Determine and prove what shape is formed for the given coordinates for ABCD, and then find the perimeter and area as an exact value.

A(24, −8), B(12, −17), C(3, −5), D(15, 4)

The shape is a .

The perimeter of ABCD is .

The area of ABCD is .

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:

3 feet long

Step-by-step explanation:

Answer: Brad's shadow is 3 feet long

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:

The length of the field is 84 yards

Step-by-step explanation:

Since the area of a rectangle is width times the length, we can write a simple equation.

Let w be the width

Let l be the length

W*L = 4284

Since we know the width of the rectangle is 51 yards, we can plug it in.

51L = 4284

Dividing 51 on both sides,

L = 84

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:

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:

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 blue and green marbles are evenly matched, fifty and fifty

each time a single colored pair is drawn, there is now an "extra" unpaired marble of the other color this extra marble must eventually be drawn along with one of the same color

the number of marbles in the A and B boxes can be anywhere from zero to fifty, but it is the SAME for both boxes

Answer:

10 friends

Step-by-step explanation:

we know that

The formula of the sum is equal to

[tex]sum=\frac{n}{2}[2a1+(n-1)d][/tex]

where

a1 is the first term

n is the number of terms (number of friends)

d is the common difference in the arithmetic sequence

In this problem we have

[tex]sum=275\ stickers[/tex]

[tex]a1=5\ stickers[/tex]

[tex]d=5[/tex] ----> the common difference

substitute in the formula and solve for n

[tex]275=\frac{n}{2}[2(5)+(n-1)(5)][/tex]

[tex]550=n[10+5n-5]\\ \\550=10n+5n^{2} -5n\\ \\5n^{2}+5n-550=0[/tex]

Solve the quadratic equation by graphing

The solution is n=10

see the attached figure

therefore

She had 10 friends who got stickers

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)

https://brainly.com/question/33114105

#SPJ12

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.

https://brainly.com/question/34566674

#SPJ11

Answer:

It is the last one. It is clearly less than 90 degrees.

Step-by-step explanation:

i would pick D as the answer.

The events of Lynn choosing a shirt and choosing a necklace are independent. The choice of one item does not affect the choice of the other. The correct answer is D) Independent - Randomly selecting a shirt does not affect randomly selecting a necklace.

The question asks whether the events of Lynn choosing a shirt and necklace are dependent or independent. In general, events are called independent if the outcome of one does not affect the outcome of another. In this scenario, picking a shirt does not change the options of necklaces and vice versa. Thus, the selection of a shirt has no effect on the selection of a necklace.

The correct answer is therefore: Independent. Randomly selecting a shirt does not affect randomly selecting a necklace. Even if Lynn picked a red, blue, or yellow shirt first, it wouldn't affect her choice of necklace. She will still have the same two necklaces to choose from regardless of what shirt she picks.

https://brainly.com/question/30905572

#SPJ3

Answer:

D)

Step-by-step explanation:

Answer:

32.39

Step-by-step explanation:

27.84+15=42.84

42.84-10.45= THE ANSWER

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:

$665.36

Step-by-step explanation:

2^16 or...

2x2x2x2x2x2x2x2x2x2x2x2x2x2x2x2 = 66536 pennies

Answer:

12ºC

Step-by-step explanation:

Morning = -3ºC

At 7pm, temperature dropped ny 9ºC.

Temperature = -3 - 9 = -12ºC

Answer:

  Equation 3: y = 12x + 17

Step-by-step explanation:

Only a linear equation (degree 1) will have a graph that is a straight line.

The degree of the equation is the highest power of any of the variables. If different variables are in the same term, it is the sum of the powers of those variables.

Answer:

Equation 3: y = 12x + 17

Step-by-step explanation:

It is the answer because all of the variables have an exponent of 1.

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:

x=2

Step-by-step explanation:

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:

  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 D

Step-by-step

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

= 14a^3b^4/7

=2a^3b^4

Answer:

#3. (x-7)(x-4)   #5. (2k+7)(k-2)

Step-by-step explanation:

For #3 you have to first get all the terms on one side of the equals sign and set it equal to 0.  So that gives us

[tex]x^{2} -11x+28=0[/tex].

Our a value is 1 and our c value is 28, so the product of those is 28.  Find the factors of 28 and the combination of those factors that add up to equal the linear term -11x is the combination we need for our problem.  The factors of 28 are:  1, 28; 2, 14; 4, 7.  4 and 7 give us 11 when we add them, but since we need a -11, we have to use the negative of both the factors since -7 + -4 = -11.  Set up your equation now using the -7 and the -4, "larger" number first (the absolute value which makes the 7 larger):

[tex]x^{2} -7x-4x+28[/tex].  This is the pattern that you will use to factor the next problem, as well.

Group the terms in groups of 2 to get:

[tex](x^{2} -7x)-(4x-28)=0[/tex].  Notice the sign change in front of the 28 in the second set of parenthesis.  This is because if I distribute the negative infront of the parenthesis back in, negative times a negative will give us the +28 in the original problem.  The same will apply again in #5 when we get there.

Now factor out whatever is common from each set of parenthesis:

[tex]x(x-7)-4(x-7)=0[/tex].

Now the common term is the factor (x-7) so that can be factored out now, leaving behind:

(x-7)(x-4).  That's the answer for #3.

Now for #5:

We will start by getting everything on one side (I am changing the k's to x's):

[tex]2x^{2} +3x-14=0[/tex].

The product of our a value and c value is again 28.  Find the combination of the factors of 28 that will add to give us the middle (linear) term of 3:  That is again 7 and 4, with the 7 needing to be positive and the 4 needing to be negative since 7 - 4 = 3.  Set up our expanded quadratic as follows, "larger" number (the absolute value of) first:

[tex]2x^{2} +7x-4x-14=0[/tex].

Group them into groups of 2 again:

[tex](2x^{2} +7x)-(4x+14)=0[/tex]

Again, notice the necessary sign change so when we distribute the negative back into the parenthesis we get the -14 we started with in the original problem.

Now factor out what is common from each set of parenthesis:

[tex]x(2x+7)-2(2x+7)=0[/tex].

What's common now is the factor (2x+7) so that can be factored out leaving behind

(2x+7)(x-2)=0

And you're done!!!

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.

https://brainly.com/question/35538480

#SPJ3

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!

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

#SPJ2

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

Answer:

0.3

Step-by-step explanation:

To calculate the dilation factor of two similar figures, you simply need to establish the ratio of the dilated figure (in our case PQRS) to the original figure (ABCD) using the same relative side.  

In our case, we'll take the longest of the sides provided in each figure.

So, we divide 12.6 by 42 to know the dilation factor of the longest side, it will be the same for all other sides:

d = 12.6 / 42 = 0.3

Answer:

D. 5000 kg

Step-by-step explanation:

We assume the second train was standing still and that momentum is conserved. The the product of mass and velocity before the collision is

(5000 kg)·(100 m/s) = 500,000 kg·m/s.

After the collision, where M is the mass of the second train, the momentum is ...

((5000+M) kg)·(50 m/s) = 500,000 kg·m/s

Dividing by 50 m/s and subtracting 5000 kg, we have ...

(5000 +M) kg = 10,000 kg

M kg = 5000 kg

The mass of the second train is 5000 kg.

Answer:

D. 5000 kg

Step-by-step explanation: