the measure of angle abd is (0.12x + 67) deg and the measure of angle cbd is (0.13x + 26) deg find the value of x.

Solving the quadratic equation we can see that the object will be 6.5 seconds falling down.

We know that the height is modeled by the quadratic equation

h=-16t²+676

The object will reach the ground when the height is zero, so we need to solve:

-16t²+676 = 0

Solving this for t we will get:

16t² = 676

t² = 676/16

t = √(676/16)

t = 6.5

The object will be 6.5 seconds falling down.

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Answer:

This statement is false as it suggests a negative correlation between average annual income and the record time to run 1 mile. A negative correlation means that as one variable increases, the other decreases. However, it is unlikely that there is any correlation between these two variables as they are not directly related. One's ability to run a mile does not necessarily depend on their income, and vice versa. It is possible that both variables have been changing over time, but it does not necessarily mean that there is a correlation between them.

Step-by-step explanation:

Answer: To compare the rates of change of the line y = -2x + 6 at the points (2, 2) and (1, 6), we need to find the slope (rate of change) of the line at each point.

The slope of a line is given by the formula:

slope = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are two points on the line.

Using the points (2, 2) and (1, 6), we can find the slopes of the line at these points:

At point (2, 2):

slope = (y₂ - y₁) / (x₂ - x₁)

= (2 - 6) / (2 - 1)

= -4

So the slope of the line at point (2, 2) is -4.

At point (1, 6):

slope = (y₂ - y₁) / (x₂ - x₁)

= (6 - 2) / (1 - 2)

= 4

So the slope of the line at point (1, 6) is 4.

Since the slope at (1,6) is positive and greater than the slope at (2,2), we can conclude that the rate of change of y with respect to x is increasing as we move from point (2,2) to (1,6). In other words, the line is getting steeper as we move from right to left along the line.

Step-by-step explanation:

Answer:

80

Step-by-step explanation:

The variable "n" is not defined, but rather "k" is used in the summation. Assuming that "n" was meant to be "k", the expression should be:

s4 = ∑ k=1 to 4 of 2(3^(k-1))

To evaluate this expression, we need to substitute each value of k from 1 to 4 into the expression 2(3^(k-1)), and then sum up the results.

Starting with k = 1:

2(3^(1-1)) = 2(3^0) = 2(1) = 2

Moving on to k = 2:

2(3^(2-1)) = 2(3^1) = 2(3) = 6

Next, k = 3:

2(3^(3-1)) = 2(3^2) = 2(9) = 18

Finally, k = 4:

2(3^(4-1)) = 2(3^3) = 2(27) = 54

Now we add up these four results:

s4 = 2 + 6 + 18 + 54 = 80

Therefore, the value of s4 is 80.

The simplified form of csc²t/ [csc²(t) - 1] is sec²t  

Trigonometric identities are mathematical relationships between trigonometric functions such as sine, cosine, tangent, cotangent, secant, and cosecant. These identities are used to simplify trigonometric expressions and solve trigonometric equations.

To solve the given expression we used the following trigonometric identities  

Here we have

csc²t/ [csc²(t) - 1]  

As we know csc θ = 1/sin θ  

=> sinθ = 1/cscθ

Take csc² t as common from numerator and denominator

=> csc²t [1]/ csc²t [ 1 - (1/csc²t)]

=> 1/ 1 - (1/csc²t)     [ csc²t will be canceled ]

=> 1/ 1 - sin²t  

From trigonometric identities  1 - sin²t  = cos²t

=> 1/ cos²t

= sec²t

Therefore,

The simplified form of csc²t/ [csc²(t) - 1] is sec²t

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Joe wants to hang a flag weighing 800 N because Cable A, tied to the store, forms a 30.0° angle, and Cable B, attached to a neighboring building, is horizontal. Cable B is thus under 461 N of stress.

First, a [+ object] measuring the weight of something or someone in order to determine its size (someone or something) Every day she weighs herself. To weigh the bananas, he used a scale.

Let's refer to the tensile stress in cables A or B, respectively, as T A and T B. With an 800 N magnitude, the sign's weight is acting downward. The weight must be balanced by the vertical tension force components because the sign also isn't moving vertically:

800 N when T A is cos 30°

T A = 800 N/cos (30°)

T A = 922 N

Without any horizontal acceleration, a horizontal component of the tensile stress must also be balanced. T A's horizontal component is T A sin 30°, while T B's horizontal component is 0 (since cable B is horizontal). Therefore:

A sin (30°) = T b

T A = T B sin (30°)

T B ≈ 922 N x 0.5, or 461 N.

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The joint frequencies are the cells where the categories for two variables in a two-way frequency table intersect, while the marginal frequencies are the sums of the row or column in a two-way frequency table.

Joint frequencies represent the frequency counts of each combination of categories for two variables, while marginal frequencies represent the frequency counts of each category for one variable, regardless of the other variable.

For example, in a two-way frequency table of gender and political affiliation, the joint frequency for "Male" and "Republican" would represent the number of individuals who are both male and Republican.

The marginal frequency for "Male" would represent the total number of males in the sample, regardless of their political affiliation. The marginal frequency for "Republican" would represent the total number of individuals who identify as Republican, regardless of their gender.

Understanding the difference between joint and marginal frequencies is important for analyzing the relationship between two variables and identifying any patterns or trends.

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Answer:

We can start by using the given information to find the initial value of the gift card, which is the value before Alexander started using it:

Initial value of gift card = $9 + 4 x $1.50 = $15

Let's use A to represent the amount of money remaining on the card after buying x cups of coffee. Each cup of coffee costs $1.50, so the amount spent on coffee after buying x cups is 1.5x. The remaining balance on the card can be found by subtracting the amount spent on coffee from the initial value of the card:

A = $15 - 1.5x

Therefore, the equation for A in terms of x is:

A = $15 - 1.5x

Step-by-step explanation:

To create the cylindrical travel case, we need approximately 588.75 square inches of leather.

What is a cylinder?

A cylinder is a three-dimensional geometric shape that consists of two parallel circular bases connected by a curved surface. The curved surface is formed by connecting every point on the edge of one circular base to the corresponding point on the other circular base with straight lines that are perpendicular to the bases. The distance between the two circular bases is the height of the cylinder.

Now,

The surface area of a cylinder is given by the formula:

A = 2πr² + 2πrh

where r is the radius of the circular base, h is the height of the cylinder, and π is the mathematical constant pi, which is approximately equal to 3.14.

In this case, we are given that the cylinder has a diameter of 15 inches, so the radius is 7.5 inches (half of the diameter). We are also given that the height of the cylinder is 5 inches. Using these values in the formula, we can calculate the surface area of the cylinder as:

A = 2π(7.5)² + 2π(7.5)(5)

= 2π(56.25) + 2π(37.5)

= 2(π)(93.75)

= 187.5π

≈588.75

Therefore,

                  we need approximately 588.75 square inches of leather to create the travel case.

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Elvira's walking speed is 5 miles per hour, and Aletheia's walking speed is approximately 4.17 miles per hour.

Let's denote Elvira's walking speed as x (in miles per hour).

Then, Aletheia's walking speed is x - 0.6 (since she walks 0.6 miles per hour slower than Elvira).

We know that Elvira walked to the coffee shop in half an hour, covering a distance of 2.5 miles.

This means her speed can be calculated as:

x = distance / time = 2.5 miles / 0.5 hours = 5 miles per hour

Now, we can find Aletheia's speed using the equation we found earlier:

x - 0.6 = Aletheia's speed

We also know that Aletheia walked to the coffee shop in three-fifths of an hour (which is 0.6 hours), covering the same distance of 2.5 miles.

This means her speed can be calculated as:

x - 0.6 = distance / time = 2.5 miles / 0.6 hours ≈ 4.17 miles per hour

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Part A: The intersection of these two lines is (-1,-4).  Since x=-1, this is also the solution to 4x=2x-2 as per the graph.

Part B: Table attached.

Part C: Graph attached.

Graphing a function involves tracing the curve on a coordinate plane that corresponds to the function. If the curve represents the function for the curve, then each point on the curve will satisfy the function equation. The graph below shows the linear function f(x) = -x+ 2 as an illustration.

In the question,

The expression 4x = 2x - 2 reduces to x = -1.  This is true for all values of y, since y is not a factor in this expression.

The two other equations intersect at point (-1, -4).  See the attached graph (Solutions2)

y = 4x

y = 2x−2

Mathematically, we can substitute the value of y from the first equation into the second:

y = 2x−2

(4x) = 2x−2

2x = -2

x = -1

The intersection of these two lines is (-1,-4).  Since x=-1, this is also the solution to 4x=2x-2.

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Answer:

D

Step-by-step explanation:

..

Step-by-step explanation:

45+6x+15=180

60+6x=180

6x=180-60

6x/6=120/6

X=20

The value of x from the same side interior angles in a parallel lines is x = 20

Angles in parallel lines are angles that are created when two parallel lines are intersected by another line. The intersecting line is known as transversal line.

We can conclude three factors determining parallel lines ,

Alternate angles are equal

Corresponding angles are equal

Co-interior angles add up to 180°

Given data ,

Let the parallel lines be represented as A and B

Now , the measures of angles are

p = 45°

And , q = ( 6x + 15 )°

where p and q are same side interior angles

And , the same side interior angles add up to 180°

On simplifying , we get

45 + 6x + 15 = 180

6x + 60 = 180

Subtracting 60 on both sides , we get

6x = 120

Divide by 6 on both sides , we get

x = 20

Hence , the angles in parallel lines are solved and x = 20

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The given inequality 2x² 25x + 33 < 0 has been proved by using properties of parallelogram.

The formula A = bh, where b is the parallelogram's base and h is its height, determines the area of a parallelogram.

Let's use x + 6 cm for the parallelogram's base and 2x + 3 cm for its height in the diagram. The parallelogram's area can therefore be represented as follows:

A = (x + 6)(2x + 3)

By extending this phrase, we get:

A = 2x² + 15x + 18x + 18

A = 2x² + 33x + 18

We now know that the parallelogram's area is more than 10.5 cm2. As a result, we can create the disparity shown below:

2x² + 33x + 18 > 10.5

By taking away 10.5 from both sides, we arrive at:

2x² + 33x + 18 - 10.5 > 0

By condensing the left side, we obtain:

2x² + 22.5x + 7.5 > 0

When we multiply both sides by 2, we obtain:

4x² + 45x + 15 > 0

By taking 3 away from both sides, we arrive at:

4x² + 45x + 12 < 0

As a result, we have demonstrated that 2x² 25x + 33 < 0.

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The answer of tan B can be calculated only by knowing all the sides of triangle by using Pythagorean theorem and the answer is 8/15.

What is Pythagorean theorem?

The pythagorean theorem says In a right angle triangle if any of the two sides is known then the third side can be calculated.

                                      c²= a²+ b²

Now in the following to calculate 3rd side, let us say x

                                                 => 8²+x² = 17²

                                                     x= 15

                              Tan B= opposite /adjacent

                                          = 8/15

                     So the tan B value will be 8/15

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By answering the above question, we may state that The chance of probability drawing a non-red marble is just the probability of drawing a non-red marble. As a result, the probability is 209/210.

Probabilistic theory is a branch of mathematics that calculates the likelihood of an event or proposition occurring or being true. A risk is a number between 0 and 1, with 1 indicating certainty and a probability of around 0 indicating how probable an event appears to be to occur. Probability is a mathematical term for the likelihood or likelihood that a certain event will occur. Probabilities can also be expressed as numbers ranging from 0 to 1 or as percentages ranging from 0% to 100%. In relation to all other outcomes, the ratio of occurrences among equally likely alternatives that result in a certain event.

The likelihood of drawing a red marble is 210, which indicates that one red marble is drawn out of every 210 in the jar. This probability may be expressed as a fraction:

P(Red) = 1/210

The likelihood of drawing a non-red marble is the inverse of the probability of drawing a red marble:

P(Not Red) = 1 minus P(Red) = 1 minus 1/210 = 209/210

As a result, the chance of drawing a marble that is not red is 209/210.

The chance of drawing a non-red marble is just the probability of drawing a non-red marble. As a result, the probability is 209/210.

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To solve the exponential equation 4^(3+x) = 25 for x, we need to use logarithms. We can use any base of the logarithm to solve this equation. However, some bases may be more convenient or easier to use than others. The possible bases of the logarithm that we can use to solve this equation are:

Log base 4: If we take the logarithm of both sides of the equation with base 4, we get:

log₄(4^(3+x)) = log₄(25)

(3+x)log₄(4) = log₄(25)

3+x = log₄(25)/log₄(4)

3+x = 2.3219

x ≈ -0.6781

Log base 25: If we take the logarithm of both sides of the equation with base 25, we get:

log₂₅(4^(3+x)) = log₂₅(25)

(3+x)log₂₅(4) = 1

3+x = 1/log₂₅(4)

3+x = 1/1.3219

x ≈ -1.6781

Log base x: If we take the logarithm of both sides of the equation with base x, we get:

logₓ(4^(3+x)) = logₓ(25)

(3+x)logₓ(4) = logₓ(25)

3+x = logₓ(25)/logₓ(4)

3+x = log₄(25)/log₄(x)

x = log₄(25)/log₄(x) - 3

Log base 10: If we take the logarithm of both sides of the equation with base 10, we get:

log₁₀(4^(3+x)) = log₁₀(25)

(3+x)log₁₀(4) = log₁₀(25)

3+x = log₁₀(25)/log₁₀(4)

3+x = 1.3979

x ≈ -1.6021

Therefore, we can use any of the above logarithm bases to solve for x in the given equation.

Answer:

To find the cost equation, we can use the two data points given:

(100, 1400) and (400, 4100)

We can use the point-slope form of a linear equation, where the slope is the change in cost over the change in quantity:

slope = (4100 - 1400) / (400 - 100) = 2700 / 300 = 9

Using the point-slope form with the first data point:

y - 1400 = 9(x - 100)

y - 1400 = 9x - 900

y = 9x + 500

So the cost equation is y = 9x + 500.

To find the cost to manufacture 150 shoes, we can plug in x = 150 into the cost equation:

y = 9(150) + 500 = 1850

So the cost to manufacture 150 shoes is $1850.

To find the revenue function, we multiply the number of shoes sold by the price per shoe:

Revenue = price x quantity = 19x

To determine the number of items needed to break even, we need to find the quantity where revenue equals cost. Let C(x) be the cost function and R(x) be the revenue function. The break-even point occurs when:

C(x) = R(x)

9x + 500 = 19x

500 = 10x

x = 50

So the company needs to sell 50 designer shoes to break even.

Step-by-step explanation:

Answer:

Cost equation:  y = 9x + 500

It costs $1,850 to manufacture 150 shoes.

Revenue function:  R(x) = 19x

The number of items that need to be sold to break even is 50.

Step-by-step explanation:

Definition of variables

If it costs $1,400 to manufacture 100 designer shoes:

If it costs $4,100 to manufacture 400 designer shoes:

Assuming the relationship between the number of shoes and the cost to manufacture them is linear, the slope of the equation of the line the models the relationship can be found by dividing the change in y-values by the change in x-values of the two data points:

[tex]\implies \sf Slope\;(m)=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{4100-1400}{400-100}=9[/tex]

Substitute the found slope and one of the points into the point-slope form of a linear equation to create an equation that gives the total cost to manufacture the shoes in terms of the number of shoes (x):

[tex]\implies \sf y-y_1=m(x-x_1)[/tex]

[tex]\implies \sf y-1400=9(x-100)[/tex]

[tex]\implies \sf y-1400=9x-900[/tex]

[tex]\implies \sf y=9x+500[/tex]

Therefore, the cost equation is y = 9x + 500.

To calculate the cost to manufacture 150 shoes, substitute x = 150 into the cost equation:

[tex]\begin{aligned}\implies \sf y&=\sf 9(150)+500\\&= \sf 1350+500\\&= \sf 1850\end{aligned}[/tex]

Therefore, it costs $1,850 to manufacture 150 shoes.

The revenue is the income a company generates before any expenses are subtracted. Therefore, the revenue function is simply the selling price of the item multiplied by the number of items sold.

Given the product sells for $19 per item, the revenue function is:

[tex]\implies \sf R(x)=19x[/tex]

The break even point is the point at which the total revenue equals the total cost, so there is neither profit nor loss.

To determine the number of items that should be sold to break even, equate the cost equation and the revenue function and solve for x.

[tex]\begin{aligned}\sf R(x)&=\sf y\\\implies \sf19x&=\sf 9x+500\\\sf 10x&=500\\\sf x&=50\end{aligned}[/tex]

Therefore, the number of items that need to be sold to break even is 50.

Answer: D 11.9

Step-by-step explanation:

To find the height of the flower pot 3 seconds after it falls, we need to substitute t=3 in the given function H(t) and calculate the value.

H(3) = 56 - 4.9(3)^2

H(3) = 56 - 4.9(9)

H(3) = 56 - 44.1

H(3) = 11.9

Therefore, the height of the flower pot, in meters, 3 seconds after it falls is 11.9 meters.

The answer is option D, 11.9.

Answer:

Krish saves 9 minutes by cycling to his office.

Step-by-step explanation:

Krish used to take 30 minutes to walk to his office. After he bought a new cycle, he takes only 7/10 of the time he used to take to reach the office. Let's call the new time he takes to reach the office "t" (in minutes).

We can set up a proportion to find the value of t:

30 / 1 = t / (7/10)

To solve for t, we can cross-multiply:

30 * 7/10 = t

21 = t

So, Krish now takes 21 minutes to reach his office by cycle.

To find how many minutes he saves by cycling, we can subtract the new time from the old time:

30 - 21 = 9

Krish saves 9 minutes by cycling to his office.

Option A. To translate a figure 4 units to the left, we need to subtract 4 from the x-coordinates of each vertex.

The original vertices are:

(4, 4), (4, 2), and (6, 3)

Subtracting 4 from the x-coordinates, we get the new vertices:

(0, 4), (0, 2), and (2, 3)

To understand how to translate a figure, it is helpful to visualize the original and new positions of the figure in a coordinate plane. In this case, the original triangle has vertices (4, 4), (4, 2), and (6, 3) as shown below:

   (4,4)

     / \

    /   \

(4,2)----- (6,3)

To translate this triangle 4 units to the left, we need to subtract 4 from the x-coordinates of each vertex. This will shift the entire triangle to the left by 4 units.

So, the new x-coordinates of the vertices become:

x-coordinate of (4, 4) - 4 = 0

x-coordinate of (4, 2) - 4 = 0

x-coordinate of (6, 3) - 4 = 2

The y-coordinates of the vertices remain the same.

Therefore, the new vertices of the triangle are:

(0, 4), (0, 2), and (2, 3)

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A triangle with vertices (4, 4), (4, 2), and (6, 3) is translated a distance of 4 units to the left. What are the new coordinates?

A. (0, 4), (0, 2), (2, 3)

B. (4, 0), (4, -2), (6, -1)

C. (4, 8), (4, 6), (6, 7)

D. (8, 4), (8, 2), (10, 3)

Answer:

To convert kilometers to miles, we can use the conversion factor 1 km = 0.621371 miles.

First, let's convert the total distance of 5.632704 km to miles:

5.632704 km * 0.621371 miles/km = 3.497672 miles

Next, let's convert the distance already hiked, 4.425969 km, to miles:

4.425969 km * 0.621371 miles/km = 2.746203 miles

To find out how many miles Lisa has left, we can subtract the distance already hiked from the total distance:

3.497672 miles - 2.746203 miles = 0.751469 miles

So Lisa has 0.751469 miles left to hike.

Step-by-step explanation:

The minimum annual interest rate for Kyoko to reach her goal, using daily compounding, is 9.903%.

The interest rate is computed using an online finance calculator that sets the following parameters.

The compounding period involved is 2,555 days for 7 years, based on the assumption that there are 365 days in a year.

N (# of periods) = 2555 days (7 years x 365 days)

PV (Present Value) = $10,000

PMT (Periodic Payment) = $-0

FV (Future Value) = $-20000

Results:

I/Y = 9.903% if interest compound 365 times per year (APR)

I/Y = 10.409% if interest compound once per year (APY)

I/period = 0.027% interest per period

Total Interest = $10,000.00

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Answer:

Step-by-step explanation:

We can simplify the expression on the left-hand side using trigonometric identities:

tan(-A) = -tan(A) (since tan(-θ) = -tan(θ))

sin(180°+A) = -sin(A) (since sin(180°+θ) = -sin(θ))

sec(270°-A) = -cos(A) (since sec(270°-θ) = -cos(θ))

cos²(90°+A) = sin²A (since cos²(90°+θ) = sin²θ)

Substituting these values, we get:

-x.sin(A).cos(A).tan(A) = x.sin(-A) - sin²A.sin(A)

-x.sin(A).cos(A).tan(A) = -x.sin(A) - sin³A

Now, we can simplify this equation by moving all the terms to one side:

-x.sin(A).cos(A).tan(A) + x.sin(A) + sin³A = 0

Factoring out sin(A), we get:

sin(A) (-x.cos(A).tan(A) + x + sin²A) = 0

Since sin(A) ≠ 0, we can divide both sides by sin(A):

-x.cos(A).tan(A) + x + sin²A = 0

Multiplying both sides by cos(A), we get:

-x.sin(A) + x.cos²(A) + sin²A.cos(A) = 0

Using the identity cos²(θ) + sin²(θ) = 1, we can write this as:

-x.sin(A) + x(1 - sin²A) + sin²A.cos(A) = 0

Simplifying and rearranging, we get:

x = sin(A)/(cos(A) - sin²(A))

Therefore, the value of x is given by the expression sin(A)/(cos(A) - sin²(A)).

Answer and Step-by-step explanation:

We can see that because Line K is acting as an Angle Bisector (a line that divides an angle into two angles of equal measures) on angle ∠LMJ, the angles ∠LMK and ∠JMK are equal to each other.

This allows us to solve for angle ∠LMK.

We do this by first setting the angles equal to each other.

10x - 22 = 2x + 66

Now, we want to isolate x by itself. We will subtract 2x from both sides of the equation, and then add 22 to both sides of the equation.

10x - 22 - 2x = 2x + 66 - 2x

8x - 22 = 66

8x - 22 + 22 = 66 + 22

8x = 88

Finally, we will divide both sides of the equation by 8.

8x ÷ 8 = 88 ÷ 8

x = 11

We aren't finished yet, as the question asks to find ∠LMK.

Plug 11 in for x into the equation 10x - 22.

10(11) - 22 =

110 - 22 =

= 88°

So, the measure of angle ∠LMK is 88°.

Have a good day!

Answer:

k = 16.806404 and k = 95.174404

Step-by-step explanation:

To determine the value of k in which the graph touches, you set both equations equal to each other

4x + k = -x^2 +8x + 20

k = -x^2+4x+20

You can use the quadratic formula -b^2+-(sqrt(b^2-4(a)(c))) / 2a

You will get the values of x = -2.898 and 6.898 for your answers

My bad, I forgot to add also input the values of x in your equation and get k = 16.806404 and k = 95.174404

Answer:

To find the x-intercept, we set y = 0 and solve for x:

r(x) = 2x^2 + 10x - 12 / x^2 + x - 6

0 = (2x^2 + 10x - 12) / (x^2 + x - 6)

0 = 2(x-1)(x+6) / (x-2)(x+3)

This gives us x-intercepts of x = 1 and x = -6.

To find the y-intercept, we set x = 0 and solve for y:

r(0) = 2(0)^2 + 10(0) - 12 / (0)^2 + (0) - 6

r(0) = -12/-6 = 2

So the y-intercept is y = 2.

To find the vertical asymptotes, we set the denominator of the function equal to zero and solve for x:

x^2 + x - 6 = 0

(x+3)(x-2) = 0

This gives us vertical asymptotes at x = -3 and x = 2.

To find the horizontal asymptote, we look at the degrees of the numerator and denominator. Since the degree of the numerator is 2 and the degree of the denominator is also 2, we divide the leading coefficient of the numerator by the leading coefficient of the denominator:

2 / 1 = 2

So the horizontal asymptote is y = 2.

Therefore, the x-intercepts are x = 1 and x = -6, the y-intercept is y = 2, the vertical asymptotes are x = -3 and x = 2, and the horizontal asymptote is y = 2.

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Answer:

[tex]\textsf{Degree:}\quad 6[/tex]

[tex]\textsf{Leading\;coefficient:}\quad \dfrac{1}{3456}[/tex]

[tex]\textsf{Zeroes:}\quad -6, 2, 5, 8[/tex]

[tex]\textsf{Factors:}\quad (x + 6), (x - 2), (x - 5), (x - 8)[/tex]

[tex]\textsf{Factors\;with\;multiplicity:}\quad (x + 6)^3(x - 2)(x - 5)(x - 8)[/tex]

[tex]\textsf{Equation\;of\;function:}\quad f(x)=\dfrac{1}{3456}(x+6)^3(x-2)(x-5)(x-8)[/tex]

Step-by-step explanation:

The zeroes are the x-values of the points where the function crosses the x-axis, so the x-values when f(x) = 0.

Therefore, the zeroes of the given function are:

[tex]\hrulefill[/tex]

According to the factor theorem, if f(x) is a polynomial, and f(a) = 0, then (x - a) is a factor of f(x). Therefore, the factors of the function are the x-values that satisfy f(x) = 0.

Therefore, the factors of the given function are:

[tex]\hrulefill[/tex]

The multiplicity of a factor is the number of times the factor appears in the factored form of the equation of the polynomial.

If the behaviour of the x-intercept is like that of a line, i.e. the curve passes directly through the intercept, its multiplicity is one.

Therefore, the factors (x - 2), (x - 5) and (x - 8) have multiplicity one.

The behaviour of the x-intercept at x = -6 is like that of a cubic function (S-shape). There, this zero has multiplicity 3: (x + 6)³.

[tex]\hrulefill[/tex]

So far we have found the zeros, factors and their multiplicities, so we can write a factored form of the function:

[tex]\implies f(x)=a(x+6)^3(x-2)(x-5)(x-8)[/tex]

The degree of the function is the highest exponent value of the variables in the polynomial. Therefore, to find the degree of the function, simply sum the exponents of the factors:

[tex]\implies \textsf{Degree}=3 + 1 + 1 + 1 = 6[/tex]

Therefore, the degree of the function is 6.

[tex]\hrulefill[/tex]

From inspection of the given graph, the y-intercept is (0, -5).

Therefore, to find the leading coefficient (value of a), substitute (0, -5) into the equation and solve for a:

[tex]\begin{aligned}\implies f(0)=a(0+6)^3(0-2)(0-5)(0-8)&=-5\\a(216)(-2)(-5)(-8)&=-5\\-17280a&=-5\\a&=\dfrac{1}{3456}\end{aligned}[/tex]

Therefore, the leading coefficient of the function is 1/3456.

[tex]\hrulefill[/tex]

Putting everything together, the function in factored form is:

[tex]f(x)=\dfrac{1}{3456}(x+6)^3(x-2)(x-5)(x-8)[/tex]

In standard form:

[tex]f(x)=\dfrac{1}{3456}x^6+\dfrac{1}{1152}x^5-\dfrac{1}{36}x^4-\dfrac{37}{432}x^3+\dfrac{17}{24}x^2+\dfrac{13}{8}x-5[/tex]

The volume of the cylinder with a diameter of 8cm and height of 2m is 100.48 cubic meters.

This question is incomplete, the complete question is:

Find the volumes of each of the following;

7) A cylinder with a diameter of 8cm and height of 2m?

A cylinder is simply a 3-dimensional shape having two parallel circular bases joined by a curved surface.

The volume of a cylinder is expressed as;

V = π × r² × h

Where r is radius of the circular base, h is height and π is constant pi ( π = 3.14 )

Given that;

Plug the given values into the above formula and solve for V.

V = π × r² × h

V = 3.14 × ( 4m )² × 2m

V = 3.14 × 16m² × 2m

V = 100.48 m³

Therefore, the volume is 100.48 cubic meters.

Learn more on volume of cylinder here: brainly.com/question/16788902

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