help pls thank you!!

Help Pls Thank You!!

Answers

Answer 1
10/2.5= 4
2/3*4= 8/3

2 2/3 is your answer (C)
Answer 2

Answer:

C 2+2/3

Step-by-step explanation:

10/2.5= 4

2/3*4= 2+2/3


Related Questions

can someone help asap

can someone help asap

Answers

-6=(2/3)(9)+c
-6=6+c
-12=c

equation:
y=2/3x-12
Please look at the picture for a step by step explanation. Hope it helps!
can someone help asap

The path of an amusement park ride is a circle with a diameter of 9.5 feet. Melanie is at the point marked on the path, as shown below. 9.5 ft Which is closest to the distance, in feet, Melanie will travel in one complete turn of the ride?​

Answers

Answer: 36.11 Ft

Step-by-step explanation:

Circumference formula

C = 2 • pie,3.14 • R

C = 2 • 3.14 • 4.75 = 36.11

Which side lengths form a right triangle?

Which side lengths form a right triangle?

Answers

Answer:

a is the answer

hope this will help you

PROBLEM 2 Ten kids line up for recess. The names of the kids are: {Alex, Bobby, Cathy, Dave, Emy, Frank, George, Homa, Ian, Jim}. Let S be the set of all possible ways to line up the kids. For example, one order might be:
(Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby) The names are listed in order from left to right, so Frank is at the front of the line and Bobby is at the end of the line. Let T be the set of all possible ways to line up the kids in which George is ahead of Dave in the line. Note that George does not have to be immediately ahead of Dave. For example, the ordering shown above is an element in T.
Now define a function f whose domain is S and whose target is T. Let x be an element of S, so x is one possible way to order the kids. If George is ahead of Dave in the ordering x, then f(x) = x. If Dave is ahead of George in x, then f(x) is the ordering that is the same as x, except that Dave and George have swapped places. (a) What is the output of f on the following input? (Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby) (b) What is the output of f on the following input? (Emy, Ian, Dave, Homa, Jim, Alex, Bobby, Frank, George, Cathy) (C) Is the function f a k-to-1 correspondence for some positive integer k? If so, for what value of k? Justify your answer. (d) There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. That is, |S| = 3628800. Use this fact and the answer to the previous question to determine T

Answers

The required value of |T| = 3628800/2 = 1814400. All the required answers of the questions are below.

(a) The input order is: Frank, George, Homa, Jim, Alex, Dave, Cathy, Emy, Ian, Bobby. Since George is ahead of Dave in this ordering, the output of f(x) is x, which is the same as the input order.(b) The input order is: Emy, Ian, Dave, Homa, Jim, Alex, Bobby, Frank, George, Cathy. Since Dave is ahead of George in this ordering, the output of f(x) is the ordering that is the same as x, except that Dave and George have swapped places. The new order would be: Emy, Ian, George, Homa, Jim, Alex, Bobby, Frank, Dave, Cathy.(c) Yes, the function f is a 2-to-1 correspondence. This is because for any input order x, there are exactly two output orders, depending on whether George is ahead of Dave or vice versa. To see why it cannot be a higher k-to-1 correspondence, consider the input order where George and Dave are adjacent in the ordering (there are 9 possible such orders). In this case, there is only one possible output order, since swapping George and Dave would result in the same ordering. Thus, the function cannot be a k-to-1 correspondence for k>2.(d) There are 3628800 ways to line up the 10 kids with no restrictions on who comes before whom. Since the function f is a 2-to-1 correspondence, this means that there are half as many possible output orders as there are input orders. Thus, |T| = 3628800/2 = 1814400.

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help pleaaase


The lengths (in feet) of two sides of a regular quadrilateral are represented by the

expressions 8x 6 and 4x 22. Find the length of a side of the quadrilateral.

Answers

Answer:

The lengths ( in feet ) of two sides of a regular quadrilateral are represented by th expresion

8x-6 and 4x+22 . find the length of a side of the quadrilateral

Answer by Alan3354(67447) (Show Source): You can put this solution on YOUR website!

A regular quadrilateral is a square, so

8x-6 = 4x + 22

Can you finish it?

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The length of a side of the quadrilateral is 38 feet.

Given that, The lengths (in feet) of two sides of a regular quadrilateral are represented by the expressions 8x + 6 and 4x + 22.

We need to find the length of a side of the quadrilateral.

What is the regular quadrilateral?

The regular quadrilateral is a 4-sided polygon which is equiangular and equilateral. Square is a quadrilateral with all the sides equal and all angles equal to 90°. Hence, the square is a regular quadrilateral.

Now, 8x+6=4x+22

⇒8x-4x=22-6

⇒4x=16

⇒x=4

Then, 8x+6=38 feet and 4x+22=38 feet

Therefore, the length of a side of the quadrilateral is 38 feet.

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The time constant of RC circuit is the time in which the current decreases to _____ of its initia value
a. 1/10
b. ½
c. 1/√2
d. 1/╥
e. 1/e

Answers

The correct solution to this problem is option e. 1/e. We can find it in the following manner.

The time constant of an RC circuit is the time in which the current decreases to 1/e (approximately 0.368) of its initial value.

This can be derived from the equation for the current in an RC circuit:

\(I=10 * e^({-t/RC} )\)

where I is current at time t, I0 is the initial current, R is the resistance, C is the capacitance, and e is the mathematical constant approximately equal to 2.71828.

To find the time constant, we set the exponent to -1:

\(e^({-t/RC} ) = 1 /e\)

Solving for t/RC, we get:

t/RC = 1

Therefore, the time constant is equal to RC, and the current decreases to 1/e of its initial value in one time constant.

So the answer is e. 1/e.

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At a carnival game, it cost Noah $12 to toss 30 rings. Fill out a table of equivalent ratios and plot the points on the coordinate axes provided.

At a carnival game, it cost Noah $12 to toss 30 rings. Fill out a table of equivalent ratios and plot

Answers

Answer: See attached.

Step-by-step explanation:

     First, we will find the unit rate by dividing.

30 rings / $12 = 2.5 rings per dollar

     Then, we can fill out the other values using this unit rate.

2 dollars * 2.5 rings per dollar = 5 rings

45 tosses / 2.5 rings per dollar = 18 rings

       Lastly, we will use these two equivalent ratios we found above to fill out the table and then plot the points. See attached.

At a carnival game, it cost Noah $12 to toss 30 rings. Fill out a table of equivalent ratios and plot

Draw the image of ABC under a dilation whose center is P and scale factor is 1/3

Draw the image of ABC under a dilation whose center is P and scale factor is 1/3

Answers

Answer:

there

Step-by-step explanation:

Draw the image of ABC under a dilation whose center is P and scale factor is 1/3

Answer:

khan

Step-by-step explanation:

Draw the image of ABC under a dilation whose center is P and scale factor is 1/3

which answer is the simplest form of the expression.

which answer is the simplest form of the expression.

Answers

Answer:

yellow box

Step-by-step explanation:

Answer:

c or the orange option. m^2n/2

Step-by-step explanation:

i hope this helps :)

Emily bought some ice cream cones for noche buena which costs 24 she paid M and got back 6 of change. Find M

Answers

Emily paid M = 30 for the ice cream cones.

To find the amount Emily paid (M), we can set up an equation using the given information.
1. The cost of the ice cream cones is 24.
2. She got back 6 in change.
Using these facts, we can write the equation:

M - 24 = 6
Now, let's solve for M:
Step 1: Add 24 to both sides of the equation.
M - 24 + 24 = 6 + 24
Step 2: Simplify.
M = 30.

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Can you please help me out with a question

Answers

Vertex of an angle on a circle

We have that in this case, we have that the measure of an angle on a circle is half the measure of its intercepted arc.

We have that the whole circle corresponds to a 360º, then:

because 360º - 214º = 144º

Then, the the intercepted arc by the angle K is 144º.

Then,

\(\angle K=\frac{144º}{2}=72º\)Answer: D. 72º

Can you please help me out with a question

the letters a,b and c, represent nonzero constants. Solve ax+b=c for x

Answers

Answer:

\(x = \frac{c-b}{a}\)

Step-by-step explanation:

To solve for x, x needs to be on one side while all the other variables are on the other

To do this first subtract b from each side making the equation,

ax = c - b

Now divide by a on both sides. This means that to solve for x the equation must be ...

\(x = \frac{c-b}{a}\)

A 25.5 foot ladder rests against the side of a house at a point 24.1 feet above the ground. The foot of the ladder is x feet from the house. Find the value of x to one decimal place.

Answers

Answer:

8.3

Step-by-step explanation:

Use the pythagorean theorem: a2 + b2 = c2 a2 would be 24.1 and c2 would be 25.5. square them then subtract a2 from b2. then find the square root of your answer which is 8.3

suppose a certain drug test is 96% sensitive, that is, the test will correctly identify a drug user as testing positive 96% of the time, and 86% specific, that is, the test will correctly identify a non-user as testing negative 86% of the time. suppose a corporation decides to test its employees for drug use, and that only 0.3% of the employees actually use the drug. what is the probability that, given a positive drug test, an employee is actually a drug user? round your answer to 3 digits after the decimal point.

Answers

Answer:

222

Step-by-step explanation:

1x2222

2

2

2

2

2

Determine the domain of the function h(x)=9x/x(x2-49).

Answers

The domain of the function h(x)=9x/x(x²-49) is (-∞, -7) ∪ (-7, 0) ∪ (0, 7) ∪ (7, ∞).

In math the term called the domain of a function is the set of its possible inputs which is the set of input values where for which the function is defined.

Here we have know the function is h(x)=9x/x(x²-49)

And we need to find the domain of the function.

Here we know that this function is defined for values where the denominator is not equal to zero.

Which is written as,

=> x(x²-49) ≠ 0

When we elaborate the equation based on the difference between the two squares property, it can be written as,

=> x(x + 7)(x - 7)  ≠ 0

Based on this we have identified that the value of x must not be

=> x  ≠ -7, 7, 0

Then the domain of the function is written as,

=> (-∞, -7) ∪ (-7, 0) ∪ (0, 7) ∪ (7, ∞).

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1. Which is an example of categorical data?A eye colorB ageC weight

Answers

Age and you can also use weight too

you can easily group people by their age and weight but not by eye color

example

age people

5 - 10 14

10 - 15 5

the table above shows that we have 14 people within the range of 5 to 10 years

and likewise the weight of the people can serve the same purpose

Choose a type of corrosion that affects your life or that you feel presents a significant risk to health and safety or the environment. Provide pictures or video identifying your chosen example of corrosion Explain how that type of corrosion affects your life. Research and explain the exact electrochemical process involved in that type of corrosion In addition, include the following: Identify the electrodes and electrolyte. Show both half reactions and indicate which reaction is the oxidization half reaction and which is the reduction half reaction. Show the balanced chemical equation. Rate of corrosion: a Explain why the corrosion is occurring? b. Estimate the time it took for the object (your example) to corrode. Identity and explain two techniques that could be used to prevent the type of corrosion you have chosen. Many corrosion prevention techniques have environmental or health issues, for example, oil disposal or inhalation hazards. Identify and explain any such issues related to the above prevention methods. Explain how one of the following environmental conditions affects the rate AND extent of the type of corrosion you have chosen: a. acid rain OR b. climate change (warm vs. cold) OR C. de-icing technique (road salt vs. sand)

Answers

1.  Iron rusting influences in many ways.

2. Iron rusting involves the formation of iron oxide by an electrochemical process on the surface, where iron oxidizes and oxygen reduces to form rust.

3. Anode is iron, and the cathode is oxygen,

4.  The half-reactions involved in iron rusting are:

- Anodic response: Fe(s) →\(Fe^2+ (aq) + 2e^-\)

- Cathodic reaction: \(O2(g) + 2H2O(l) + 4e^-\)→ \(4OH^- (aq)\)

5. The balanced chemical equation for iron rusting is:

\(- 4Fe(s) + 3O2(g) + 6H2O(l)\) → \(4Fe(OH)3(s)\)

\(- 4Fe(OH)3(s)\) → \(2Fe2O3.H2O(s) + 4H2O(l)\)

6. The corrosion of iron takes place because iron is a reactive metal, water, etc.

7.  Two techniques that might be used to prevent the sort of corrosion I have selected are:- Protective coatings, Cathodic safety.

8. One environmental circumstance that affects the fee and extent of iron rusting is: Acid rain

1. Iron rusting influences my existence in lots of methods. Some of the effects are:

- It reduces the strength and durability of iron items, which includes bridges, pipes, cars, equipment, and so forth., making them liable to failure and injuries.- It reasons aesthetic damage and lack of value to iron gadgets, consisting of fixtures, sculptures, ornaments, and many others., making them look antique and ugly.- It increases the upkeep and replacement expenses of iron items, as they need to be repaired or replaced greater often because of corrosion.- It contributes to environmental pollution and waste, as rusted iron items release poisonous substances into the soil and water, and occupy landfills.

2. The precise electrochemical process worried in iron rusting is as follows:

- When iron is uncovered to moist air, it forms a thin layer of iron oxide on its floor. This layer is porous and allows oxygen and water to penetrate deeper into the steel.- The iron atoms on the floor lose electrons and end up oxidized to form iron(II) ions. This is the anodic response.- The oxygen molecules within the air or water benefit electrons and grow to be decreased to shape hydroxide ions. This is the cathodic reaction.- The iron(II) ions and the hydroxide ions react to shape iron(II) hydroxide, which similarly reacts with oxygen to shape iron(III) hydroxide. This compound dehydrates and oxidizes to form iron(III) oxide-hydroxide, which is a reddish-brown substance called rust.

3. The electrodes and electrolyte worried in iron rusting are:

- The anode is the iron metal itself, in which oxidation takes place.- The cathode is the oxygen molecule, wherein reduction takes place.- The electrolyte is the water or moisture that includes dissolved oxygen and other ions.

4. The half-reactions involved in iron rusting are:

- Anodic response: Fe(s) →\(Fe^2+ (aq) + 2e^-\)

- Cathodic reaction: \(O2(g) + 2H2O(l) + 4e^-\)→ \(4OH^- (aq)\)

5. The balanced chemical equation for iron rusting is:

\(- 4Fe(s) + 3O2(g) + 6H2O(l)\) → \(4Fe(OH)3(s)\)

\(- 4Fe(OH)3(s)\) → \(2Fe2O3.H2O(s) + 4H2O(l)\)

6. Rate of corrosion:

a. The corrosion of iron takes place because iron is a reactive metal that tends to lose electrons and form positive ions in aqueous solutions. Iron additionally has a high affinity for oxygen and paperwork stable oxides that adhere to its floor.

The presence of water or moisture facilitates the transport of electrons and ions between the anode and the cathode, as a consequence accelerating the corrosion procedure.

B. The time it took for the object (your example) to corrode depends on many elements, such as the sort, size, form, and composition of the item, the environmental situations (temperature, humidity, acidity, salinity, etc.), and the presence or absence of protective coatings or inhibitors. Therefore, it's miles difficult to estimate a genuine time for corrosion without knowing that information.

7. Two techniques that might be used to prevent the sort of corrosion I have selected are:

- Protective coatings: Applying a layer of paint, plastic, or steel on the floor iron can prevent or lessen the touch between iron and the corrosive agents (oxygen and water). This can slow down or forestall the corrosion manner. - Cathodic safety: Connecting iron to a more electropositive metal (such as zinc or magnesium) can save you or reduce the corrosion of iron.

8. One environmental circumstance that affects the fee and extent of iron rusting is:

- Acid rain: Acid rain is rainwater that contains acidic pollutants together with sulfur dioxide and nitrogen oxides from commercial emissions or volcanic eruptions. Acid rain lowers the pH of the electrolyte (water or moisture) and increases its conductivity.

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Choose a type of corrosion that affects your life or that you feel presents a significant risk to health

Polygons in the coordinate

Polygons in the coordinate

Answers

In order to know if a triangle is a right triangle on a coordinate plane, you can find the lengths of all three sides of the triangle using the distance formula and apply the Pythagorean theorem.

How to know if it's a triangle

Find the lengths of the three sides of the triangle using the distance formula.

Once you have the lengths of the sides, check if any of the three sides satisfy the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In other words, if a² + b² = c², where c is the longest side, then the triangle is a right triangle.

If one of the sides satisfies the Pythagorean theorem, then the triangle is a right triangle.

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find the function f(x) described by the given initial value problem. f'(x) = sinx, f(0) = 2

Answers

The function f(x) that satisfies the given initial value problem is f(x) = -cos(x) + 3.

To find the function f(x) described by the given initial value problem, F'(x) = sin(x), and F(0) = 2, follow these steps:

1. Integrate F'(x) with respect to x to find f(x): ∫sin(x) dx = -cos(x) + C, where C is the constant of integration.
2. Use the initial condition F(0) = 2 to find the value of C: -cos(0) + C = 2 => C = 2 + cos(0) = 2 + 1 = 3.
3. Replace C with the found value in the function: f(x) = -cos(x) + 3.

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How do the expressions 8+k2−6 and 8(k+2)−4k compare when k = 7?

Drag a symbol into the box to correctly complete the statement.
8+k2−6 Response area 8(k+2)−4k when k = 7.
=< >

Answers

Answer:

The answer is > :)

Step-by-step explanation:

How do the expressions 8+k26 and 8(k+2)4k compare when k = 7?Drag a symbol into the box to correctly

the study of hypnosis and its relationship to hysteria was the starting point for

Answers

The study of hypnosis and its relationship to hysteria was the starting point for the development of psychoanalysis by Sigmund Freud.

The study of hypnosis and hysteria in the late 19th century by Sigmund Freud and his contemporaries led to the development of psychoanalysis, a form of psychotherapy that emphasizes the role of unconscious thoughts and feelings in shaping behavior. Through his work with patients suffering from hysteria, Freud became interested in the idea that psychological conflicts could be traced back to early childhood experiences, and that these conflicts could be resolved through the exploration of unconscious thoughts and feelings. This eventually led to the development of psychoanalysis as a theory and practice for the treatment of psychological disorders.

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What is the value of 0 in the following image

What is the value of 0 in the following image

Answers

We will determine the value of theta as follows:

\(\begin{gathered} cos(\theta)=\frac{10}{21}\Rightarrow\theta=cos^{-1}(\frac{10}{21}) \\ \\ \Rightarrow\theta\approx61.6 \end{gathered}\)

So, the measurement of the angle is approximately 61.6°.

6th grade math

5x - 2

Answers

Answer:

-10

Step-by-step explanation:

Answer: 3x

verry eazy just subtract and keep the x

Can someone help me with this please

Can someone help me with this please

Answers

Answer: The system of equations is

40x + 55y = 920

40x + 65y = 1000

x is the cost of the adult ticket; y is the cost of the child ticket

Step-by-step explanation:

If you have to solve this, elimination is a good method.

Subtract the top equation from the second equation.

40x + 65y = 1000

-40x + 55y = 920  x cancels  Solve for y

0  + 10y  =  80    Divide both sides by 10

y  = 8 .  Substitute 8 for y in either equation and solve for x

40x + 65(8) = 1000

40x + 520 = 1000   Subtract 520 from both sides

  40x = 1000 - 520

40x = 480   Divide both sides by 40

x = 12  

Use Cramer’s rule to solve for x: x + 4y − z = −14 5x + 6y + 3z = 4 −2x + 7y + 2z = −17

Answers

Looks like the system is

x + 4y - z = -14

5x + 6y + 3z = 4

-2x + 7y + 2z = -17

or in matrix form,

\(\mathbf{Ax} = \mathbf b \iff \begin{bmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -14 \\ 4 \\ -17 \end{bmatrix}\)

Cramer's rule says that

\(x_i = \dfrac{\det \mathbf A_i}{\det \mathbf A}\)

where \(x_i\) is the solution for i-th variable, and \(\mathbf A_i\) is a modified version of \(\mathbf A\) with its i-th column replaced by \(\mathbf b\).

We have 4 determinants to compute. I'll show the work for det(A) using a cofactor expansion along the first row.

\(\det \mathbf A = \begin{vmatrix} 1 & 4 & -1 \\ 5 & 6 & 3 \\ -2 & 7 & 2 \end{vmatrix}\)

\(\det \mathbf A = \begin{vmatrix} 6 & 3 \\ 7 & 2 \end{vmatrix} - 4 \begin{vmatrix} 5 & 3 \\ -2 & 2 \end{vmatrix} - \begin{vmatrix} 5 & 6 \\ -2 & 7 \end{vmatrix}\)

\(\det \mathbf A = ((6\times2)-(3\times7)) - 4((5\times2)-(3\times(-2)) - ((5\times7)-(6\times(-2)))\)

\(\det\mathbf A = 12 - 21 - 40 - 24 - 35 - 12 = -120\)

The modified matrices and their determinants are

\(\mathbf A_1 = \begin{bmatrix} -14 & 4 & -1 \\ 4 & 6 & 3 \\ -17 & 7 & 2\end{bmatrix} \implies \det\mathbf A_1 = -240\)

\(\mathbf A_2 = \begin{bmatrix} 1 & -14 & -1 \\ 5 & 4 & 3 \\ -2 & -17 & 2 \end{bmatrix} \implies \det\mathbf A_2 = 360\)

\(\mathbf A_3 = \begin{bmatrix} 1 & 4 & -14 \\ 5 & 6 & 4 \\ -2 & 7 & -17 \end{bmatrix} \implies \det\mathbf A_3 = -480\)

Then by Cramer's rule, the solution to the system is

\(x = \dfrac{-240}{-120} \implies \boxed{x = 2}\)

\(y = \dfrac{360}{-120} \implies \boxed{y = -3}\)

\(z = \dfrac{-480}{-120} \implies \boxed{z = 4}\)

Answer:

in photo attached.

Step-by-step explanation:

Use Cramers rule to solve for x: x + 4y z = 14 5x + 6y + 3z = 4 2x + 7y + 2z = 17

3. What is the value of x in the figure
below?

3. What is the value of x in the figurebelow?

Answers

Answer:

3 = 14 and 4 = 8

Step-by-step explanation:

The reason as to why this is, is because of the line going down the middle. If you were to open one of the lengths as a door starting from the middle line it would fit perfectly with the top one.

The endpoints of the diameter of a circle are (6, 0) and (-6, 0). What are the coordinates of the center of this circle?

Answers

Answer:

(0,0)

Step-by-step explanation:

a diameter means The average of two measurements of the diameter taken at right angles to each other. so if you try graphing you will be able to find out its (0,0) is the center of the circle coordinates

The coordinates of the center of this circle is (0,0).

What are the coordinates of a center of a circle?

The center of a circle is the midpoint of the diameter. So, by using the midpoint formula, if the endpoints of the diameter are (a, b) and (c, d), then the coordinates of the center of circle are [(a + c)/2, (b + d)/2].

According to the question

The endpoints of the diameter of a circle are (6, 0) and (-6, 0).

now ,

(a,b) =  (6, 0)

(c, d)= (-6, 0)

coordinates of the center of circle = [ \(\frac{a+c}{2} ,\frac{b+d}{2}\) ]

\(\frac{6-6}{2} ,\frac{0+0}{2}\)

⇒ (0,0)

Hence, the coordinates of the center of this circle is (0,0).

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I NEED HELP!!!!
\(3(7+7i)+i(7+7i)\)

Thanks!:)​

Answers

Solve the equation:-

\(3(7 + 7i) + i(7 + 7i) \\take \: common \: (7 + 7i) \: out\\ (7 + 7i) (3 + i) \: is \: your \: answer \)

Open the equation:-

\(3(7 + 7i) + i(7 + 7i) \\ 21 + 21i + 7i + 7 {i}^{2} \\ 7 {i}^{2} + 28i + 21 \: \: answer\)

Find the value of h(-67) for the function below.

h(x) = -49x − 125

A.
-3,408
B.
3,158
C.
3,283
D.
-1.18

Answers

Answer:

B.   3,158

Step-by-step explanation:

h(x) = -49x − 125

Let x = -67

h(-67) = -49(-67) − 125

          =3283-125

          = 3158

Answer:

Answer B

Step-by-step explanation:

To find the value of h(-67) for the function h(x) = -49x - 125,

we substitute -67 for x in the function and evaluate it.

h ( - 67 ) = - 49 ( - 67 ) - 125

Now we can simplify the expression:

h ( -67 ) = 3283 - 125

h ( -67 ) = 3158

A lattice point is an ordered pair (x, y) where both x and y are integers. A triangle is formed
by the three points (1, 1), (9, 1), and (9, n). For what integer value of n > 0 are there exactly 560 lattice
points strictly in the interior of the triangle?

Answers

To find the integer value of n that results in exactly 560 lattice points strictly in the interior of the triangle, use the equation: n = (560 + 2)/8.

This equation is derived by counting the lattice points on each side of the triangle, starting with the side between points (1,1) and (9,1). This side has a length of 8, so it contains 8 lattice points, including the two endpoints. The remaining sides have lengths of 8 + n and 8 + n, which combined have 8 + 2n lattice points. The total number of lattice points in the triangle is then 8 + 8 + 2n = 16 + 2n.  

Solving for n when the total number of lattice points is 560 gives us n = (560 + 2)/8. Therefore, the integer value of n is 70.

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