Solve this question please for me

f (x) - 2/3x - 7 find the inverse

The distance between point T (-5, 1) and point I (-1, 1) is 4 units.

To find the distance between two points, we can use the distance formula, which is derived from the Pythagorean theorem. The formula is:

Distance = √((x2 - x1)² + (y2 - y1)²)

Let's apply this formula to find the distance between point T (-5, 1) and point I (-1, 1):

x1 = -5, y1 = 1 (coordinates of point T)

x2 = -1, y2 = 1 (coordinates of point I)

Plugging these values into the formula, we have:

Distance = √((-1 - (-5))² + (1 - 1)²)

        = √(4² + 0²)

        = √(16 + 0)

        = √16

        = 4

Therefore, the distance between point T (-5, 1) and point I (-1, 1) is 4 units.

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solve for which is greater:

volume of cylinder - volume of cone

942.48 cm³ - 314.16 cm³

628.32 cm³

       Therefore, cylinder has larger volume by 628.32 cm³

Answer:

volume of a cylinder: \(\pi r^2h\)

volume of a cone: \(\dfrac13\pi r^2h\)

(where r is the radius and h is the height)

\(\pi r^2\) is the formula for the area of a circle

Cylinder

\(\implies V=\pi r^2h=\pi \times 5^2 \times 12=300\pi \ \textsf{cm}^3\)

Cone

\(\implies V=\dfrac13\pi r^2h=\dfrac13 \times \pi \times 5^2 \times 12=100\pi \ \textsf{cm}^3\)

Comparing the two volumes, the volume of the cylinder is three times the volume of the cone.

Rational expressions are often used in combining rates of work. Therefore, it's true.

It should be noted that a rational expression is simply defined by a rational fraction.

They're are used in combining rates of work. Fir example, if Mr John performs 1/2 of his work and does 1/3 on another day. This can be expressed as:

= 1/2 + 1/3

= 5/6

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\((x^2 - y^2) \, dx + 2xy \, dy = 0\)

Multiply both sides by \(\frac1{x^2}\).

\(\left(1 - \dfrac{y^2}{x^2}\right) \, dx + \dfrac{2y}x \, dy = 0\)

Substitute \(y=vx\), so \(v=\frac yx\) and \(dy=x\,dv+v\,dx\).

\((1-v^2) \, dx + 2v (x\,dv + v\,dx) = 0\)

\((1 + v^2) \, dx + 2xv \, dv = 0\)

Separate the variables.

\(2xv\,dv = -(1 + v^2) \, dx\)

\(\dfrac{v}{1+v^2}\,dv = -\dfrac{dx}{2x}\)

Integrate both sides

\(\displaystyle \int \frac{v}{1+v^2}\,dv = -\frac12 \int \frac{dx}x\)

On the left side, substitute \(w=1+v^2\) and \(dw=2v\,dv\).

\(\displaystyle \frac12 \int \frac{dw}w = -\frac12 \int\frac{dx}x\)

\(\displaystyle \ln|w| = -\ln|x| + C\)

Solve for \(w\), then \(v\), then \(y\).

\(e^{\ln|w|} = e^{-\ln|x| + C}\)

\(w = e^C e^{\ln|x^{-1}|}\)

\(w = Cx^{-1}\)

\(1 + v^2 = Cx^{-1}\)

\(1 + \dfrac{y^2}{x^2} = Cx^{-1}\)

\(\implies \boxed{x^2 + y^2 = Cx}\)

Your mistake is in the first image, between third and second lines from the bottom. (It may not be the only one, it's the first one that matters.)

You incorrectly combine the fractions on the left side.

\(\dfrac1{-2v} -\dfrac v{-2} = \dfrac1{-2v} - \dfrac{v^2}{-2v} = \dfrac{1-v^2}{-2v} = \dfrac{v^2-1}{2v}\)

Step-by-step explanation:

Set up f(x).

\( \sqrt[3]{11x - 3} = f(x)\)

Replace f(x) with y

\( \sqrt[3]{11x - 3} = y\)

Swap x and y

\( \sqrt[3]{11y - 3} = x\)

Solve for y

Cube both sides

\(11y - 3 = {x}^{3} \)

Add 3 to both sides

\(11y = {x}^{3} + 3\)

Divide both sides by 11

\(y = \frac{x {}^{3} + 3 }{11} \)

so

\(f {}^{ - 1} (x )= \frac{ {x}^{3} + 3}{11} \)

What is ratio?

A ratio is a way of comparing two or more values or quantities. It expresses the relationship between the size of one value or quantity to the size of another. Ratios are often written as fractions or as ":".

The Federal Reserve is making nickels with a constant of proportionality. This can be determined by analyzing the relationship between the amount of nickels made and the time elapsed. If the number of nickels made is directly proportional to the time elapsed, then the ratio between the number of nickels made and the time elapsed will be constant.

As we can see from the table, the ratio between the number of nickels made and the time elapsed is constant, which is 6000 nickels per hour. This means that the Federal Reserve is making nickels with a constant of proportionality.

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The Federal Reserve is making nickels with a constant of proportionality.

What is ratio?

A ratio is a way of comparing two or more values or quantities. It expresses the relationship between the size of one value or quantity to the size of another. Ratios are often written as fractions or as ":".

The Federal Reserve is making nickels with a constant of proportionality. This can be determined by analyzing the relationship between the amount of nickels made and the time elapsed. If the number of nickels made is directly proportional to the time elapsed, then the ratio between the number of nickels made and the time elapsed will be constant.

As we can see from the table, the ratio between the number of nickels made and the time elapsed is constant, which is 6000 nickels per hour. This means that the Federal Reserve is making nickels with a constant of proportionality.

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The fish tank has a total volume of 288 inch³. As a result, Jessica's new fish tank has a capacity of 288 inch³ for water.

The volume of a rectangular prism can be calculated using the formula:

V = l x b x h..........(i)

where,

V ⇒ Volume

l  ⇒ length

b ⇒ width

h ⇒ height

From the question, we are given the values,

l = 8 inches

b = 4 inches

h = 9 inches

Putting these values in equation (i), we get,

V = 8 x 4 x 9

⇒ V = 288 in³

Therefore, the fish tank has a total volume of 288 inch³. As a result, Jessica's new fish tank has a capacity of 288 inch³ for water.

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Not all intersecting lines are perpendicular.

Perpendicular lines require a 90-degree angle of intersection, creating the formation of right angles.

Nonetheless, unlike perpendicular lines that necessitate exactly 90 degrees of intersections, other types of driven lines can be at varying angles beside these.

As such, it is critical to highlight that in general intersecting lines come with different angles of intersection, and only those featuring exact 90-degree angle of intersection become normal cases of intersecting lines, becoming one among many subtypes existing today.

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the mοnthly periοdic interest rate, rοunded tο the nearest hundredth οf a percent is (C) 1.31%

Any lοans and bοrrοwings cοme with interest. the percentage οf a lοan balance that lenders use tο determine interest rates. Cοnsumers can accrue interest thrοugh lending mοney (via a bοnd οr depοsit certificate, fοr example), οr by making a depοsit intο a bank accοunt that pays interest.

We must divide its yearly percentage rate (APR) by 12 tο determine a mοnthly periοdic interest rate (the number οf mοnths in a year).

Hence, the periοdic interest rate fοr each mοnth is:

15.75% / 12 = 1.3125%

The result οf rοunding tο the clοsest hundredth οf such a percent is:

1.31%

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The dilation transformation of the triangle ABC by a scale factor of 3, with the point P as the center of dilation indicates;

Side A'B' will be parallel to side AB

Side A'C' will be parallel to side AC

Side BC will lie on the same line as side BC

A dilation transformation is one in which the dimensions of a geometric figure are changed but the shape of the figure is preserved.

The possible options, from a similar question on the internet are;

Be parallel to

Be perpendicular to
Lie on the same line as

The location of the point P, which is the center of dilation, and the lines PC and PA of dilation and the scale factor of dilation indicates that we get;

PB' = 3 × PB

PA' = 3 × PA

PC' = 3 × PC

Therefore; The side B'C' will be on the same line as the side BC

The Thales theorem, also known as the triangle proportionality theorem indicates that;

The side A'C' will be parallel to the side AC

The side A'B', will be parallel to the side AB

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Answer:

B. xy(y - x) (y + x)

Step-by-step explanation:

Solution :

Let x, y, z be the dimensions of the rectangle.

Volume of the rectangle (V) = xyz

Given that the vertex should lie on ellipse \($4x^2 + y^2+4z^2=4$\) .......(i)

So here the volume xyz must be maximum with constraints above.

We solve this using Lagranches method with variable λ

Lagranches function is

\($F : xyz + \lambda(4x^2+y^2+4z^2-4)=0$\) .....(ii)

To find λ, \($\frac{dF}{dx}=0; \frac{dF}{dy}=0;\frac{dF}{dz}=0$\)

\($\frac{dF}{dx}=0 \Rightarrow yz+\lambda(16x)=0 \Rightarrow \lambda = -\frac{yz}{16x}$\) .............(iii)

\($\frac{dF}{dy}=0 \Rightarrow xz+\lambda(2y)=0 \Rightarrow \lambda = -\frac{xz}{2y}$\) ..................(iv)

\($\frac{dF}{dz}=0 \Rightarrow xy+\lambda(16z)=0 \Rightarrow \lambda = -\frac{xy}{16z}$\) ...............(v)

Equating (iii) and (iv)

\($-\frac{yz}{16x}=-\frac{xz}{2y} \Rightarrow y^2=8x^2 \Rightarrow y = \sqrt8 x$\) ...............(vi)

Equating (iii) and (v)

\($-\frac{yz}{16x}=-\frac{xy}{16z} \Rightarrow z^2=x^2 \Rightarrow z = x$\) ....................(vii)

Substitute (vi) and (vii) in (i),

From (i),

\($4x^2 + (\sqrt8 x)^2+4x^2 = 4$\)

\($\Rightarrow 4x^2 +8x^2+4x^2 = 4 \Rightarrow x = \frac{1}{4}$\)

From (vi),

\($y = \sqrt8 x$\)

\($\Rightarrow y= \sqrt8 \times \frac{1}{4} \Rightarrow y =\frac{\sqrt8}{4}$\)

\($\Rightarrow y =\frac{\sqrt{2\times4}}{4} \Rightarrow y = \frac{1}{\sqrt2}$\)

From (vii),

z = x

\($z =\frac{1}{4}$\)

Therefore, for maximum volume the dimensions of a rectangle box are

\($x =\frac{1}{4} ; y = \frac{1}{\sqrt2} ; z =\frac{1}{4}$\)

Answer:

1191 cm²

Step-by-step explanation:

Area of parallelogram = base X vertical height.

Call the vertical line h.

h/ sin 80  =  7/ sin 10

h = (7 sin 80) / sin 10

= 39.7.

Area of parallelogram = 30 X 39.7

= 1191 cm²

The Area of Parallelogram whose side length is 30 cm is 1,190.952 square cm.

Given:

Side length = 30 cm

Using Trigonometry

tan 80 = P / Base

5.6712 = P / 7

P = 39.6984 cm

Using the formula for Area of parallelogram

= Base x height

= 30 x 39.6984

= 1,190.952 square cm.

Therefore, the area is 1,190.952 square cm.

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Answer:

The polar equation of the curve is:

\(r=8c*cos(\theta)\)

Step-by-step explanation:

In polar form x and y could be written as:

\(x=rcos(\theta)\)

\(y=rsin(\theta)\)

Taking the square of each value we have:

Let's recall that \(cos^{2}(\theta)+sin^{2}(\theta)=1\)

\(x^{2}+y^{2}=r^{2}(cos^{2}(\theta)+sin^{2}(\theta))\)

\(x^{2}+y^{2}=r^{2}\)

Then, the cartesian equation is rewritten as:

\(x^{2}+y^{2}=8cx\)

\(r^{2}=8c*rcos(\theta)\)

Therefore, the polar equation of the curve is:

\(r=8c*cos(\theta)\)

I hope it helps you!

Answer: 0.5 0.333

Step-by-step explanation:

use demos graphing calculator and find the points that go through the axis.

Answer:y=−2x3+13y=−2x3+13

Step-by-step explanation:

Answer:

a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:

r^2 - 2r - 3 = 0

Factoring, we get:

(r - 3)(r + 1) = 0

So the roots are r = 3 and r = -1.

The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:

y_h = c1e^3x + c2e^(-x)

To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = Ae^4x

Taking the first and second derivatives of y_p, we get:

y_p' = 4Ae^4x

y_p'' = 16Ae^4x

Substituting these into the original differential equation, we get:

16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x

Simplifying, we get:

5Ae^4x = e^4x

So:

A = 1/5

Therefore, the particular solution is:

y_p = (1/5)*e^4x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^3x + c2e^(-x) + (1/5)*e^4x

b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:

r^2 + r - 2 = 0

Factoring, we get:

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

So the roots are r = -2 and r = 1.

The general solution to the homogeneous equation y'' + y' - 2y = 0 is:

y_h = c1e^(-2x) + c2e^x

To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax + B)e^x

Taking the first and second derivatives of y_p, we get:

y_p' = Ae^x + (Ax + B)e^x

y_p'' = 2Ae^x + (Ax + B)e^x

Substituting these into the original differential equation, we get:

2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x

Simplifying, we get:

3Ae^x = 3xe^x

So:

A = 1

Therefore, the particular solution is:

y_p = (x + B)e^x

Taking the derivative of y_p, we get:

y_p' = (x + 2 + B)e^x

Substituting back into the original differential equation, we get:

(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x

Simplifying, we get:

-xe^x - Be^x = 0

So:

B = -x

Therefore, the particular solution is:

y_p = xe^x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^(-2x) + c2e^x + xe^x

c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:

r^2 - 9r + 20 = 0

Factoring, we get:

(r - 5)(r - 4) = 0

So the roots are r = 5 and r = 4.

The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:

y_h = c1e^4x + c2e^5x

To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax^2 + Bx + C)e^4x

Taking the first and second derivatives of y_p, we get:

y_p' = (2Ax + B)e^4x + 4Axe^4x

y_p'' = 2Ae^4x +

The equation is x=-6

Step-by-step explanation:

In standard form the equation is y=4

And this is a horizontal line so the perpendicular line will be a vertical line of the form x=a

so that is why the equation is x=-6

Answer:

= -5

Step-by-step explanation:

Answer:

$1,061

Step-by-step explanation:

2% of 1000 is 20.

If 2% is compounded yearly, then

year one- $1,020

year two- $1,040.40

year three- $1,061.21

Rounded to the nearest whole dollar, Elaine would have $1,061 in her savings account after three years.

Answer:

1061

Step-by-step explanation:

The width of the two gardens will be equal to 5 feet.

A rectangle is a quadrilateral having four sides and the sum of the angles is 180 in the rectangle the opposite two sides are equal and parallel and the two sides are at 90-degree angles.

The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the rectangle in a two-dimensional plane is called the area of the rectangle.

Given that a rectangular garden has a length twice as great as it’s width. a second rectangular garden has the same width as the first garden and a length that is 4 meters greater than the length of the first garden. the second garden has an area of 70 square meters.

For the first rectangle, the  dimensions will be,

L = 2w

For second rectangle width is same then the area is,

L x w = 70

( 2w +4 ) x w = 70

2w² + 4w - 70 = 0

w² + 2w -35 = 0

w² + 7w - 5w -35 = 0

w(w + 7 ) -5(w+7) = 0

w =-7 and  w = 5

Hence, the width is w = 5 ft.

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Answer:

Step-by-step explanation:

Let's retype that as y = b^x.  The inverse of this function is log    y = x  

                                                                                                        b

which you could also write as  y = ln     x

                                                             b

If you had y = e^x, the inverse function would be y = ln x.

Note that if you let ln x be the input to y = e^x, the result would be just 'x,' which confirms that these two functions are inverses of one another.

The function that grows at the fastest rate for increasing values of x is f(x) = 4×2ˣ.

We can see this by comparing the growth rates of the three functions for larger and larger values of x.

As x gets larger, the exponential function f(x) grows much faster than the other two functions, which are both polynomial functions.

if we plug in x = 10, we get:

f(10) = 4×2¹⁰ = 4×1024 = 4096

h(10) = 9×10² + 25 = 925

g(10) = 15×10 + 6 = 156

As we can see, f(10) is much larger than h(10) and g(10), indicating that f(x) grows at a much faster rate than the other two functions.

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Answer:

6610

Step-by-step explanation:

We have tan(X) = opposite/ adjacent

tan(QBP) = PQ/BQ

tan(35) = PQ/BQ    ---eq(1)

tan(QAP) = PQ/AQ

tan(20) = \(\frac{PQ}{AB +BQ}\)

\(=\frac{1}{\frac{AB+BQ}{PQ} } \\\\=\frac{1}{\frac{AB}{PQ} +\frac{BQ}{PQ} } \\\\= \frac{1}{\frac{230}{PQ} + tan(35)} \;\;\;(from\;eq(1))\\\\= \frac{1}{\frac{230 + PQ tan(35)}{PQ} } \\\\= \frac{PQ}{230+PQ tan(35)}\)

230*tan(20) + PQ*tan(20)*tan(35) = PQ

⇒ 230 tan(20) = PQ - PQ*tan(20)*tan(35)

⇒ 230 tan(20) = PQ[1 - tan(20)*tan(35)]

\(PQ = \frac{230 tan(20)}{1 - tan(20)tan(35)}\)

\(= \frac{230*0.36}{1 - 0.36*0.7}\\\\= \frac{82.8}{1-0.25} \\\\=\frac{82.8}{0.75} \\\\= 110.4\)

PQ = 110.4

≈110

Height above sea level = 6500 + PQ

6500 + 110

= 6610

Answer:

\(8x^{2} +10x\).

Step-by-step explanation:

Split them into sections.

The first one is \(x(2x+1)\), which is \(2x^{2} +x\).

The second one is \((2x+x)(2x+3)\), which is \(6x^{2} +9x\).

Add \(2x^{2} +x\) and \(6x^{2} +9x\), which is equal to \(8x^{2} +10x\).

Hope I helped!

The total distance covered is 15000 m. The total time taken is 5220 seconds and the speed is 2.87 m/s.

We know that speed has to do with the ratio of the distance that has been covered to the time taken. The speed is a scalar quantity and as such we do not consider the direction hence we would not consider that going back in the opposite direction. The total distance can be obtained algebraically.

The total distance that the student travelled can be obtained from;

5000 m + 5000 m + 5000 m

= 15000 m

The total time taken in seconds = 1800 seconds + 1800 seconds + 1620 seconds

= 5220 second

Speed = Distance/ Time

= 15000 m/5220 second

= 2.87 m/s

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Answer:

x = 4.76°

Step-by-step explanation:

\( \tan x = \frac{1}{12} \\ \\ x = { \tan}^{ - 1} \bigg(\frac{1}{12} \bigg) \\ \\ x = 4.7636416907 \degree \\ \\ x = 4.76\degree \)

Answer:

10

Step-by-step explanation:

The 5th term in the expansion is 126 * (4x³y²)^5 * (-1/2y)^4, which simplifies to 126 * 4^5 * (x³)^5 * (y²)^5 * (-1/2)^4 * y^4.

To find the 5th term in the expansion of (4x³y² - 1/2y)^9, we can use the binomial theorem. The binomial theorem allows us to expand a binomial raised to a positive integer exponent.

The binomial theorem states that for any binomial (a + b)^n, the general term in the expansion can be written as C(n, k) * a^(n-k) * b^k, where C(n, k) represents the binomial coefficient, also known as "n choose k."

In our case, the binomial is (4x³y² - 1/2y)^9. The first term has an exponent of 9, so the last term will have an exponent of 0. Since we are looking for the 5th term, the exponent of y will be 9 - 5 = 4. The exponent of x will be 5, as 4x³ raised to the power of 5 gives x^15.

Using the binomial coefficient, we have C(9, 5) = 126.

Simplifying further, we get 126 * 1024 * x^15 * y^10 * (1/16) * y^4, which can be simplified as 126 * 64 * x^15 * y^14 * (1/16). The final result is 8064 * (x^15 * y^14) * (1/16), which can be further simplified if necessary.

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Answer:

))))))

Step-by-step explanation:

43+2(3-2)=43+2*3-2*2=43+6-4=45

Answer:

45

Step-by-step explanation:

43 + 2(3-2)

43 +2 (1)

43 + 2

45