1
Select the correct answer.
Which expression is equivalent to x+y+x+y+ 3(y+5)?
OA. 2x+5y+5
OB. 2x+y+30
OC. 2x+ 5y + 15
OD. 2x+3y + 10

The solution to the equation is x = 1.62

The equation is given as:

[tex]x =\sqrt{x\ -\ \frac{1}{x}}+\sqrt{1-\ \frac{1}{x}}[/tex]

Next, we split the equations as follows:

y = x

[tex]y =\sqrt{x\ -\ \frac{1}{x}}+\sqrt{1-\ \frac{1}{x}}[/tex]

Next, we graph the equations (see attachment)

From the attached graph, the equations intersect at

(x, y)= (1.62, 1.62)

Remove the y value

x = 1.62

Hence, the solution to the equation is x = 1.62

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Rationalize the right side first. Expand and simplify.

[tex]x = \sqrt{x - \dfrac1x} + \sqrt{1 - \dfrac1x}[/tex]

[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = \left(\sqrt{x - \dfrac1x} + \sqrt{1 - \dfrac1x}\right) \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right)[/tex]

[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = \left(\sqrt{x - \dfrac1x}\right)^2 - \left(\sqrt{1 - \dfrac1x}\right)^2[/tex]

[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = \left(x - \dfrac1x\right) - \left(1 - \dfrac1x\right)[/tex]

[tex]x \left(\sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}\right) = x - 1[/tex]

Rearrange terms as

[tex]\dfrac{x - 1}x = \sqrt{x - \dfrac1x} - \sqrt{1 - \dfrac1x}[/tex]

[tex]\dfrac{x - 1}x + \sqrt{\dfrac{x - 1}x} = \sqrt{x - \dfrac1x}[/tex]

Substitute

[tex]y = \sqrt{\dfrac{x-1}x} = \sqrt{1 - \dfrac1x} \\\\ \implies y^2 = 1 - \dfrac1x \implies x = \dfrac1{1-y^2}[/tex]

(noting that [tex]y>0[/tex]) so that

[tex]x - \dfrac1x = \dfrac1{1-y^2} - (1-y^2) = \dfrac{2y^2 - y^4}{1-y^2}[/tex]

and the equation transforms to

[tex]y^2 + y = \sqrt{\dfrac{2y^2 - y^4}{1-y^2}}[/tex]

[tex]y + 1 = \sqrt{\dfrac{2 - y^2}{1-y^2}}[/tex]

Solve for [tex]y[/tex].

[tex](y+1)^2 = \left(\sqrt{\dfrac{2-y^2}{1-y^2}}\right)^2[/tex]

[tex]y^2 + 2y + 1 = \dfrac{2 - y^2}{1 - y^2}[/tex]

[tex]-y^4 - 2y^3 + 2y + 1 = 2 - y^2[/tex]

[tex]y^4 + 2y^3 - y^2 - 2y + 1 = 0[/tex]

[tex]\left(y^4 + 2y^3 + y^2\right) - 2y^2 - 2y + 1 = 0[/tex]

[tex](y^2 + y)^2 - 2(y^2 + y) + 1 = 0[/tex]

[tex]\left(y^2 + y - 1\right)^2 = 0[/tex]

[tex]y^2 + y - 1 = 0[/tex]

[tex]y = \dfrac{-1 \pm \sqrt5}2[/tex]

Since [tex]y>0[/tex], we take the positive root. Then

[tex]y = \sqrt{\dfrac{x-1}x} = \dfrac{-1 + \sqrt5}2[/tex]

Solve for [tex]x[/tex].

[tex]\left(\sqrt{\dfrac{x-1}x}\right)^2 = \left(\dfrac{-1 + \sqrt5}2\right)^2[/tex]

[tex]\dfrac{x-1}x = \dfrac{6 - 2\sqrt5}4 = \dfrac{3 - \sqrt5}2[/tex]

[tex]x - 1 = \dfrac{3-\sqrt5}2x[/tex]

[tex]\dfrac{-1+\sqrt5}2 x = 1[/tex]

[tex]x = \dfrac2{-1+\sqrt5}[/tex]

Rationalize to clean up the answer.

[tex]x = \dfrac2{-1+\sqrt5}\cdot\dfrac{-1-\sqrt5}{-1-\sqrt5} = \boxed{\dfrac{1+\sqrt5}2} \approx 1.6108[/tex]

i.e. the golden ratio constant.

Answer:

12

Step-by-step explanation:

Factoring out a 1/4 from 1/4a - 3, you get 1/4 (a - 12) as 12 * 1/4 = 3. Thus, ? = 12.

Answer:

1053

Step-by-step explanation:

The analysis of the kidney exists named urology.  The kidney exists as a bean-shaped organ. The Kidney exists metanephric in nature and is located retroperitoneal in position.

The organ current in the body which regulates the fluid concentration of the body exists named the kidney.

The nephron exists as the structural and operational unit of the kidney.

The sequence of the appearance of the kidney exists as pursues:-

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By using what we know about right triangles, we conclude that the height is 11ft.

We can think of this as a right triangle.

Where the length of the escalator is the hypotenuse

We know that the angle of depression is 42.51°, then the top angle of our right triangle is:

90° - 42.51° = 47.49°

Now, the height of the top of the escalator would be the adjacent cathetus to said angle, then we can use the relation:

cos(a) = (adjacent cathetus)/(hypotenuse)

Replacing what we know:

cos(47.49°)  = height/16ft

cos(47.49°) *16ft = height = 10.8ft

Rounding to the nearest foot, we get:

height = 11ft

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Answer:

Step-by-step explanation:

To find the x intercepts, substitute in 0 for y and solve for x

So,

2x^2+2x-4 = 0

2(x^2+x-4) = 0

2/2(x^2+x-4) = 0/2

x^2+x-4 = 0

(x-1) (x+2) = 0

now,

x-1 = 0

x = 1

or

x+2 = 0

x = -2

The x intercepts for the parabola (1,0), (-2,0)

Answer:

$1.40

Step-by-step explanation:

5.00-0.80=4.20

4.20 ÷ 3= 1.40

Answer:

84m

Step-by-step explanation:

The perimeter of a triangle is the sum of the length of the external sides.

Therefore, the perimeter of the triangle = 21m + 28m + 35m = 84m.

Hey there!

What is the formula for finding the perimeter of a triangle?
a + b + c = perimeter

What does the labels mean?
a = side 1

b = base

c = side 2

What should the equation look like?
21m + 28m + 35m = perimeter

How do should it be solve?
21m + 28m + 35m = perimeter

49m + 35m = perimeter

84m = perimeter

What is the answer to the question?

84m

Good luck on your assignment and enjoy your day!

~Amphitrite1040

Answer:

B

Step-by-step explanation:

if an angle is congruent, it is the same measurement

The mold the candlemaker can use to make the larger-sized candle is 480 cubic centimeters.

A cylinder is a surface created by projecting a closed two-dimensional curve along an axis intersecting the plane of the curve.

larger-sized candle = 2(smaller-sized candle)

= 2(240 cubic centimeters)

= 480 cubic centimeters

Therefore, the mold the candlemaker can use to make the larger-sized candle is 480 cubic centimeters.

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Answer:

A cylinder with a height that is double the height of the original mold, and a radius that is the same as the radius of the original mold.

Hope this helps!

Step-by-step explanation:

The last digit of the product of all the numbers between 11 and 29 is 0.

A product is something that has undergone one or more multiplications.

Here, we're looking for the final digit of the product of all whole numbers greater than 11 and less than 29.

Next, we have the product as: 12*13*14*15*16*17*18*19*20*21*22*23*24*25*26*27*28

Now take note of the 20 that is present.

Any number multiplied by 20 will result in a zero, so:

Product = 20*(12*13*14*15*16*17*18*19*21*22*23*24*25*26*27*28)

Using only that, we can infer that 0 represents the final digit in the product of all the numbers between 11 and 29.

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An example of the real world example of a rational function would be when a person would decide to share a cookie with their friend. Each of them would get half of the cookie each.

This is the term that is used to refer to the fraction that is represented by the use of a rational fraction. That is there is denominator and a numerator.

The denominator must have a variable and this variable has to be of degree 1. Hence the real world example of a rational function would be when a person would decide to share a cookie with their friend. Each of them would get half of the cookie each.

First you have to know if it is an inequality or not. Then you have to assign x and y variables to numbers. From here you form an equation. This equation is plotted on a graph.

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Concluding, there are (2 + 12) boxes of pencils left.

We know that Ms. Bloom had (4 + 7/12) boxes of pencils, and we also know that (2 + 1/12) of the boxes were broken.

The number of boxes of pencils left is given by the difference between the two above numbers, then we get:

N = (4 + 7/12) - (2 + 1/12) = (4 - 2) + (7/12 - 1/12) = 2 + 6/12

Now we can simplify the fraction:

N = 2 + 1/2

Concluding, there are (2 + 12) boxes of pencils left.

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the interval in which g(x) is positive is [3, ∞].

Here we have the function:

g(x) = ∛(x - 3)

The interval in which the function is positive is defined by:

g(x) > 0.

Then we have:

∛(x - 3) > 0.

We need to solve that inequality for x.

∛(x - 3) > 0

The cube root respects the sign of the argument, then:

x - 3 > 0

x > 3

Then the interval in which g(x) is positive is [3, ∞].

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The most appropriate model for a given situation is exponential.

The most appropriate model for a given situation is exponential.

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Answer:

  (a)  (f +g)(x) = √(x-2) +√(x)

Step-by-step explanation:

The sum of two functions is the sum of their definitions.

  (f +g)(x) = f(x) +g(x)

  (f +g)(x) = √(x -2) +√(x)

Note: the radicals can only be combined if they can be reduced to a form that has the same expression under the radical. Here, that is not the case, so there is no way to simplify the sum.

The Laplace of the given equation is [tex]y=-\frac{15e^2}{8}e^{2t}+\left(-3e^2+\frac{15e^2}{4}\right)e^{2t}t+\frac{e^2t^4}{4}+e^2t^3+\frac{9e^2t^2}{4}+3e^2t+\frac{15e^2}{8}[/tex].

According to the statement

we have given that the equation and we have to solve this problem with the help of the Laplace transform.

So, According to the statement

the given equation is

y'' − 4y' + 4y = t^3e^2t, y(0) = 0, y'(0) = 0

And the Laplace transform is an integral transform method which is particularly useful in solving linear ordinary differential equations.

Firstly fill the value of the Laplace of the second order derivative and put x = 0 in the equation

And same it with the first order of the derivative.

And then the Laplace of the equation become

[tex]y=-\frac{15e^2}{8}e^{2t}+\left(-3e^2+\frac{15e^2}{4}\right)e^{2t}t+\frac{e^2t^4}{4}+e^2t^3+\frac{9e^2t^2}{4}+3e^2t+\frac{15e^2}{8}[/tex]

So, The Laplace of the given equation is [tex]y=-\frac{15e^2}{8}e^{2t}+\left(-3e^2+\frac{15e^2}{4}\right)e^{2t}t+\frac{e^2t^4}{4}+e^2t^3+\frac{9e^2t^2}{4}+3e^2t+\frac{15e^2}{8}[/tex]

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Answer: 120

Step-by-step explanation:

Answer:

B. 60°

Step-by-step explanation:

Angle 4 and angle 8 are equal. Find the measure of angle 4 to know the measure of angle 8.

∠4=∠8

180-120=60

∠4=60°

Since angle 4 equals 60 degrees, angle 8 also equals 60 degrees.

Hope this helps!

Answer:

g(1) = 1

Step-by-step explanation:

Since 1 ≠ -2  and  1 ≠ -1

Then we must use the expression:

g(x) = x³ - x² + 1  ,to calculate g(1).

Therefore

g(1) = (1)³ - (1)² + 1

     = 1 - 1 + 1

     = 0 + 1

     = 1

The factoring the equation (a + 1)² - 4b² we get (a+1+2b)(a+1-2b).

Given:  (a + 1)² - 4b²

Apply Perfect Square Formula, and we get

(a + b)² = a² + 2 a b + b²

(a + 1)² = a² + 2a1+1²

= a² + 2a1 + 1² - 4 b²

simplifying the equation, we get

a² + 2a1 + 1² = a² + 2a + 1

= a² + 2a + 1 - 4b²

Factor, a² + 2a + 1 = (a + 1)²

= (a + 1)² - 4 b²

Simplify 4 b² = (2 b)²

= (a + 1)² - (2b)²

Apply the Difference of Two Squares Formula, and we get

x² - y² = (x + y)(x - y)

(a + 1)² - (2b)² = ((a + 1) + 2b)((a + 1) - 2 b)

simplifying the equation, we get

= ((a + 1) + 2b)(a + 1) - 2 b)

= (a + 1 + 2b)(a + 1 - 2b)

Therefore, the correct answer is (a+1+2b)(a+1-2b).

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Answer:

17

Step-by-step explanation:

the pattern

x is the blank

1+5 = 6

5+6 = 11

6+11 = x

11 + x = 28

so x = 17

The spherical coordinate (4,6π,2π/3) is (5,2π3,0.927) in spherical coordinates.

I’m going to assume that your values are in the form of:

(r,θ,z)=(4,6π,2π/3)

where

(r,θ)are the polar coordinates of the point’s projection in the xy-plane

z is the usual z-coordinate in the Cartesian coordinate system

Now solving for Cartesian coordinates:

x=rcosθ=4cos(6π)=4(0.86)=3.44

y=rsinθ=4sin(6π)=2

Now for spherical transformation, note that:

(ρ,θ,ϕ)

ρ is the distance between P and the origin

θ is the same angle used to describe the location in cylindrical coordinates

ϕ is the angle formed by the positive z-axis and line segment from the origin and point in space, note that0≤ϕ≤π

OK, now that we have that out of the way, let’s compute:

ρ=r2+z2 = 5

θ=θ=6π

ϕ=arccos(zr2+z2−−−−−−√)=arccos(35)≈0.927rad

The spherical coordinate (4,6π,2π/3) is (5,2π3,0.927) in spherical coordinates.

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Answer:

1413.86 ft^2.

Step-by-step explanation:

The diameter of the circle = 50 - 2(18)

= 50-36

=  14 ft.

So its radius is 7 ft.

The total area of the blue shape

= 2(35 * 18) + 3.14*7^2

= 1413.86 ft^2.

Answer: 1547.10

Step-by-step explanation: Since you are not in a physics class, in which case requires the usage of the radioactive decay formula, definite integrals, and natural logarithms) but an Algebra class which deploys the usage of percentages, this should be rather simple.

I believe they want you to find 27% of the half life of Carbon-14 since it is below 50%. To find the percent of a number (in this case 27%), you multiply the number by 0.27.

5730 * 0.27 = 1547.10

Hope this helped! And if I'm wrong, please don't hesitate to correct me.

The price of the product before promotional sales was 500,000.

To find what the previous price of the product was, we must take into account that we know that 87% of the value is equal to 435,000. So to find out how much the 13% (discount) that the product had is equivalent to, we do the following operation:

435,000 × 100 ÷ 87 = 500,000

So we know that the 13% discount was 65,000

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The height is changing by 0.5 units per minute as evident in the given table.

It follows from the task content that the table given indicates that at, 0min, the height was 150.

While at 1 min, the height is 150.5. On this note, it follows that the height increases by 0.5 units for after every minute.

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Answer:

To find the y-intercept, begin at the origin and move vertically to the graphed line. To find the slope, use two ordered pairs on the line and substitute into the equation m = StartFraction y 2 minus y 1 Over x 2 minus x 1 EndFraction.

The total surface area of the shape is 10053 cm²

From the question, we are to determine the surface area of the shape

The surface area of the shape = Surface area of hemispherical part + Surface are of the cylindrical part

∴ Surface area of the shape = 2πr² + 2πrh + πr²

Where r is the radius

and h is the height of the cylindrical part

From the given information.

r = 20

h = 50

Putting the parameters into the equation, we get

Surface area of the shape = 2π(20)² + 2π(20)(50) + π(20)²

Surface area of the shape = 2π×400+ 2π×1000 + π×400

Surface area of the shape = 800π+ 2000π + 400π

Surface area of the shape = 3200π cm²

Surface area of the shape = 10053.096 cm²

Surface area of the shape ≈ 10053 cm²

Hence, the total surface area of the shape is 10053 cm²

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