Find all values of k for which the equation 3x2−(k+2)x+k−1=0 has no solutions.

Answer:

D)

Step-by-step explanation:

Answer:

  B)  24/25

Step-by-step explanation:

You're asked to find the exact value of the composition ...

  sin(2·arccos(3/5))

You know the double angle formula for the sine is ...

  sin(2x) = 2sin(x)cos(x)

and you know the sine can be derived from the cosine by ...

  sin(x) = √(1 -cos(x)^2)

Since you're given the cosine as 3/5, this means you need to find the value of ...

  sin(2·arccos(3/5)) = 2·√(1 -(3/5)^2)·(3/5) = 2·(√(25-9))/5·3/5 = 2·4/5·3/5

  sin(2·arccos(3/5)) = 24/25 . . . . . . matches the marked answer

_____

If you do this on your calculator, you will find the result to be 0.96, which is the decimal value of 24/25.

Answer:

[tex]\pi\dfrac{12.1}{3}\left(\dfrac{12.1}{\tan 57^{\circ}}\right)^2[/tex]

Step-by-step explanation:

You are interested in two of the sides of the right triangle, the leg AC and the leg BC. The trig function that relates their values to the angle shown is told you by the mnemonic SOH CAH TOA, which reminds you ...

  Tan = Opposite/Adjacent

The side opposite the angle, BC is given as 12.1; the side adjacent is AC, designated r. Then the above relation tells you ...

  tan(57°) = 12.1/r

Rearranging, we have ...

  r = 12.1/tan(57°)

The volume of the cone is given by the formula ...

  V = (1/3)πr²h . . . . . where h = 12.1

Filling in what we know, this is ...

  V = (1/3)π(12.1/tan(57°))²·12.1

This can be rearranged to the form shown in your answer choices:

  V = π(12.1/3)(12.1/tan(57°))² . . . . . . matches the lower-right choice

Answer:

a.

Step-by-step explanation:

Answer:

  the appropriate choice is the last one

Step-by-step explanation:

[tex]7^{\frac{4}{5}}=(7^4)^{\frac{1}{5}}=\sqrt[5]{7^4}[/tex]

This is equivalent to b^(pq) with b=7, p=4, q=1/5. Here, q is a rational number, as suggested by the last answer choice.

Answer:

Step-by-step explanation:

The 9th term is the next-to-last term, where the left term of the binomial is raised to the first power, the right term of the binomial is raised to the 8th power (9-1=8), and the multiplier is 9C8 = 9!/(8!·1!) = 9. This product is ...

  9·(3x)^1·(-2y)^8 = 6912xy^8

By Stokes' theorem, the line integral of [tex]\vec F[/tex] over [tex]C[/tex] is equivalent to the surface integral of the curl of [tex]\vec F[/tex] over [tex]S[/tex], where [tex]S[/tex] is the part of the plane [tex]7x+6y+z=1[/tex] in the first octant, with [tex]S[/tex] having positive/upward orientation.

Parameterize [tex]S[/tex] by

[tex]\vec s(u,v)=\dfrac{(1-u)(1-v)}7\,\vec\imath+\dfrac{u(1-v)}6\,\vec\jmath+v\,\vec k[/tex]

with [tex]0\le u\le1[/tex] and [tex]0\le v\le1[/tex].

Take the normal vector to [tex]S[/tex] to be

[tex]\vec s_u\times\vec s_v=\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k[/tex]

The curl of [tex]\vec F[/tex] is

[tex]\nabla\times\vec F(x,y,z)=(x-y)\,\vec\imath-y\,\vec\jmath+\vec k[/tex]

Then the line integral is equivalent to

[tex]\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S[/tex]

[tex]=\displaystyle\int_0^1\int_0^1\left(\frac{(6-13u)(1-v)}{42}\,\vec\imath-\dfrac{u(1-v)}6\,\vec\jmath+\vec k\right)\cdot\left(\dfrac{1-v}6\,\vec\imath+\dfrac{1-v}7\,\vec\jmath+\dfrac{1-v}{42}\,\vec k\right)\,\mathrm du\,\mathrm dv[/tex]

[tex]=\displaystyle\frac1{252}\int_0^1\int_0^1(12-6v-19u+19uv)(1-v)\,\mathrm du\,\mathrm dv=\boxed{\frac{11}{1512}}[/tex]

First, find the normal vector of the plane, and then find the curl of the vector field to obtain a new vector field to be integrated over the given surface. Then perform the surface integral of this resultant vector field over the surface to get the required output.

Using Stokes' Theorem, we first find the normal vector of the plane 7x + 6y + z = 1 by extracting the coefficients, i.e., the vector is (7,6,1). Next, find the curl of the vector field F(x, y, z) = i + (x + yz)j + (xy - sqrt(z ))k. This provides the vector field, let's say G, that you are going to integrate over the surface that C bounds. The curl is calculated as follows:

The curl of F = ∇ x F= ( ∂/∂y [(xy - sqrt(z))] - ∂/∂z [(x + yz)] )i - ∂/∂x [(xy - sqrt(z))]j + ∂/∂x [(x + yz)]k.

Then perform the surface integral of this resultant vector field, G, over the surface bounded by C to get your answer. The surface integral is ∫∫ G·dS.

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Answer:

8 E -28

Step-by-step explanation:

Use the 2nd button and the comma button to get the "EE" which puts numbers into scientific notation for you.  I use parenthesis and this is what my screen looks like right before I hit "enter" to get the answer:

[tex](4.4 E-10)/(5.5E17)[/tex]

Hit enter and you get 8 E -28

Answer:

8E -28

.

Step-by-step explanation:

Answer:

12ºC

Step-by-step explanation:

Morning = -3ºC

At 7pm, temperature dropped ny 9ºC.

Temperature = -3 - 9 = -12ºC

Answer:

25.

Step-by-step explanation:

44:11*100 = 25

Answer:

25%

Step-by-step explanation:

Of means multiply and is means equals

P *44 =11

Divide each side by 44

P *44/44=11/44

P = 1/4

P = .25

Change from decimal form to percent form by multiplying by 100

P = 25%

ANSWER

[tex]\frac{41\pi}{90} [/tex]

EXPLANATION

To find how many degrees are in 82°,

We convert 82° to radians.

To do that we multiply by

[tex] \frac{\pi}{180 \degree} [/tex]

This implies that,

[tex]82 \degree =82 \degree \times \frac{\pi}{180 \degree} = \frac{41\pi}{90} [/tex]

Hence there are

[tex] \frac{41\pi}{90} [/tex]

radians in 82 degrees.

To solve this integral, one would establish parametric equations for the line segment between the points given, substitute the parametric equations into the vector field, and then compute the integral.

To solve the integral Integral from (2, 1, 2) to (6, 7, -5) y dx + x dy + 7 dz, it is necessary to establish the parametric equations for the line. The equation for a path in a three-dimensional space between two points can be defined as: r(t) = (1 - t)A + tB, where A and B represent the initial and final points respectively, and t is the parameter that varies between 0 and 1.

For our case the points A = (2, 1, 2) and B = (6, 7, -5), so r(t) would become r(t) = (1 - t)(2, 1, 2) + t(6, 7, -5) = (2 + 4t, 1 + 6t, 2 - 7t).

Subsequently, you need to calculate the line integral of F along this path. Here, F = yi + xj + 7k, and when the values of x, y, and z from the function r(t) are substituted into F, the result is F = (1 + 6t)i + (2 + 4t)j + 7k.

Finally, evaluate the line integral. Due to the conservative nature of the field, the value of the integral would be the same over any path leading from (2,1,2) to (6,7,-5)

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This answer calculates a line integral of a conservative vector field using the given parametric equations. The answer lies in plugging the parametric equations into the integral and calculating it out.

This question involves calculating a line integral for a vector field. The vector field represented here is F = yi + xj + 7k. Parametric equations for the line segment from (2,1,2) to (6,7,-5) can be written as x = 2 + 4t, y = 1 + 6t, z = 2 -7t with 0 ≤ t ≤ 1.

Next, we plug into the integral ∫F.dr from t=0 to t=1, where dx = 4dt, dy = 6dt and dz = -7dt. This gives us the integral ∫_{0}^{1} (ydx + xdy + 7dz). Inserting the parametric equations into this integral and calculating it out gives the value of the line integral.

This procedure embodies the idea that conservative vector fields have the property that the line integral is independent of the path, meaning the value of the integral depends only on the endpoints of the path and not on the specific path taken.

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Answer:

32.39

Step-by-step explanation:

27.84+15=42.84

42.84-10.45= THE ANSWER

[tex]\displaystyle\\\text{We have the points:}\\E(5,~8)\\F(12,~0)\\G(-5,~0)\\H(-12,~8)\\\\EF=\sqrt{(5-12)^2+(8-0)^2}=\sqrt{(-7)^2+8^2}=\\=\sqrt{49+64}=\sqrt{113}\approx\boxed{\bf10.63}\\\\FG=12-(-5)=12+5=\boxed{\bf17}\\\\GH=\sqrt{(-5-(-12))^2+(0-8)^2}=\sqrt{7^2+(-8)^2}=\\=\sqrt{49+64}=\sqrt{113}\approx\boxed{\bf10.63}\\\\HE=|-12-5|=|-17|=\boxed{\bf17}\\\\P=EF+FG+GH+HE=10.63+17+10.63+17=55.26\approx\boxed{\bf55.3}\\\\\text{Correct answer: }~B)[/tex]

Answer:

Step-by-step explanation:

Interest earned is proportional to the interest rate, so if the interest earned is the same, the amounts invested must be inversely proportional to the interest rates. That is, for the 3% and 2% accounts, the ratio of money invested is 2:3.

In other words, 2/5 of the money ($3200) was invested at 3%, and 3/5 of the money ($4800) was invested at 2%.

_____

If you need an equation, you can let x represent the amount invested at the highest rate. Then 8000-x is the amount invested at the lower rate. For the interest in the two accounts to be equal, we have ...

  3%·x = 2%·(8000-x) . . . . . the amounts of interest earned are the same

  3/2·x = 8000 -x . . . . . . . . divide by 2%

  5/2·x = 8000 . . . . . . . . . . . add x and simplify

  x = 8000·(2/5) = 3200  . . . multiply by the inverse of the x coefficient

  8000-x = 8000 -3200 = 4800 . . . . the amount invested at the lower rate

Tamara invested $3200 in the 3% account and $4800 in the 2% account.

_____

She earned $96 in each account for the year.

Answer:

The area of one side of the tank is [tex]25ft^{2}[/tex]

Step-by-step explanation:

A cube is a regular Polyhedron limited by six equal squares. It is also known as the regular hexahedron, or simply hexahedron.

The cube is a ortohedron, that is, it has its six faces straight and perpendicular to each other. In addition to being their rectangle faces, they are also square.

It has the following properties:

*Number of faces: 6

*Number of vertices: 6

*Number of edges: 12

*Number of edges from one vertex: 3

Being the length of the edge of the cube, then its Volume (V) is expressed by the formula: [tex]v=a^{3}[/tex] Being a the length of the edge of the cube

A fish tank is the shape of a cube. Its volume is [tex]125ft^{3}[/tex], to calculate the are of one side of the tank, we know that each faces has the form of a square of area [tex]a^{2}[/tex].

To calculate the value of a.

[tex]a=\sqrt[3]{125ft^{3} } =5ft[/tex]

Then the area of square is [tex]a^{2}[/tex]

Substituting the value of a

[tex](5ft)^{2} =25ft^{2}[/tex]

Answer:

25 feet squared

Step-by-step explanation:

Answer:

$665.36

Step-by-step explanation:

2^16 or...

2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2 = 66536 pennies

Answer:

9.78 u² . . . . . best matches selection B

Step-by-step explanation:

The formula for the area of a segment is ...

A = (1/2)r²(θ-sin(θ)) . . . . . θ is the central angle in radians, r is the radius

Filling in the given values, we have ...

A = (1/2)·(6√3)²(π/3 -sin(π/3)) = 54(π/3 -√3/2) ≈ 9.7832960 u²

_____

Inappropriate rounding of intermediate computation values will give a different result. The value of π/3-sin(π/3) is the small difference of relatively larger numbers, so minor errors in the values of those numbers can have a great effect on the difference.

Answer:

The correct options are:

So, these functions have the same range.

The functions have the same base.

The functions have the same domain.

g(x) is a translation left 1 unit.

Step-by-step explanation:

According to the table, both are exponential functions.

We have that

[tex]f(x) = 3^{x}[/tex]

[tex]g(x) = 3^{x+1}[/tex]

Lets see each affirmation:

The functions have the same base.

An exponential function [tex]a^{x}[/tex] has base a.

In this problems, both f and g have base 3.

The functions have the same range.

The range of f are all the values that f can assume. That is, all the positive numbers.

The range of g are all the values that g can assume. That is, also all the positive numbers.

So, these functions have the same range.

The functions have the same exponent.

An exponential function [tex]a^{x}[/tex] has exponent x.

f has exponent x and g has exponent x + 1. So those functions do not have the same exponent.

The functions have the same domain.

Yes, they both have x = {0,1,2,3} as domain.

g(x) is a translation left 1 unit.

g(x) = f(x+1). So yes, g(x) is a translation left 1 unit.

g(x) is a translation right 2 units.

g(x) is not f(x-2). So g(x) is not a translation right 2 units.

g(x) is a translation up 2 units.

g(x) is not f(x) + 2. So g(x) is not a translation up 2 units.

Answer:

A B D E

Step-by-step explanation:

Just did it!

90 degrees you are looking to your side

180 degrees you are looking behind you

around origin of 0,0

the image is flipped into the negative world if it is in posiitve or vice versa

The coordinates of the triangle after a rotation of 180° counterclockwise is given by P' ( -3 , 2 ) , Q' ( -8 , 2 ) , R' ( -5 , 5 )

The measure of the amount a figure is rotated about the center of rotation is called the angle of rotation. The angle of rotation is usually measured in degrees. We specify the degree measure and direction of a rotation.

90° clockwise rotation: (x,y) becomes (y,-x)

90° counterclockwise rotation: (x,y) becomes (-y,x)

180° clockwise and counterclockwise rotation: (x, y) becomes (-x,-y)

270° clockwise rotation: (x,y) becomes (-y,x)

270° counterclockwise rotation: (x,y) becomes (y,-x)

Given data ,

Let the rotation angle be represented as A

Now , the value of A is A = 180° counterclockwise

Let the triangle be represented as PQR

Now , the coordinates of the triangle PQR is

The coordinate of P = P ( 3 , -2 )

The coordinate of Q = Q ( 8 , -2 )

The coordinate of R = R ( 5 , -5 )

And , when 180° clockwise and counterclockwise rotation: (x, y) becomes (-x,-y)

Substituting the values in the equation , we get

The coordinate of point P' = P' ( -3 , 2 )

The coordinate of point Q' = Q' ( -8 , 2 )

The coordinate of point R' = R' ( -5 , 5 )

Hence , the coordinates of the triangle after rotation is

P' ( -3 , 2 ) , Q' ( -8 , 2 ) , R' ( -5 , 5 )

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Answer:

The average rate of change is [tex]=-4[/tex]

Step-by-step explanation:

The average rate of change from x=a to x=b of f(x) is given by;

[tex]=\frac{f(b)-f(a)}{b-a}[/tex]

We want to find the average rate of change of the function represented in the table  from x=-2 to x=2.

[tex]=\frac{f(2)-f(-2)}{2--2}[/tex]

From the table f(-2)=25 and f(2)=9

The average rate of change from x=-2 to x=2 is

[tex]=\frac{9-25}{2--2}[/tex]

[tex]=\frac{-16}{4}[/tex]

[tex]=-4[/tex]

Answer:

-4

Step-by-step explanation:

Answer:

⁵/₁₂

Step-by-step explanation:

1. The probability or rolling a number > 1 (given all sides have equal chances):

Possible outcomes: 1, 2, 3, 4, 5, 6; Wanted outcomes: 2, 3, 4, 5, 6

5/6 of the possible outcomes are wanted.

2. Chance of flipping heads (given both sides have an equal chance):

Possible outcomes: heads, tails; Wanted outcomes: heads

1/2 of the possible outcomes are wanted.

3. Probability of both: ⁵/₆ * ¹/₂ = ⁵/₁₂

Answer:

Step-by-step explanation:

There are a few formulas that are useful for this:

You will notice that for lateral area purposes, a pyramid or cone is equivalent to a prism or cylinder of height equal to half the slant height. And for volume purposes, the volume of a pyramid or cone is equal to the volume of a prism or cylinder with the same base area and 1/3 the height.

Since the measurements are given in cm, we will use cm for linear dimensions, cm^2 for area, and cm^3 for volume.

___

The heights of the cones at the top of the towers can be found from the Pythagorean theorem.

  (slant height)^2 = (height)^2 + (radius)^2

  height = √((slant height)^2 - (radius)^2) = √(10^2 -5^2) = √75 = 5√3

The heights of the pyramids can be found the same way.

  height = √(13^2 -2^2) = √165

___

Area

The total area of the castle will be ...

  total castle area = castle lateral area + castle base area

These pieces of the total area are made up of sums of their own:

  castle lateral area = cone lateral area + pyramid lateral area + cylinder lateral area + cutout prism lateral area

and ...

  castle base area = cylinder base area + cutout prism base area

So, the pieces of area we need to find are ...

Here we go ...

Based on the above discussion, we can add 1/2 the slant height of the cone to the height of the cylinder and figure the lateral area of both at once:

  area of one cone and cylinder = π·10·(18 +10/2) = 230π

  area of cylinder with no cone = top area + lateral area = π·1^2 +π·2·16 = 33π

  area of one pyramid = 4·4·(13/2) = 52

The cutout prism outside face area is equivalent to the product of its base perimeter and its height, less the area of the rectangular cutouts at the top of the front and back, plus the area of the inside faces (both vertical and horizontal).

  outside face area = 2((23+4)·11 -3·(23-8)) = 2(297 -45) = 504

  inside face area = (3 +(23-8) +3)·4 = 84

So the lateral area of the castle is ...

  castle lateral area = 2(230π + 52) +33π + 504 + 84 = 493π +692

  ≈ 2240.805 . . . . cm^2

The castle base area is the area of the 23×4 rectangle plus the areas of the three cylinder bases:

  cylinder base area = 2(π·5^2) + π·1^2 = 51π

  prism base area = 23·4 = 92

  castle base area = 51π + 92 ≈ 252.221 . . . . cm^2

Total castle area = (2240.805 +252.221) cm^2 ≈ 2493.0 cm^2

___

Volume

The total castle volume will be ...

  total castle volume = castle cylinder volume + castle cone volume + castle pyramid volume + cutout prism volume

As we discussed above, we can combine the cone and cylinder volumes by using 1/3 the height of the cone.

  volume of one castle cylinder and cone = π(5^2)(18 + (5√3)/3)

  = 450π +125π/√3 ≈ 1640.442 . . . . cm^3

 volume of flat-top cylinder = π·1^2·16 = 16π ≈ 50.265 . . . . cm^3

The volume of one pyramid is ...

  (1/2)4^2·√165 = 8√165 ≈ 102.762 . . . . cm^3

The volume of the entire (non-cut-out) castle prism is the product of its base area and height:

  non-cutout prism volume = (23·4)·11 = 1012 . . . . cm^3

The volume of the cutout is similarly the product of its dimensions:

  cutout volume = (23 -8)·4·3 = 180 . . . . cm^3

so, the volume of the cutout prism is ...

  cutout prism volume = non-cutout prism volume - cutout volume

  = 1012 -180 = 832 . . . .  cm^3

Then the total castle volume is ...

  total castle volume = 2·(volume of one cylinder and cone) + (volume of flat-top cylinder) +2·(volume of one pyramid) +(cutout prism volume)

  = 2(1640.442) + 50.265 +2(102.762) +832 ≈ 4368.7 . . . . cm^3

The total area of the castle is approximately 2493.026 square centimeters, and the total volume is around 4368.7 cubic centimeters.

The total area of the castle is the sum of its lateral and base areas. The lateral area is composed of the lateral areas of two cones, two pyramids, one cylinder, and the lateral area of a cutout prism. The base area consists of the bases of three cylinders and the base of the cutout prism.

For the lateral area:

Cone and Cylinder: The lateral area of one cone and cylinder combined is 230π square centimeters.

Flat-Top Cylinder: The area of the cylinder with a flat top is 33π square centimeters.

Pyramid: The lateral area of one pyramid is 52 square centimeters.

Cutout Prism: The outside face area is 504 square centimeters, and the inside face area is 84 square centimeters.

The total lateral area of the castle is 493π + 692 square centimeters, which is approximately 2240.805 square centimeters.

For the base area:

Cylinder Bases: The combined area of the bases of the three cylinders is 51π square centimeters.

Prism Base: The base area of the rectangular prism is 92 square centimeters.

The total base area of the castle is 51π + 92, approximately 252.221 square centimeters.

Therefore, the total area of the castle is the sum of the total lateral area and the total base area, which is approximately 2240.805 + 252.221 square centimeters, totaling around 2493.026 square centimeters.

For the volume:

Cylinder and Cone: The volume of one cylinder and cone is 450π + 125π/√3, approximately 1640.442 cubic centimeters.

Flat-Top Cylinder: The volume of the cylinder with a flat top is 16π, approximately 50.265 cubic centimeters.

Pyramid: The volume of one pyramid is 8√165, approximately 102.762 cubic centimeters.

Prism (Non-Cutout): The volume of the non-cutout prism is 23 × 4 × 11, equal to 1012 cubic centimeters.

Cutout Prism: The volume of the cutout prism is 1012 - 180, which is 832 cubic centimeters.

The total volume of the castle is the sum of the volumes of the components, totaling approximately 4368.7 cubic centimeters.

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Answer:

the answer is D

Step-by-step

14ab^3/7a^-2b^-1

= 14a^3b^4/7

=2a^3b^4

   Image points of A, B, C and D by reflecting them about x-axis will be A'(1, -3), B'(-2, 2), C'(-4, -5) and D'(2, 5).

        P(h, k) → P'(h, -k)

Following the rule for the reflection,

Image points by reflecting the points A, B, C and D will be,

A(1, 3) → A'(1, -3)

B(-2, -2) → B'(-2, 2)

C(-4, 5) → C'(-4, -5)

D(2, -5) → D'(2, 5)

        Therefore, image points of A, B, C and D by reflecting them about x-axis will be A'(1, -3), B'(-2, 2), C'(-4, -5) and D'(2, 5).

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Answer:

option D

2x4; -11

Step-by-step explanation:

Order of matrix is in form (m x n), here m is the row and n is the column of the matrix.

So this matrix have 2 rows and 4 columns

1)Order of matrix

2x4

2)[tex]a_{12}[/tex]

here 1 is the row and 2 is the column

-11

Each elements of the matrix can be identity as below

[tex]\left[\begin{array}{cccc}x_{11} &x_{12} &x_{13} &x_{14} \\x_{21} &x_{22} &x_{23} &x_{24} \\\end{array}\right][/tex]

The answers are:

First image:

The answer is the second option, the angles is [tex]53\°[/tex]

Second image:

The answer is the third option:

[tex]\frac{5}{13}[/tex]

Third image:

The length of the adjacent leg is the first option:

[tex]8\sqrt{2}units[/tex]

Fourth image:

The answer is the fourth option, [tex]72\°[/tex]

Fifth image:

The answer is the fourth option, DF (hypothenuse) is equal to 25 units.

To solve these problems, we need to use the following trigonometric identities and the Pythagorean Theorem, since we are working with right triangles.

[tex]Tan(\alpha)=\frac{y}{x}\\\\(Tan(\alpha))^{-1} =(\frac{y}{x})^{-1}\\\\\alpha =Arctan(\frac{y}{x})[/tex]

[tex]Sin(\alpha)=\frac{opposite}{hypothenuse}[/tex]

Pythagorean Theorem:

[tex]c^{2}=a^{2} +b^{2}[/tex]

So, solving we have:

First image:

We are given a right triangle that has the following lengths:

[tex]base=x=6units\\height=y=8units\\hypothenuse=10units[/tex]

Then, calculating we have:

[tex]\alpha =Arctan(\frac{y}{x})\\\\\alpha =Arctan(\frac{8}{6})\\\\\alpha =Arctan(1.33)\\\\\alpha =53\°[/tex]

Hence, the answer is the second option, the angles is [tex]53\°[/tex]

Second image:

We are given a right triangle that has the following lengths:

[tex]base=x=12units\\height=y=5units\\hypothenuse=13units[/tex]

Then calculating the sin ratio, we have:

[tex]Sin(\alpha)=\frac{opposite}{hypothenuse}[/tex]

[tex]Sin(\alpha)=\frac{5}{13}[/tex]

Thence, the answer is the third option:

[tex]\frac{5}{13}[/tex]

Third Image:

We are given the following information:

[tex]hypothenuse=16units\\\\\alpha =45\°[/tex]

Then, calculating one of the angle legs, since both will have the same length, using the sine trigonometric identity, we have:

[tex]Sin(\alpha)=\frac{Opposite}{Hypothenuse}\\ \\Sin(45\°)=\frac{Opposite}{16}\\\\Opposite=Sin(45\°)*16\\\\Opposite=\frac{\sqrt{2} }{2}*16=8\sqrt{2}[/tex]

Hence, the answer is the first option the length of the adjacent leg is

[tex]Opposite=\frac{\sqrt{2} }{2}*16=8\sqrt{2}units[/tex]

Fourth image:

We are given the following information:

[tex]base=x=9units\\height=y=3units[/tex]

To calculate the angle at the B vertex, first, we need to calculate the angle at the C vertex, and then, calculate the B vertex by the following way:

Since the sum of all the interior angles of a triangle are equal to 180°, we have that:

[tex]180\°=Angle_{B}+Angle{C}+90\°[/tex]

[tex]Angle_{B}=180\° -90\°-Angle_{C}[/tex]

So, calculating the angle at the C vertex, we have:

[tex]\alpha =Arctan(\frac{y}{x})[/tex]

[tex]\alpha =Arctan(\frac{3}{9})[/tex]

[tex]\alpha =Arctan(0.33)=18.26\°[/tex]

Then, calculating the angle at the B vertex, we have:

[tex]Angle_{B}=180\° -90\°-18.26\°=71.74\°=71.8\°=72\°[/tex]

Hence, the answer is the fourth option, [tex]72\°[/tex]

Fifth image:

We are given the following information:

[tex]base=x=24units\\height=y=7units[/tex]

Now, to calculate the distance DF (hypothenuse) we need to use the Pythagorean Theorem:

[tex]c^{2}=a^{2} +b^{2} \\\\hypothenuse^{2}=adjacent^{2}+opposite^{2}\\\\\sqrt{hypothenuse^{2}}=\sqrt{adjacent^{2}+opposite^{2}}\\\\hypothenuse=\sqrt{adjacent^{2}+opposite^{2}}[/tex]

Then, substituting we have:

[tex]hypothenuse=\sqrt{24^{2}+(7)^{2}}[/tex]

[tex]hypothenuse=\sqrt{576+49}=\sqrt{625}[/tex]

[tex]hypothenuse=\sqrt{625}[/tex]

[tex]hypothenuse=25units[/tex]

Hence, the answer is the fourth option, DF (hypothenuse) is equal to 25 units.

Have a nice day!

Answer:

  11.2 miles

Step-by-step explanation:

The mnemonic SOH CAH TOA reminds you that ...

  Tan = Opposite/Adjacent

The side you are given is adjacent to the given angle, and the side you are asked to find is opposite the given angle. You can fill in the given values and solve for the required measure.

  tan(66°) = BC/(5 mi)

Multiplying by 5 mi gives ...

  BC = (5 mi)·tan(66°) = (5 mi)·2.246 = 11.23 mi ≈ 11.2 mi

Answer:

the answer is 1/5 because y/x gives you k

The constant of proportionality [tex]k[/tex] is [tex]\frac{1}{5}[/tex].

What is constant of proportionality.

The constant of proportionality is the constant value of the ratio between two proportional quantities.

                                                [tex]$k=\frac{y}{x}$[/tex]

Where [tex]k[/tex] is the constant of proportionality.

It is given that the table which shows the relationship [tex]y=kx[/tex] is,

x y

60 12

45 9

30 6

15 3

We have to find the value of [tex]k[/tex].

According to the formula of constant of proportionality,

Let,

[tex]${{k}_{1}}=\frac{{{y}_{1}}}{{{x}_{1}}}$[/tex]

[tex]${{k}_{1}}=\frac{12}{60}[/tex]

[tex]$\therefore {{k}_{1}}=\frac{1}{5}$[/tex]

Now,

Let,

[tex]${{k}_{2}}=\frac{{{y}_{2}}}{{{x}_{2}}}$[/tex]

[tex]${{k}_{2}}=\frac{9}{45}[/tex]

[tex]$\therefore {{k}_{2}}=\frac{1}{5}$[/tex]

The third is,

[tex]${{k}_{3}}=\frac{6}{30}[/tex]

[tex]$\therefore {{k}_{3}}=\frac{1}{5}$[/tex]

and

[tex]${{k}_{4}}=\frac{3}{15}[/tex]

[tex]$\therefore {{k}_{4}}=\frac{1}{5}$[/tex]

Hence, the Option [tex]$C$[/tex] is correct answer.

Learn more about constant of proportionality:

https://brainly.com/question/19775117?referrer=searchResults

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Answer:

  see the attachment

Step-by-step explanation:

• When two secants intersect a circle the measure of the angle between them is half the difference of the intercepted arcs. Tangent DH is a degenerate case of a secant in which both points of intersection are the same point (D). Thus ∠GHD is half the difference of arc GD and ED.

• We assume points B and C are on radii DA and EA, respectively. Hence arc DE has the same measure as ∠BAC, and ∠DGE has half that measure.

• Tangent DH will intersect radius DA at right angles. Hence ∠ADH = 90°.

• ∠DHE = ∠DHG, whose measure is given by the formula in the first answer choice. The formula in the 4th choice is different (and wrong).

• ∠DGE and ∠BFC both intercept arcs with the same measure. Neither is greater than the other: they are equal in measure.

• Inscribed triangle DGE cannot be a right triangle because none of its sides is a diameter of the circumcircle.

Answer:

ghd bac adh

Step-by-step explanation:

Answer is lots of god Which equation is the inverse of y = 100 – x2?