The **maximum** **displacement** of this cosine wave function is equal to 8 inches.

The **frequency** of this cosine wave function is equal to 1/8 Hertz.

The **time** (t) required for one cycle is equal to 8 seconds.

Mathematically, a cosine wave simply refers to an equation of simple harmonic motion (SHM) and it is modelled by this mathematical expression:

y = asinωt = asin2πft ......equation 1.

Where:

a represents the amplitude orFrom the information provided, the equation for this simple harmonic motion (SHM) is given by:

d = -8cos(π/4)t ......equation 2.

By comparing equation 1 and equation 2, we have the following parameters:

**Maximum** **displacement **or amplitude, a = |-8|

Taking the absolute value of a, we have:

**Maximum** **displacement **or amplitude, a = 8

For the **frequency**, we have:

Angular velocity, ω = 2πf

Making **frequency **(f) the subject of formula, we have:

Frequency** **(f) = ω/2π

**Frequency **(f) = (π/4)/2π

**Frequency **(f) = π/8π

**Frequency **(f) = 1/8 Hertz.

Finally, we would calculate the amount of **time** (t) required for one cycle as follows:

Time (t) = 1/f

Time (t) = 1/(1/8)

**Time** (t) = 8 seconds.

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Complete Question:

An object moves in simple harmonic motion described by the equation d = -8cos(π/4)t, where t is measured in seconds and d in inches. Find the following:

maximum displacement

the frequency

the time required for one cycle.

What quadrant is 0 And. . -1 1/2 in or is it on a y- axis or x-axis.

The** coordinate pair** is between the third and fourth **quadrants**, on the y-axis.

Remember that the **quadrants **are:

First quadrant: x > 0, y > 0.

Second quadrant: x < 0, y > 0.

Third quadrant: x < 0, y < 0.

Fourth quadrant: x > 0, y < 0.

In this case our **coordinate pair **is (0, -1/2).

So it will be between the third and fourth quadrants.

And yes, because one of the **variables **is zero, it is on the y-axis (just between the two **quadrants**).

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I BEG YOU FOR HELP!!! Determine the relationship between the two triangles and whether or not they can be proven to be congruent.

**Answer:**

**Step-by-step explanation:**

The triangles are congruent because they follow the SSS triangle congruence postulate. As the SSS postulate says that all 3 sides of one triangle are congruent to another triangle's sides, these triangles shown have all three of their sides congruent to each other.

**Answer: I am no sure but the only way to decide whether a pair of triangles are congruent would be to measure all of the sides and angles, and these triangles do not look the same so I would say that these tringles are not congruent.**

**Step-by-step explanation:**

Question 1. Find the center and radius of the circunscribed circle.

In order to find the center of the circunscribed circle, we can use the midpoint theorem because the center point is in the middle of any two vertices

that is, if we take points (9,23) and (8,16) the midpoint C is given as

[tex]C=(\frac{8+9}{2},\frac{16+23}{2})[/tex]which gives

[tex]C=(8.5,19.5)[/tex]So the **center of the circle is the point (8.5,19.5)**

On the other hand, the radius is equal to the distance from any vertex to the center. If we take the vertex (8,16), we get

[tex]r=\sqrt[]{(8.5-8)^2+(19.5-16)^2}[/tex]which gives

[tex]\begin{gathered} r=\sqrt[]{0.5^2+3.5^2} \\ r=\sqrt[]{0.25+12.25} \\ r=\sqrt[]{12.5} \\ r=3.5355 \end{gathered}[/tex]so, **the radius measure 3.54 units.**

Now, lets prove that the answer are correct. In order to do that, we can choose the other vertices and apply the same procedure as above.

So the vertices are (5,20) and (12,19). Again, the center is the midpoint between these points and is given as

[tex]\begin{gathered} C=(\frac{5+12}{2},\frac{20+19}{2}) \\ C=(\frac{17}{2},\frac{39}{2}) \\ C=(8.5,19.5) \end{gathered}[/tex]which is the same center as above.

Now, the distance from the center to vertec (5,20) is

[tex]\begin{gathered} r=\sqrt[]{(8.5-5)^2+(20-19.5)^2} \\ r=\sqrt[]{3.5^2+0.5^2} \\ r=\sqrt[]{12.25+0.25} \\ r=\sqrt[]{12.5} \\ r=3.5355 \end{gathered}[/tex]which is the same radius obtained above. **Then, the answers are correct.**

Question 2-22

A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eatin

5

36

35

36

O

●

cakes/week

1

01 cakes/week

35

1

1 cakes/week

4

< Previous

cakes/week

2022-2023 T-Math-Gr7Reg-T2-CBT: Section 2-...

Jake's weekly** rate of eating** of cake slices is 11.6

Given,

Number of equal **slices of cake** = 12

Number of slices Jake eaten after 3 days = 5

We have to find the weekly rate of eating;

Here,

**Number of days** in a week = 7

Jake eaten 5 slices in 3 days so, 7 - 3 = 4

Again after 3 days 5 slices.

Then,

4 - 3 = 1

That is, Jake eaten 10 slices of cake in 6 days.

Number of cake slices eaten in 1 day = 5/3 = 1.6

Therefore,

Weekly rate of eating is 10 + 1.6 = 11.6

That is,

Jake's weekly** rate of eating** of cake slices is 11.6

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In reference to 3x + 2 ≥-4Is 0 a solution?Yes or No?

**ANSWER :**

**yes**

**EXPLANATION :**

From the problem, we have the inequality :

[tex]3x+2\ge-4[/tex]To check if a number is a solution, substitute it to the inequality and check the truthfulness.

Check if 0 is a solution :

[tex]\begin{gathered} 3(0)+2\ge-4 \\ 2\ge-4 \end{gathered}[/tex]Since 2 is greater than -4, therefore 0 is a solution

Which of the following is the correct mathematical expression for:

The difference between three times a number and 4

A. 3x + 4

B. 3x - 4

C. 1/3x + 4

D. 1/3x - 4

**Answer:**

B

**Step-by-step explanation:**

if the number is x then 3 times the number is 3x

the difference is the subtraction of 3x and 4 , that is

3x - 4

20 students started the class. Then 2 students dropped the

class.

What percent of the students have dropped the class?

If 20 was at the beginning and 2 students drop the class it would be 10%

10a-4a simplify

PLEASEE I NEED HELLPPP

they answer is 6a because you mines 10-4 and you get 6 then just put the loke 6a

There are 2.54 centimeters in 1 Inch. There are 100 centimeters in 1 meter. To the nearest inch, how many inches are in 7 meters? Enter the answer in the box. inches

Based on the given equivalences, you have:

**7 m = 7(100 cm) = 700 cm**

**700 cm = 700 (2.54 in) = 1,778 in**

**Hence, there are 1,778 inches in 7 meters**

Answer the questions below.

(a) Amount of data entered

The amount of data entered affects his pay at the end of the week.(b) R

The independent variable is the input.(c) Average temperature

The average temperature depends on the city's distance from the equator.jamals lawn is shaped like a square with an area of 224.9 ft2. which measurement is closest to the side length of his lawn in feet?

Given:

Area of the square shaped lawn = 224.9 ft²

A square has all four side lengths equal.

To find the side length of this lawn, use the formula for area of a square below:

[tex]\text{Area = }L^2[/tex]Take the square root of both sides to find the sile length L:

[tex]\begin{gathered} \sqrt[]{Area\text{ }}=\sqrt[]{L^2} \\ \\ \sqrt[]{Area}\text{ = L} \\ \\ \sqrt[]{224.9}\text{ = L} \\ \\ 14.99\text{ ft = L} \end{gathered}[/tex]Therefore, the measurement that is closest to the side length in feet is 15 ft

**ANSWER:**

**15 ft**

equati 1. v = 3.1 + 14 y = -1

y= 3x +14 (a)

y=-4x (b)

Replace the value of y (b) on equation (a), and solve for x

-4x = 3x+14

-4x-3x = 14

-7x = 14

x= 14/-7

**x = -2**

Replace the value of x on any initial equation and solve for y:

y= -4 (-2) = 8

**y=8**

Answer:

1*v=3,0.1+14y=-1 : v=3,y=-0.07857

[1.v=3}

[0.1+14y=-1]

v=3,y=-0.07857

Step-by-step explanation:

Suppose you buy two sweaters. Each sweater cost the same amount. The amount of money you spend varies directly with the number of sweaters you buy. If you spent $37.50 for two sweaters, what is the constant of variation? A. 2.00 B. 37.50 C. 0.053 D. 18.75

**Answer:**

D

**Step-by-step explanation:**

37.50 ÷ 2 = 18.75. jdjdjfbffcv

When you mix two colors of paint in equivalent ratios, the resulting color is always the same. Complete the table as you answer the questions.

How many cups of yellow paint should you mix with 1 cup of blue paint to make the same shade of green? Explain or show your reasoning.

Make up a new pair of numbers that would make the same shade of green. Explain how you know they would make the same shade of green.

row 1

cups of blue paint

cups of yellow paint

row 2

2

10

row 3

1

5

row 4

3

15

What is the proportional relationship represented by this table?

What is the constant of proportionality? What does it represent?

By Calculating the **Constant of Proportionality,** we get,

1. 5 cups of yellow paint is required for 1 cup of blue paint.

2. For 4 cups of blue paint, 20 cups of **yellow paint is required.**

For 5 cups of blue paint, 25 cups of yellow paint are required.

3. Cups of Yellow Paints = 5* Cups of Blue Paints

4. The constant of Proportionality is 5.

Let the cups of **blue paint **= x

and cups of **yellow paint **= y

From the table, we can infer that to make the same shade of green, we need to mix 5 cups of yellow color with 1 cup of blue.

We have, To make the same shade of green,

we need to mix 5 cups of yellow color with 1 cup of blue.

we need to mix 10 cups of yellow color with 2 cups of blue.

we need to mix 15 cups of yellow color with 3 cups of blue.

So, here we can see a relationship between the **two colors,** blue (x) and green(y)

Let k be the constant of **proportionality, **then, we have :

10 = k *2

k =[tex]\frac{10}{2} = 5[/tex]

Hence, For one cup of blue paint, we need 5 cups of yellow paint to make the same shade of green.

And the equation of the same is y =kx, that is y =5x....equation(1)

To make new pair of** numbers** that would make the same shade of green.

We can use the equation 1,

for x = 4, we need y = 20

foe x =5, we need y = 25

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You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 8.4% compounded monthly.

The desired **monthly** yield at the retirement **time** will be equal to $565,714.28.

**Compound Interest** may be defined as the interest earned by the bank on the basis of principle and also accumulated **interest **which increases **exponentially** and not linearly with respect to time. In calculating compound interest, the amount earned at the end of first year becomes principle for the next year and so on. Compound interest can be calculated, **annually**, half-yearly or **quarterly** etc.

Time for which work is planned = 40 years, **Principle** = $4500 and APR = 8.4% = 0.084/12 = 0.007.

The value of n = 12 × 25 = 300

The **amount** can be calculated by the formula A = P/r [1 - (1 + r) ⁻ⁿ]

A = (4500/0.007) [1 - (1 + 0.007) ⁻³⁰⁰]

A = 642,857.14 [1 - 0.12]

A = 642,857.14 × 0.88

A = $565,714.28 which is required amount.

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**Complete Question:**

**You plan to work for 40 years and then retire using a 25-year annuity. You want to arrange a retirement income of $4500 per month. You have access to an account that pays an APR of 8.4% compounded monthly. What monthly deposits are required to achieve the desired monthly yield at retirement?**

Hi I need help with this! I completed most of it, but I need help with the last portion. Thank you!

**The equation of the axis of symmetry is x = 2**

The equation of the axis of symmetry is the value of x that divides the parabola into equal halves (mirror images).

The value of x which gives the mirror images is the value of x coordinate of the vertex. The vertex is at (2, 0). The x coordinate = 2

From the graph, we can also see it occurred at x = 2

The axis of symmetry: x = x-coordinate of the vertex

**The equation of the axis of symmetry is x = 2**

Find the area of a triangular window with the given base and height base = 10 ft height =7 ft

The area of a triangular shape is given by the following formula

[tex]A=\frac{bh}{2}[/tex]where,

base, b = 10 ft

height, h = 7 ft

therefore,

[tex]A=\frac{10\cdot7}{2}=\frac{70}{2}=35[/tex]thus, the answer is 35 ft^2

Graph the y = 2x ^ 2 - 12x + 15 Plot five points on the parabolathe vertextwo points to the left of the vertex, and two points to the right of the vertexThen click on the graph-a-function button

**Explanation: **

Given the function:

[tex]y=2x^2-12x+15[/tex]First, we find the vertex of the parabola.

Vertex

The equation of the axis of symmetry is calculated using the formula:

[tex]x=-\frac{b}{2a}[/tex]From the function: a=2, b=-12

[tex]\implies x=-\frac{-12}{2(2)}=\frac{12}{4}=3[/tex]Substitute x=3 into y to find the y-coordinate at the vertex.

[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(3)^2-12(3)+15 \\ =18-36+15 \\ =-3 \end{gathered}[/tex]**The vertex is at (3, -3).**

Two points to the left of the vertex

When x=2

[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(2)^2-12(2)+15=8-24+15=-1 \\ \implies(2,-1) \end{gathered}[/tex]When x=1

[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(1)^2-12(1)+15=2-12+15=5 \\ \implies(1,5) \end{gathered}[/tex]Two points to the right of the vertex

When x=4

[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(4)^2-12(4)+15=32-48+15=-1 \\ \implies(4,-1) \end{gathered}[/tex]When x=5

[tex]\begin{gathered} y=2x^2-12x+15 \\ =2(5)^2-12(5)+15=50-60+15=5 \\ \implies(5,5) \end{gathered}[/tex]**Answer:**

Plot these points on the graph: (3, -3), (2,-1), (1,5), (4,-1), and (5,5).

Trying to find the length of the robot arm given the xy point along with angle theta and angle theta r

8a. The **length **of the robot arm, and the **angle **in radian and degrees are as follows

8b. The **length **of the robot arm, and the angle in radian and degrees are as follows

**8a**

i. let the length of the robot arm be r

The length of the lever arm using the points is done by

r^2 = ( -26 )^2 + ( -15 )^2

r^2 = 676 + 225

r^2 = 900

r = √ 900

r** = 30**

ii. theta in radian

tan θ = opposite / adjacent

tan θ = y direction / x direction

tan θ = -15 / -26

tan θ = 15/26

θ = Arc tan ( 15/26 )

θ = 29.9816 degrees

θ ≈ 30 degrees

**θ ≈** 30 degrees + 180 =** 210 degrees**

In radian

π = 180 degrees

? = 210 degrees

we cross multiply to get

? * 180 degrees = π * 210 degrees

? = π * 210 degrees / 180 degrees

= **7π/6 radian**

**8b**

i. let the length of the lever arm be r

The length of the lever arm using the points is done by

r^2 = ( -30 )^2 + ( 52 )^2

r^2 = 900 + 2704

r^2 = 3604

r = √ 3604

r = 60.033

**r ≈ 60**

ii. theta in radian

tan θ = opposite / adjacent

tan θ = y direction / x direction

tan θ = -30 / 52

tan θ = -30/52

θ = Arc tan ( -30/52 )

θ = -29.9816 degrees

θ ≈** -30 degrees **(clockwise direction)

**In radian**

π = 180 degrees

? = 30 degrees

we cross multiply to get

? * 180 degrees = π * 30 degrees

? = π * 30 degrees / 180 degrees

= **π/6 radian**

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-3-23x=-14(x-21)-15(x+33)

Find (x)

Please I keep messing up somewhere so please show step by step

The value of **(x)** that **satisfy** the equation → **- 3 - 23x = - 14(x - 21) - 15(x + 33) **is **x = - 33.**

What is Equation?

An equation is a mathematical statement with an **'equal to'** symbol between **two expressions** that have **equal values.**

Given is the following **equation** -

**- 3 - 23x = - 14(x - 21) - 15(x + 33)**

The given **equation** is -

- 3 - 23x = - 14(x - 21) - 15(x + 33)

Simplifying for **(x)**, we get -

- 3 - 23x = - 14x + 294 - 15x - 495

- 23x + 14x + 15x = 3 + 294 - 495

6x = - 198

**x = - 33**

Therefore, the value of **(x)** that **satisfy** the equation → **- 3 - 23x = - 14(x - 21) - 15(x + 33) **is **x = - 33.**

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a box contains 3 white balls and 4 black balls. a ball is drawn at random the color is recorded and then the ball is put back in the box. Then a second ball is drawn at random from the same box. find the probability of the event that at least one of the balls is white

The box has 3 white balls and 4 black balls.

Total number of balls = 3 + 4 = 7

**First draw:**

The probability of getting a white ball is given by

[tex]\begin{gathered} P(white)=\frac{\text{number of white balls}}{total\text{ number of balls}} \\ P(white)=\frac{3}{7} \end{gathered}[/tex]**Second draw:**

Notice that after the first draw the ball is put back in the box.

The probability of getting a white ball is given by

[tex]P(white)=\frac{3}{7}[/tex]At least one of the balls is white means that one white ball or two white balls.

[tex]P(x\ge1)\; =P(x=1)+P(x=2)_{}[/tex]We have already found the probability of getting one white ball that is P(x=1) = 3/7

The probability of getting two white balls is

[tex]\begin{gathered} P(two\; white)=P(white)\times P(white) \\ P(two\; white)=\frac{3}{7}\times\frac{3}{7}=\frac{9}{49} \end{gathered}[/tex]Finally, the probability of at least one white ball is

[tex]\begin{gathered} P(x\ge1)\; =P(x=1)+P(x=2)_{} \\ P(x\ge1)\; =\frac{3}{7}+\frac{9}{49} \\ P(x\ge1)\; =\frac{30}{49} \end{gathered}[/tex]Therefore, the probability of the event that at least one of the balls is white is **30/49**

4. Manu determines the roots of a polynomial equation by applying the theorems he knows. He organizes the results of these theorems.

From the fundamental theorem of algebra, Manu knows there are 3 roots to the equation.

From Descartes’ rule of sign, Manu finds no sign changes in and 3 sign changes in .

The rational root theorem yields as the list of possible rational roots.

The lower bound of the polynomial is .

The upper bound of the polynomial is 1.

What values in Manu’s list of rational roots should he try in synthetic division in light of these findings?

The values in **Manu’s **list of rational roots he should try in the **synthetic division**, in light of these findings is

"**Manu **should try first +1/4,+1/2 and +1."

This is further explained below.

What is the synthetic division?Generally, Because the lower and higher **bonds **are located between -6 and 1, **Manu **should have only analyzed the potential rational zeros that fall between those two numbers.

A lower bond indicates that the feasible rational zero cannot be lower than -6, and therefore the higher number that may be achieved cannot be more than 1.

In this manner, **Manu **will get an opportunity to test. This approach of rationalization based on bonds aims to reduce the number of potential solutions.

Therefore, if **Manu **discovers through the synthetic division that the possible roots are +1/4,+1/2,+1,+5/4,+2,+5/2,+4,+5,+10,+20, then he should only consider those inside the intervals marked by the lower and upper **bounds**, which are +1/4,+1/2,+1, because the rest is greater than 1. This is because the rest of the possible **roots **are higher than 1.

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CQ

Manu determines the roots of a polynomial equation p(x)=0 by applying the theorems he knows. He organizes the results of these theorems. From the fundamental theorems of algebra, Manu knows there are 3 roots to the equation. From Descartes' rule of sign, Manu finds no sign changes in p(x) and 3 sign changes in p(-x). The rational root theorem yields +1/4,+1/2,+1,+5/4,+2,+5/2,+4,+5,+10,+20 as a list of possible rational roots. The lower bound of the polynomial is -6. The upper bound of the polynomial is 1. What values in Manu's list of rational roots should he try in synthetic division in light of these findings?

The graph shows the reciprocal parent function.Which statement best describes the function?O A. The function is negative when x < 0.OB. The function is never negative.C. The function is negative when x > 0.O D. The function is negative when x < 0 and also when x > 0.PREVIOUS

Let's begin by identifying key information given to us in the graph:

The reciprocal parent function is given as **1/f(x)**. This is better written as shown below

When the value for **x **is greater than zero, the function is positive as shown below:

When the value of **x **is lesser than zero, the function is negative as shown below:

**Therefore, the correct answer is option A (The function is negative when x < 0)**

The nutty professor sells cashews for $6.20 per pound and Brazil nuts for $4.30 per pound. How much of each type should be used to make a 32 pound mixture that sells for $5.37 per pound?

Let's call C the weight of the Cashews and B the weight of the Brazil nuts.

Since the professor wants to sel a 32 pound mixture, we have the following equation:

[tex]C+B=32.[/tex]Now, since the price of the mixture will be $5.37 per pound, this means the price of the whole mixture will be

[tex]32\cdot5.37=171.84.[/tex]This leads us to the followin equation:

[tex]6.20C+4.30B=171.84.[/tex]Now we have a system of equations:

[tex]C+B=32[/tex][tex]6.20C+4.30B=171.84.[/tex]To solve it, let's isolate one of the variables in the first equation by subtracting B from both sides of it:

[tex]C=32-B\text{.}[/tex]Now, let's use this value of C in the second equation:

[tex]6.20(32-B)+4.30B=171.84,[/tex][tex]198.4-6.20B+4.30B=171.84,[/tex][tex]198.4-1.9B=171.84.[/tex]To solve this equation, let's subtract 198.4 from both sides of it:

[tex]-1.9B=-26.56.[/tex]Then, let's divide both sides by -1.9:

[tex]B\approx13.98.[/tex]Using this value of B in the first equation will give us:

[tex]C=32-13.98=18.02.[/tex]**These would be the values of B and C round to two decimals.** If we want, we can also write them as integer numbers. so **the mix would need to have 18 pounds of cashews and 14 pounds of Brazil nuts. **

i need help with this asap please

Answer:

Given that,

[tex]2+(-8)+32+(-128)+.\ldots_{}[/tex]To find the sum of the first 5 terms.

First, to find the first 5 terms of the given sequence.

The given sequence is 2,-8,32,-128,...

It follows geometric series with initial term 2, and common ratio as -4

The explicit formula of the given sequence is,

[tex]t_n=2(-4)^{n-1}_{}_{}[/tex]To find the 5th term of the sequence,

Put n=5 in the above equation we get,

[tex]t_5=2(-4)^{5-1}[/tex][tex]t_5=2(-4)^4[/tex][tex]t_5=2(256)[/tex][tex]t_5=512[/tex]Since common ratio is less than 1, we get the sum of the series formula as,

[tex]S_n=\frac{a(1-r^n)}{1-r}[/tex]Substituting the values we get,

[tex]S_5=\frac{2(1-(-4)^5)}{1+4}[/tex][tex]=\frac{2(1+1024)}{5}[/tex][tex]=\frac{2(1025)}{5}[/tex][tex]=2(205)[/tex][tex]=410[/tex]The sum of the first 5 terms of the given series is 410.

**Answer is: option B: 410**

Use y = 3x2 + 18x - 2 to answer the following question(1, 19) is a point on the graph. What point is the reflection of (1, 19) across the axis of symmetry of the parabola?

Since y is a parabolla, there will be two values for y = 19. We already know that x = 1 is one value, to find the other, we can substitute y = 19 on the equation and solve for x to get the following:

[tex]\begin{gathered} 19=3x^2+18x-2 \\ \Rightarrow3x^2+18x-2-19=0 \\ \Rightarrow3x^2+18x-21=0 \\ \Rightarrow3(x^2+6x-7)=0 \\ \Rightarrow3(x-1)(x+7)=0 \end{gathered}[/tex]the solutions of the equation are x = 1 and x = -7. Since we already have that (1,19) is a point on the graph, then we have that the other point is** (-7,19)**

What is the slope of the line passing through (-2, 4)

and (3, -4)?

**Answer:**

[tex]\frac{-8}{5}[/tex]

**Step-by-step explanation:**

Slope is the change in y over the change in x

[tex]\frac{y_{2 - y_{1} } }{x_{2 - y_{1} } }[/tex]

(-2,4) is ([tex]x_{1}[/tex], [tex]y_{1}[/tex])

(3. -4) is ([tex]x_{2}[/tex],[tex]y_{2}[/tex])

[tex]\frac{-4 - 4}{3 - -2}[/tex] = [tex]\frac{-8}{3+ 2}[/tex] = [tex]\frac{-8}{5}[/tex]

**Answer:**

-8 / 5

**Step-by-step explanation:**

We can use the **slope formula**

m = ( y2-y1)/(x2-x1)

= ( -4 -4)/( 3 - -2)

= -8/ ( 3+2)

= -8 / 5

Solve the system of inequalities by graphing.

y ≥ 2

y < 4

Select a line to change it between solid and dotted. Select a region to shade it.

The** area **between the solid line (y=x+4) and the dotted line (y=-2x-2) represents the system of** inequalities **solution set.

Two linear** inequalities **system on a **coordinate plane**. The first has a solid line graphed with a negative slope of one, a negative y-intercept, and a **shaded** origin area. The area encompassing the origin is shaded, and the second is a dashed vertical line 3 units to the left of the origin.

x ≥ –3; y ≥ x – 2

x > –3; 5y ≥ –4x – 10

x > –3; y ≥ –x + 1

x > –2; y ≥ –x – 1

The solution set for the system of** inequalities** is represented by the region between the solid line (y=x+4) and the dotted line (y=-2x-2).

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Determine if the line passing through A(7,5) and B(-14, -9) is parallel, or perpendicular to the line passing through C(0,1) and D(4, -5).

To solve this problem, we will use the two pair of points to find the slope of the equation of each line. Then, by comparing these slopes, we can determine either if they are perpendicular or parallel.

**Slope calculations**

To calcula the slopes, given the pairs of points, we are going to use the following formula: Given points (a,b) and (c,d) the slope of the line that passes through them is given by the formula

[tex]m=\frac{d\text{ - b}}{c\text{ -a}}=\frac{b\text{ - d}}{a\text{ -c}}[/tex]Let us calculate first the slope of the line that passes through the points (7,5) and (-14,-9). In this case, we have a=7,b=5,c=-14 and d=-9. So we get

[tex]m=\frac{5\text{ - (-9)}}{7\text{ - (-14)}}=\frac{14}{21}=\frac{2\cdot7}{3\cdot7}=\frac{2}{3}[/tex]Now, let us calculate the slope of the line that passes through the points (0,1) and (4,-5). In this case, we have a=0,b=1,c=4 and d=-5. So we get

[tex]m=\frac{1\text{ -(-5)}}{0\text{ - 4}}=\frac{6}{\text{ -4}}=\text{ -}\frac{3\cdot2}{2\cdot2}=\text{ -}\frac{3}{2}[/tex]**Slope comparison**

Now, we compare the slopes to determine if the lines are perpendicular or parallel. Recall that two lines are parallel if they have the same slope and they are perpendicular if the product of their slopes is -1. From our calculations, we can see that the slopes are not equal. Let us confirm that they are perpendicular. To do so, we multiply both slopes. So we get

[tex]\frac{2}{3}\cdot(\text{ -}\frac{3}{2})=\text{ -1}[/tex]Since their product is -1, this confirms that both lines are perpendicular.

I need help with part b.

log ( √2.86668684 ) is value of **logarithm 18 .**

**What is logarithm used for in math?**

[tex]log_{b} \sqrt{18}[/tex] = [tex]log_{b} = log_{b} \sqrt{15} + log_{b} \sqrt{3}[/tex]

= log( √15 + √3 )

= log ( √1.0986 * 1.6094 + √1.0986 )

= log ( √1.76808684 + √1.0986 )

= log ( √2.86668684 )

Learn more about **logarithm **

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