a parabola has a directrix at the line y=6 and a focus at -8 , 9. write an equation for the parabola

Answer:

a. ? = 12

b. ? = -4

c. ? = 4

d. ? = -3

e. ? = -3

Step-by-step explanation:

The following parts can be solved by the concept of Area.

1. The area of the sprinkled lawn is 9π square feet, where π is approximately 3.14.

2. The area that the storm will affect is 113.1 square miles, where the value of π is approximately 3.14.

1. To find the area of the sprinkled lawn, we need to find the area covered by the sprinkler in one rotation and then multiply it by the number of rotations. The sprinkler sprays water 3 feet in every direction, so it covers a circular area with a radius of 3 feet. The area of a circle with a radius of 3 feet is given by the formula A = πr^2, where r is the radius of the circle.

Therefore, the area covered by the sprinkler in one rotation is 9π square feet (A = π x 3^2 = 9π). If the sprinkler rotates n times, then the total area covered by the sprinkler is n times the area covered in one rotation.

2. To find the area that the storm will affect, we need to find the area of the circle with a radius of 6 miles. The area of a circle is given by the formula A = πr^2, where r is the radius of the circle. Therefore, the area that the storm will affect is 36π square miles (A = π x 6^2 = 36π). To get a numerical value, we can use the approximation of π as 3.14, which gives us an area of approximately 113.1 square miles (A = 3.14 x 36 = 113.1).

Thus, the storm will affect an area of approximately 113.1 square miles.

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The length of each side of the octagon is 6 cm

It is a polygon with eight sides and eight angles. There are a total of 20 diagonals in it. All its interior angles sum up to 1080°.

Given that, a regular octagon has an area of 288 square centimeters and an apothem of 12 cm.

We need to find the side of the octagon,

We know that area of the octagon = 8/2 × side × apothem

Therefore,

288 = 8/2 × side × 12

side = 288 / 48

side = 6

Hence, the length of each side of the octagon is 6 cm

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Answer:

3

Step-by-step explanation:

here is explain.

2*3+3 = 9

number 3 from the replacement set makes the equation

the equation y = 4x - 3 is also not a solution for the ordered pairs y > -1/2x + 5 and y ≤ 3x - 2.

What is inequality?

Mathematical expressions with inequalities on both sides are known as inequalities. In an inequality, we compare two values as opposed to equations. In between, the equal sign is changed to a less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

To determine which equation is not a solution for the ordered pairs y > -1/2x + 5 and y ≤ 3x - 2, we can substitute different ordered pairs into the equations and see if they satisfy the inequalities.

For example, the ordered pair (0, 0) satisfies the inequality y > -1/2x + 5, since 0 > (-1/2)(0) + 5 = 5, and it satisfies the inequality y ≤ 3x - 2, since 0 ≤ 3(0) - 2 = -2.

Now, let's try the equations:

y = -x + 4

y = 4x - 3

Substituting (0, 0) into y = -x + 4, we get y = -0 + 4 = 4, which does not satisfy the inequality y > -1/2x + 5, since 4 is not greater than (-1/2)(0) + 5 = 5. Therefore, the equation y = -x + 4 is not a solution for the ordered pairs y > -1/2x + 5 and y ≤ 3x - 2.

On the other hand, substituting (0, 0) into y = 4x - 3, we get y = -3, which also does not satisfy the inequalities, since -3 is not greater than 5 or less than or equal to -2.

Therefore, the equation y = 4x - 3 is also not a solution for the ordered pairs y > -1/2x + 5 and y ≤ 3x - 2.

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Answer:

2/3 + 2/3 + 1/3.

Step-by-step explanation:

You can use equivalent fractions. 6 is 2x more than 3. You can use three as the denominator.

Mark decompose 5/6 into three fractions using fractions : 1/3, 5/18, 2/9

In Mathematics, fractions are represented as a numerical value, which defines a part of a whole. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.

Given, fraction

5/6

Multiplying 3 with both numerator and denominator

15/18

= (6 + 5 + 4)/18

= 6/18 + 5/18 + 4/18

= 1/3 + 5/18 + 2/9

Hence, 1/3, 5/18, 2/9 are fractions Mark used to decompose 5/6.

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Answer:

\(y^{5}\) + 11y³ - 2y² - 22

Step-by-step explanation:

(y³ - 2)(y² + 11)

each term in the second factor is multiplied by each term in the first factor.

y³(y² + 11) - 2(y² + 11) ← distribute both parenthesis

= \(y^{5}\) + 11y³ - 2y² - 22

Answer:

A = $6,625.00

Step-by-step explanation:

A = P(1 + rt)

A = 5000(1 + (0.065 × 5)) = 6625

Answer:

118/32

Step-by-step explanation:

To convert a fraction into an improper fraction, you need to take the denominator and multiply it by the whole number then add the numerator. This new number will be the numerator to your improper fraction while your old denominator will still stay your denominator. With this information, let's find what 3 11/16 is as an improper fraction:

Numerator: 16 * 3 + 11 = 48 + 11 = 59

Denominator: 16

Improper fraction: 59/16

Next, it wants us to change the denominator to 32. Since we would have to multiply our old denominator, 16, by 2 to get the new denominator, we will also need to multiply the numerator by 2.

Numerator: 59 * 2 = 118

Denominator: 16 * 2 = 32

Improper fraction in 32nds: 118/32

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The estimated cost in year 1 is $526,400.

The initial cost is $430,000, and the cost increases uniformly according to an arithmetic gradient of $10,000 per year. At an interest rate of 18% per year, the estimated cost in year 1 is $526,400.

The arithmetic gradient is the fixed amount added to the previous value to arrive at the new value. An example of an arithmetic gradient is an investment or a payment that grows at a consistent rate. The annual increase in cost is $10,000, and this value remains constant throughout the five-year period.

The formula for arithmetic gradient is:

Arithmetic gradient = (Final cost - Initial cost) / (Number of years - 1)

The interest rate, or the cost of borrowing, is a percentage of the amount borrowed that must be repaid along with the principal amount. We will use the simple interest formula to calculate the estimated cost in year 1 since it is not stated otherwise.

Simple interest formula is:

I = Prt

Where: I = Interest amount

P = Principal amount

r = Rate of interest

t = Time period (in years)

Calculating the estimated cost in year 1 using simple interest:Initial cost = $430,000

Arithmetic gradient = $10,000

Number of years = 5

Final cost = Initial cost + Arithmetic gradient x (Number of years - 1)

Final cost = $430,000 + $10,000 x (5 - 1)

Final cost = $430,000 + $40,000

Final cost = $470,000

Principal amount = $470,000

Rate of interest = 18%

Time period = 1 yearI = PrtI = $470,000 x 0.18 x 1I = $84,600

Estimated cost in year 1 = Principal amount + Interest amount

Estimated cost in year 1 = $470,000 + $84,600

Estimated cost in year 1 = $554,600 ≈ $526,400 (rounded to the nearest dollar)

Therefore, the estimated cost in year 1 is $526,400.

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Answer:

Gradient = -5

Step-by-step explanation:

The gradient of a line is the slope of the line.

The standard form of the line equation is y = mx  + b

where m is the slope ( or gradient) b is the y - intercept.

Given :

2y = -10x + 1

\(y = -\frac{10}{2} x + \frac{1}{2}\\\\y = -5x + \frac{1}{2}\\\\Therefore , \ gradient \ is \ -5\)

The gradient of the line whose equation is 2y=-10x+1 is -5.

Gradient could be a measure of how steep a slope or a line is.

The equation of the road is 2y=-10x+1

y=-5x+1/2

Therefore the gradient of the line is that the coefficient of x which is -5.

Hence the gradient of the line is -5.

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Answer:

Step-by-step explanation:

-11 +23-46+ (-12) =

12 - 46 + (-12) =

-34 - 12 =

-46

Answer:

\(r=5.79898987...\\A=510.31110884...\)

Step-by-step explanation:

Challenging question!

Start by constructing a square in the lower-left quadrant of the circle (that isn't shaded) that is tangent to center C, OP, and OQ:

(See picture)

With the blue line constructed, we can conclude that the blue line (the blue line is the measure from the center of the inscribed circle to the center of the quarter circle) measures \(\sqrt{2} r\) due to the Pythagorean theorem.

The orange line is the radius of the inscribed circle, so adding the blue line and the orange line gives the radius of the quarter circle, so:

\(\sqrt{2}r +r=14\)

Solve the equation with a bit of algebra to get:

\(\sqrt{2} r+r=14\\r(\sqrt{2}+1)=14\\r=\frac{14}{\sqrt{2}+1 }\\r=5.79898987...\)

Now, we will find the area of the inscribed circle:

\(A=\pi r^2\\A=\frac{22}{7} *(5.79898987...)^2\\A=105.68889115...\)

And we will now find the area of the quarter circle...:

\(A_{2} =\pi r^2\\A_{2}=\frac{22}{7}*14^2\\A_2=616\)

...and we subtract both results...:

\(A_{3}= 616-105.68889115...\\A_3=510.31110884...\)

To get the area of the shaded region.

Not sure what does the last part ("correct to 3 s. f.") meant but that's all I can do.

Using Euler's Method with a step size of h = 0.1, the approximate values of the solution at t = 0.1, 0.2, 0.3, 0.4, and 0.5 are:

t = 0.1: y ≈ 1.1

t = 0.2: y ≈ 1.22

t = 0.3: y ≈ 1.34

t = 0.4: y ≈ 1.47

t = 0.5: y ≈ 1.61

To use Euler's Method, we start with an initial condition. In this case, the given initial condition is y(0) = 1. We can then iteratively calculate the approximate values of the solution at each desired time point using the formula:

Yn = Yn-1 + h * F(Xn-1, Yn-1)

Here, h represents the step size (0.1 in this case), Xn-1 is the previous time point (t = Xn-1), Yn-1 is the solution value at the previous time point, and F(Xn-1, Yn-1) represents the derivative of the solution function.

For the given differential equation +2y = 2 - ey, we can rearrange it to the form y' = (2 - ey) / 2. The derivative function F(Xn-1, Yn-1) is then (2 - eYn-1) / 2.

Using the initial condition y(0) = 1, we can proceed with the calculations:

t = 0.1:

Y1 = Y0 + h * F(X0, Y0)

= 1 + 0.1 * [(2 - e^1) / 2]

≈ 1 + 0.1 * (2 - 0.368) / 2

≈ 1 + 0.1 * 1.316 / 2

≈ 1 + 0.1316

≈ 1.1

Similarly, we can calculate the approximate values of the solution at t = 0.2, 0.3, 0.4, and 0.5 using the same formula and previous results.

Using Euler's Method with a step size of h = 0.1, we found the approximate values of the solution at t = 0.1, 0.2, 0.3, 0.4, and 0.5 to be 1.1, 1.22, 1.34, 1.47, and 1.61, respectively.

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Rates are used to show how a quantity measures over another.

Andy does not read at a constant rate.

The table is given as:

\(\left[\begin{array}{cccccc}x&10&20&30&40&50\\y&4&9&10&15&20\end{array}\right]\)

Pick any two points from the table:

\((x_1,y_1) = (10,4)\)

\((x_2,y_2) = (20,9)\)

The rate (m) is then calculated as:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

So, we have:

\(m = \frac{9-4}{20-10}\)

\(m = \frac{5}{10}\)

\(m = 0.5\)

Pick another point from the table

\((x_1,y_1) = (10,4)\)

\((x_2,y_2) = (30,10)\)

The rate (m) is then calculated as:

\(m = \frac{y_2 - y_1}{x_2 - x_1}\)

\(m = \frac{10 - 4}{30 - 10}\)

\(m = \frac{6}{20}\)

\(m = 0.3\)

The rates calculated are not equal

i.e. 0.3 and 0.5 are not equal

This means that;

Andy does not read at a constant rate.

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Answer:

c=65

Step-by-step explanation:

a=115

180-115=b

b=65

b=c

c=65

The simplified expression is 6x.

When we are asked to simplify an expression, we typically want to rewrite it in a more compact or easier-to-understand form. In this case, we want to remove any parentheses (symbols of grouping) and combine like terms.

The expression we're given is:

6(x − 6) + 36

To begin simplifying this expression, we can first distribute the 6 to both terms inside the parentheses:

6(x - 6) = 6x - 36

Now our expression looks like:

6x - 36 + 36

Notice that the second and third terms, -36 and 36, are additive inverses of each other. That means that when we add them together, they cancel each other out:

-36 + 36 = 0

So we can rewrite our expression as:

6x + 0

But 0 doesn't actually contribute anything to the expression, so we can just drop it:

6x

So the simplified form of the expression is just 6x.

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Answer: The answer should be A.

Step-by-step explanation: 1/2 x -16 = -8

(a) ∠ORP and ∠ORN are a linear pair .

Let us prove that option a , is the correct option by finding the value of x .

Angle ORP and angle ORN are a linear , which means their sum will be equal to 180° .

So :-

\(80 + 3x + 10 = 180\)

\(3x + 80 + 10 = 180\)

\(3x + 90 = 180\)

\(3x = 180 - 90\)

\(3x = 90\)

\(x = \frac{90}{3} \)

\(x = 30°\)

As we were able to find put the value of x only after we cam to know that ∠ORP and ∠ORN are a linear pair , we say that option a is the correct statement .

Therefore , the correct answer is :-

(a) ∠ORP and ∠ORN are a linear pair .

The value of x is equal to 30°. Hence, proved.

Angles on a straight line relate to the sum of angles that can be arranged together so that they form a straight line. Angles on a straight line add to 180°.

Given that, m∠ORP=80° and m∠ORN=(3x+10)°.

Here, m∠ORP+m∠ORN=180° (Angles on straight line adds upto 180°)

80°+(3x+10)°=180°

90°+3x=180°

3x=90°

x=90°/3

x=30°

Hence, proved.

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Answer:

v=4/7

Step-by-step explanation:

81/8 = 1/3v + 25/8 - 23/6v

-25/8            -25/8

56/8 = 1/3v -23/6v

56/8 = 24/6v

24/6 / 56/8

4/7

Since Mila has made 50% of her free-throws. The word or phrase that describes the probability that Mila will hit her next free throw is uncertain or unknown.

To determine the likelihood of successful free throws, divide the number of successful attempts by the total number of attempts, and then multiply by 100.

A forecast of an probability distribution of uncertainty is passed by a probability distribution, and this term ought to be subject to modification only upon having additional knowledge.

Due to the fact that Mila has made 50% of her free throws in all, one cannot say or predict with accuracy if she will make her next free throw

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Answer:

The measure of the indicated exterior angle is 110°

Step-by-step explanation:

Let us revise the rule of the exterior angle

In a triangle, the measure of an exterior angle at a vertex of the triangle equals the sum of the measures of the two opposite interior angles to this vertex.

We will use this rule to solve the question

∵ The ladder, the house, and the ground formed a right triangle

∵ The ladder's angle with the ground facing away from the house is

   an exterior angle of the triangle

∵ The angles of measures 90° and 20° are its opposite interior angles

→ By using the rule above

∴ The measure of the exterior angle = 90° + 20°

∴ The measure of the exterior angle = 110°

∴ The measure of the ladder's angle with the ground facing away

   from the house is 110°

∴ The measure of the indicated exterior angle is 110°

Using proportions, it is found that the hedgehog loses 3.45 ounces after 115 days of hibernation.

A proportional relationship is modeled by:

\(y = kx\)

In which k is the constant of proportionality.

In this problem, the ratio between the weighs loss after x days in hibernation is given by:

\(k = \frac{0.24}{8} = \frac{0.84}{28} = \frac{2.25}{75} = \frac{2.79}{93} = 0.03\)

Since this ratio is constant, the relationship is proportional, modeled by:

\(y = 0.03x\)

The weight lost after 115 days of hibernation is y(115), hence:

\(y = 0.03(115) = 3.45\)

The hedgehog loses 3.45 ounces after 115 days of hibernation.

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Answer:

0

Step-by-step explanation:

We can use the formula change in y divided by change in x for this, so for example we can take the points (6, 2) and (3, 2).

The change in y would be 2 - 2 = 0.

The change in x would be 3 - 6 = -3

Chang in y divided by change in x = 0/-3 = 0.

Answer:

x ≥ 7/3

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Equality Properties

Step-by-step explanation:

Step 1: Define Inequality

-15x + 4 ≤ 39

Step 2: Solve for x

Here we see that any value x greater than or equal to 7/3 would work as a solution to the inequality.

The average number of beads that the girls have is 63

Let's start by using algebra to represent the given information:

Let b be the number of beads that Sally has.

Then, Kelly has 3/4 times as many beads as Sally, which can be expressed as (3/4)b.

Also, we know that Kelly has 18 more beads than Sally, which can be expressed as (b + 18).

Putting these together, we can write the equation:

(3/4)b = b + 18

Solving for b, we get:

b = 72

So, Sally has 72 beads, and Kelly has (3/4) × 72 = 54 beads.

The average number of beads that the girls have is (72 + 54)/2 = 63 beads

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Answer:

To identify the slopes and y-intercepts of the given lines, we need to rewrite them in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

a) 5x + 9y - 25 = 0

First, isolate y:

9y = -5x + 25

y = (-5/9)x + 25/9

Slope: -5/9

Y-intercept: 25/9

b) 4x + 5y + 5 = 0

Isolate y:

5y = -4x - 5

y = (-4/5)x - 1

Slope: -4/5

Y-intercept: -1

c) 5x + 2y - 4 = 0

Isolate y:

2y = -5x + 4

y = (-5/2)x + 2

Slope: -5/2

Y-intercept: 2

d) 3x - 5y - 5 = 0

Isolate y:

-5y = -3x + 5

y = (3/5)x - 1

Slope: 3/5

Y-intercept: -1