanation:

the last graph

The negative correlation coefficient of -0.73 between consumption of vitamin A and the cancer rate of a particular type of cancer suggests that as vitamin A consumption increases, the cancer rate tends to decrease.

A correlation coefficient measures the strength and direction of the linear relationship between two variables.

In this case, a correlation coefficient of -0.73 indicates a negative correlation between consumption of vitamin A and the cancer rate.

Interpreting this correlation, it can be inferred that there is an inverse relationship between the two variables. As consumption of vitamin A increases, the cancer rate tends to decrease.

However, it is important to note that correlation does not imply causation.

It would be incorrect to conclude that consuming more vitamin A causes this type of cancer. Correlation does not provide information about the direction of causality.

Other factors and confounding variables may be involved in the relationship between vitamin A consumption and cancer rate.

To establish a causal relationship, further research, such as experimental studies or controlled trials, would be necessary. These types of studies can help determine whether there is a causal link between vitamin A consumption and the occurrence of this particular cancer.

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The given data set {(0, 3), (1, 18), (2, 52), (3, 108), (4, 324)} can be analyzed to determine whether it represents a linear, quadratic, or exponential relationship between the variables.

A linear relationship between two variables is one in which the change in one variable is proportional to the change in the other variable. A quadratic relationship is one in which the variable being measured increases at a non-constant rate. An exponential relationship is one in which the variable being measured increases or decreases at a constant percentage rate.

On plotting the given data set, we can see that the curve is not a straight line, so it is not a linear function. Also, the curve does not have the characteristics of a quadratic function since it is not a parabola. However, the curve seems to increase at an increasing rate, suggesting that it is an exponential function.

Therefore, the given data set represents an exponential relationship between the variables.

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To find the volume of a triangular prism with a right angle, multiply the area of the base triangle by the height of the prism.

Start with a triangular prism that has a right angle. The base of the prism is a right-angled triangle.

Measure the lengths of the two perpendicular sides of the right-angled triangle, which are typically referred to as the base (b) and the height (h) of the triangle.

Calculate the area of the base triangle using the formula: Area = (1/2) * base * height.

Measure the height (H) of the prism, which is the perpendicular distance between the two parallel bases.

Multiply the area of the base triangle by the height of the prism to find the volume:

Volume = Base Area * Height = (1/2) * base * height * H.

If the dimensions are given in different units, make sure to convert them to the same unit before performing the calculations.

The volume of a triangular prism with a right angle can be found by multiplying the area of the base triangle by the height of the prism. Ensure that the base dimensions and the height are measured accurately and in the same unit.

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Answer:

c

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

When solved for x=  -1<x<1

Answer:

9

Step-by-step explanation:

Answer:

9

Step-by-step explanation:

3 3/4 divided by 5/12 equals 9

hope this helps :)

Answer:

a-b

Step-by-step explanation:

You can't simply or evaluate a-b so so it still give you a-b

The value of trigonometric ratios is  \(cosA= 0.4962,cosB=0.8684\) ,   \(sinA=0.8684, sinB=0.4962, tanA=1.75, tanB=0.5714\) and all the sides are given.

In mathematics, the trigonometric ratios are real functions which relate an angle of a right-angled triangle to ratios of two side length. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics and many others.

Side is defined as one of the outer surfaces of something. It is an aspect of a geometrical shape. The shapes that we see are made up of lines(line segments) and points (vertices).

\(CosA= 4/8.06=0.4962\)

\(CosB=7/8.06=0.8684\)

\(SinA=7/8.06=0.8684\)

\(SinB=4/8.06=0.4962\)

\(TanA=7/4=1.75\)

\(TanB=4/7=0.5714\)

Therefore, The value of trigonometric ratios is  \(cosA= 0.4962,cosB=0.8684\) ,   \(sinA=0.8684, sinB=0.4962, tanA=1.75, tanB=0.5714\) and all the sides

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Answer:

Step-by-step explanation:

Numbers: HCF of 18 and 30

18: 2 * 3 * 3

30:2 * 3 * 5

HCF: 2 and 3 = 6

a: a * a and a.

HCF = a

b: b

c: c and c*c

HCF: C

Answer: 6*a*b*c(3a + 5*c)

Note: for this question one of the factors is the HCF and the other is what remains when the HCF is taken out.

The factor theorem indicates that the expression (x + 2) is a factor of the polynomial, f(x) = 2·x² - x - 14, because, f(-2) = 0, the correct option is option C.

C. (x + 2) is a factor.

The factor theorem, which is a unique example of the remainder theorem, indicates that the link between the factors and the zeros of the polynomial, such that (x - α) is a factor of the polynomial f(x), if and only if we get f(α) = 0.

The polynomial is; 3·x² - x - 14, the expression is; (x + 2)

The factor theorem indicates that we get, (x - a) is a factor of the polynomial of nth degree, where, n ≥ 1, if f(a) = 0

Whereby the polynomial is expressed as f(x) = 3·x² - x - 14 = 0, we get;

(x + 2), is a factor of the polynomial, according to the factor theorem, if we get, f(-2) = 0

Plugging in x = -2, in the polynomial, we get;

f(-2) = 3×(-2)² - (-2) - 14 = 0

Therefore, (x + 2) is a factor of the polynomial

The correct option is therefore;

C. (x + 2) is a factor

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The monomial (x + 2) is a factor of the polynomial, the correct option is C.

We want to find the relationship between (x + 2) and the polynomial:

P(x) = 3x^2 - x - 14

The monomial  (x + 2) will be a factor of the polynomial only if x = -2 is a zero of the polynomial, so let's check that.

P(-2) = 3*(-2)^2 - (-2) - 14

P(-2) = 3*4 + 2 - 14 = 0

So -2 is a zero of the polynomial, then we the monomial (x + 2) is a factor of the polynomial.

The correct option is C.

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

Explanation:

t = number of t-shirts

t = 100 shirts are in the box

Plug this into the equation and simplify to find W

W = 250+132.5t

W = 250+132.5(100)

W = 250+13250

W = 13500

The box weighs 13500 grams

This is equivalent to 13500/1000 = 13.5 kg

The area of the square is 144 \(cm^{2}\).

Let x be the side of the square. Then the length of the triangle is (x+4). Perimeter is the length of all sides of a geometric figure combined. For an equilateral triangle, it's equal to thrice the length of one side. For a square, it's four times the length of one side. The Perimeter of the Triangle is 3(x+4) & the Perimeter of the square is 4x.

We know, both these perimeters are equal. Hence,

4x = 3(x+4)

To further simplify the above equation.

4x = 3x + 12

x = 12

Hence, the length of one side of the square is 12 cm. The area of the square can be calculated as follows:

Area = \((side)^{2}\)

Area = 12 * 12

Area = 144 \(cm^{2}\)

Hence, the Area of the Square is 144 \(cm^{2}\)

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Step-by-step explanation:

The velocity of the particle at a given value of θ can be found by taking the derivative of the position vector with respect to time:

v(θ) = dr/dt = dr/dθ * dθ/dt = dr/dθ * ω

where ω is the angular velocity and dr/dθ is the derivative of the position vector with respect to θ.

The position vector r = a * e^(b * θ) * i + a * b * e^(b * θ) * j

Taking the derivative with respect to θ:

dr/dθ = a * b * e^(b * θ) * i + a * b^2 * e^(b * θ) * j

So the velocity of the particle is given by:

v(θ) = a * b * e^(b * θ) * ω * i + a * b^2 * e^(b * θ) * ω * j

Answer:

Step-by-step explanation:

v(θ) = a * b * e^(b * θ) * ω * i + a * b^2 * e^(b * θ) * ω * jj

The solution to the initial value problem is y(x) = -1/(2x + x^2/2 - 1), and the minimum value of y(x) is attained at x = -1/2. The minimum value is an exact number, but it is difficult to express it as a fraction or decimal.

The initial value problem y' = 2y^2 + xy^2, y(0) = 1 is a first-order nonlinear ordinary differential equation. The goal is to find a function y(x) that satisfies the differential equation and the initial condition.

To solve this equation, we first note that it is separable. We can rewrite the equation as:

y'/(y^2) = 2 + x

Integrating both sides with respect to x, we get:

-1/y = 2x + x^2/2 + C

where C is a constant of integration. Multiplying both sides by -y, we get:

y(x) = -1/(2x + x^2/2 + C)

To determine the value of C, we use the initial condition y(0) = 1. Substituting x = 0 and y = 1 into the equation, we get:

1 = -1/C

so C = -1. Thus, the solution to the initial value problem is:

y(x) = -1/(2x + x^2/2 - 1)

To find the minimum value of y(x), we note that y(x) is undefined for x = -2, and is negative for x < -2. Thus, we restrict our attention to x ≥ -2.

We take the derivative of y(x) and set it equal to zero to find the critical points:

y'(x) = 2(2x + 1)/(2x + x^2 - 2)^2 = 0

Solving for x, we get x = -1/2. To determine whether this is a minimum or maximum, we compute the second derivative of y(x):

y''(x) = 8(x^2 + 2x + 3)/(2x + x^2 - 2)^3

Since y''(-1/2) > 0, we conclude that x = -1/2 is a local minimum. Since there are no other critical points, we conclude that the solution attains its minimum value at x = -1/2.

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The height and radius of the cone-shaped cup that will use the smallest amount of paper, that is the maximum volume is  3.72 and  2.632 respectively.

Formula used in the solution is

Volume of the cone= (1/3)π(r^2)h

Area of the cone= πr√(l^2 + r^2)

Given, the volume of the cone is 27 cm^3.

(1/3)π(r^2)h =27

Simplifying the equation to get the value of r.

π(r^2)=81/h

r^2=81/hπ

\(r=\sqrt{\frac{81}{\pi h} }\)

Substituting the value of r in area, we get

\(A=\pi \sqrt{\frac{81}{\pi h} } \sqrt{h^{2} + \frac{81}{\pi h} }\)

\(\frac{dA}{dh} =\pi \sqrt{\frac{81}{\pi h} } \sqrt{h^{2} + \frac{81}{\pi h} }\)

dA/dh=√81 × (π - 162/\(h^{3}\)) × 1/√(πh + 81/\(h^{2}\))

At dA/dh=0, we will get,

√81 × (π - 162/\(h^{3}\)) × 1/√(πh + 81/\(h^{2}\))=0

√81 × (π - 162/\(h^{3}\))=0

\(\pi - \frac{162}{h^{3} }=0\)

Thus, \(\pi h^{3} -162=0\)

\(\pi h^{3} =162\)

\(h^{3} =\frac{162}{\pi}\)

h^3= 51.5662015618

h= 3.7221

Now, substitute h in radius,

\(r=\sqrt{\frac{81}{\pi \times 3.7221} }\)

r=2.632

Hence, the height of the cone shaped cup is 3.7221 and the radius is 2.632.

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In a rectangle, diagonals have two main properties: they bisect each other and have equal lengths. If DE is given as 15, then the other half of the diagonal (let's call it DF) will also be 15. This means the full diagonal length is 30.


A rectangle is a quadrilateral with four right angles. The opposite sides of a rectangle are equal in length, and its diagonals bisect each other. The length of a diagonal in a rectangle can be found using the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

In the case of a rectangle, the hypotenuse is the diagonal, and the other two sides are the length and width of the rectangle. Therefore, if we know the length and width of the rectangle, we can use the Pythagorean theorem to find the length of the diagonal.

Let's apply this concept to your question. You mentioned that the diagonals of a rectangle intersect at point D, and that DE has a length of 15. However, you did not provide any additional information about the rectangle, such as its length and width. Without this information, it is impossible to determine the length of the diagonal.

Therefore, I encourage you to provide more information about the rectangle so that I can help you find the length of its diagonal using the Pythagorean theorem. I hope this explanation helps you understand how to find the length of a diagonal in a rectangle.

I understand you have a question about a rectangle with intersecting diagonals at point D, and DE is given as 15. Due to some typos, I'll assume you meant to say "rectangle" instead of "heal" and "intersect" instead of "interesect."

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Integral value is finite since the given integral converges when the function is \(\int\limits^\infty_\infty {3xe^{-x^{2} } } \, dx\).

Given that,

Analyze the integral to see if it is convergent or divergent. convergent integrals should be evaluated.

\(\int\limits^\infty_\infty {3xe^{-x^{2} } } \, dx\)

We have to simplify the equation.

Finding the area of the curve's undersurface is the process of integration. To do this, draw as many little rectangles as necessary, then add up their areas.

We know that,

I= \(\int\limits^\infty_\infty {3xe^{-x^{2} } } \, dx\)

Let p = x²then xdx= dp/2

Then

I= \(\int\limits^\infty_\infty {3/2e^{-p} } \, dp\)

I= 3/2(\(e^{-p}\))infinity to minus infinity

I= 3/2 (\(e^{-x^{2} }\))infinity to minus infinity

I=0

Therefore, integral value is finite since the given integral converges when the function is \(\int\limits^\infty_\infty {3xe^{-x^{2} } } \, dx\).

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