write a quadratic equation in the form x2+bx+c=0 that has the following roots: -8±4i

Answer:

the answer would be 100 I guess

For the equations 2xy - y^2 = 1 and xy + x + y = x^2y^2 using implicit differentiation the value Dxy is given by Dxy = (1 - 2xy + 3y^2)/(x - y)^3 and Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3  respectively.

Equation 2xy - y^2 = 1,

Differentiate both sides of the equation with respect to x,

Treating y as function of x and then differentiate again with respect to x.

Using implicit differentiation,

First, differentiate both sides with respect to x,

2y + 2xy' - 2yy' = 0

Next, solve for y',

⇒2xy' - 2yy' = -2y

⇒y' (2x - 2y) = -2y

⇒y' = -y/(x - y)

Now, differentiate again with respect to x,

y''(x - y) - y'(2x - 2y) = y/(x - y)^2

Substitute the expression we obtained for y' in terms of y and x,

y''(x - y) - (-y/(x - y))(2x - 2y) = y/(x - y)^2

Simplify and solve for y'',

y''(x - y) + (2xy - 3y^2)/(x - y)^2 = 1/(x - y)^2

The expression for Dxy is,

Dxy = (1 - 2xy + 3y^2)/(x - y)^3

For the equation xy + x + y = x^2y^2,

Differentiate both sides of the equation with respect to x,

Using implicit differentiation,

First, differentiate both sides with respect to x,

⇒y + xy' + 1 + y' = 2xyy'

Solve for y',

⇒xy' - 2xyy' + y' = -y - 1

⇒y' (x - 2xy + 1) = -y - 1

⇒y' = -(y + 1)/(x - 2xy + 1)

Now, differentiate again with respect to x,

y''(x - 2xy + 1) - y'(2y - 2x y' + 1) = (y + 1)/(x - 2xy + 1)^2

Substitute the expression we obtained for y' in terms of y and x,

y''(x - 2xy + 1) - (-y - 1)/(x - 2xy + 1)^2 (2y - 2x y' + 1) = (y + 1)/(x - 2xy + 1)^2

Simplify and solve for y''

y''(x - 2xy + 1) - (2y^2 - 2xy - 2y)/(x - 2xy + 1)^2 = (y + 1)/(x - 2xy + 1)^2

The expression for Dxy is,

Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3

Therefore , the value of Dxy using implicit differentiation for two different functions is equal to

Dxy = (1 - 2xy + 3y^2)/(x - y)^3 and Dxy = (2y^2 - 2xy - 3y - 1)/(x - 2xy + 1)^3

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In this case, we'll have to carry out several steps to find the solution.

Step 1:

Data:

camera:

list price (before tax) = $459.99

sales tax = 7.25%

Step 2:

percentage:

[tex]sales \ tax = 7.25\% = 7/100 = 0.07[/tex]

[tex]total \ cost = \$459.99 + \$459.99 \times (0.07) = \$459.99 + \$32.1993 = \$492.1893[/tex]

The answer is:

$492.19

Answer:

Step-by-step explanation:

5m(4m) = 20m^2

, [tex]\frac{-21 \pi}{8}[/tex] lies in 3rd quadrant, ,[tex]\frac{-5 \pi }{9}[/tex] lies in 3rd quadrant,t, [tex]\frac{5 \pi}{6}[/tex]  radian lies in 2nd quadrant and [tex]\frac{35 \pi}{4}[/tex] lies in 1st quadrant.

What is Trigonometry?

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

To determine the quadrant in which the angle -21π/8 lies

Convert, [tex]\frac{-21 \pi}{8}[/tex]  to number

-21×180/8

-472.5 degrees lies in which quadrant we have to check.

The value is negative

Quadrant IV =  0 to -90°

Quadrant III = -90° to -180°

Quadrant II  = -180° to -270°

Quadrant I   = -270° to -360°

-21π/8 lies in 3rd quadrant

-5π/9=-100 which lies in 3rd quadrant

5π/6=150 which is positive so it lies in 2nd quadrant

35π÷4=1575 which is positive so it lies in 1st quadrant

Hence, [tex]\frac{-21 \pi}{8}[/tex]  lies in 3rd quadrant,[tex]\frac{-5 \pi }{9}[/tex]lies in the 3rd quadrant, [tex]\frac{5 \pi}{6}[/tex] lies in 2nd quadrant and [tex]\frac{35 \pi}{4}[/tex] lies in  1st quadrant.

The complete question is-

In what quadrant does the terminal ray of the angle lie?

Select Quadrant I, Quadrant II, Quadrant ​III​, or Quadrant IV.

Angle measure Quadrant I Quadrant II Quadrant ​III​ Quadrant IV

−21π8

Quadrant , I – negative fraction numerator 21 pi over denominator 8 end fraction

Quadrant , I I – negative fraction numerator 21 pi over denominator 8 end fraction

Quadrant , ​, I I I, ​ – negative fraction numerator 21 pi over denominator 8 end fraction

Quadrant , I V – negative fraction numerator 21 pi over denominator 8 end fraction

−5π9

Quadrant , I – negative fraction numerator 5 pi over denominator 9 end fraction

Quadrant , I I – negative fraction numerator 5 pi over denominator 9 end fraction

Quadrant , ​, I I I, ​ – negative fraction numerator 5 pi over denominator 9 end fraction

Quadrant , I V – negative fraction numerator 5 pi over denominator 9 end fraction

5π6

Quadrant , I – fraction numerator 5 pi over denominator 6 end fraction

Quadrant , I I – fraction numerator 5 pi over denominator 6 end fraction

Quadrant , ​, I I I, ​ – fraction numerator 5 pi over denominator 6 end fraction

Quadrant , I V – fraction numerator 5 pi over denominator 6 end fraction

35π4

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The result of the formula  [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex] for y = 3 is  [tex]-29/2[/tex]  .

When [tex]y = 3[/tex] is used, the value of the expression [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex]  has a value of  [tex]-29/2[/tex] .

To analyse an algebraic expression is to determine its value when a certain number is used in lieu of the variable. To evaluate the expression, we first replace the variable with the given number, then we use the order of operations to simplify the expression.

If  [tex]y = 3[/tex] , we can insert it into the expression & simplify as follows to evaluate   [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex] for  [tex]y = 3[/tex]  .

[tex]12 - 3(3)^2 + (√4) / (2(3) - 2)[/tex] (y = 3 replacement)

[tex]12 - 27 + 2 / 4\s-15 + 1/2\s-29/2[/tex]

Therefore, The result of the formula  [tex]12 - 3y2 + (v=4) / (2y - 2)[/tex] for y = 3 is  [tex]-29/2[/tex]  .

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(a) Therefore after t = 1.75 seconds the diver will hit the water.

(b) The velocity the diver hit the water is 56 ft/s.

From the given condition we have d(t) = 16t²

and the height is 49ft

(a) Now when the diver hit the water the equation become

16t² = 49

t² = 49/16

t = ±7/4

t = ±1.75

since time can not be negative so t = 1.75

Therefore after t = 1.75 seconds the diver will hit the water.

(b)

Now differentiating d(t) with respect to t we get

d'(t) = 32t

now putting t=7/4 we get

the velocity d'(7/4) = 32*7/4

d'(7/4) = 56ft/s

Therefore the velocity the diver hit the water is 56 ft/s.

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The complete question is :

A cliff diver plunges from a height of 49ft above the water surface. The distance the diver falls in t seconds is given by the function d(t)=16t²ft

(a) After how many seconds will the diver hit the water?

(b) With what velocity (in ft/s ) does the diver hit the water?

Answer:

32 units square

Step-by-step explanation:

Area of trapezoid = 1/2 x h x (a +b) {a and b are parallel sides}

1st parallel side = 2 + 6 + 2

                          = 10 units

2nd parallel side = 6 units

height = 4 units

Area = 1/2 x 4 x (10+6)

         = 2x16

         = 32 units square

[tex]53 - 3^2 * 3 = 35[/tex]

The value of the triple integral[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex] by using spherical coordinates [tex]2\pi(e^{-1}-e^{-9})[/tex].

Given that the triple integral is-

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

E is the region bounded by the spheres which are,

[tex]x^2+y^2+z^2=1\\\\x^2+y^2+z^2=9[/tex]

In spherical coordinates we have,

x = r cosθ sin ∅

y = r sinθ sin∅

z = r cos∅

dV = r²sin∅ dr dθ d∅

E contains two spheres of radius 1 and 3 () respectively, the bounds will be like this,

1 ≤ r ≤ 3

0 ≤ θ ≤ 2π

0 ≤ ∅ ≤ π

Then

[tex]\int \int\int _{E } \frac{e^{-(x^2+y^2+z^2)}}{\sqrt{(x^2+y^2+z^2}}\sqrt{dV}[/tex]

[tex]\int\int\int _{E} \frac{e^{-r^2}}{r}r^2Sin\phi drd\phi d\theta\\\\2\pi \int_{0}^{\pi} \int_1^3 re^{-r^2} dr d\phi\\\\2\pi \int_1^3 re^{-r^2} dr\\\\2\pi(e^{-1}-e^{-9})[/tex]

The complete question is-

Use spherical coordinates to evaluate the triple integral ∭ee−(x2 y2 z2)x2 y2 z2−−−−−−−−−−√dv, where e is the region bounded by the spheres x2 y2 z2=1 and x2 y2 z2=9.

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On Thursday, 12 students brought their lunch at school.

Using a number out of 100, a percentage is a technique to indicate a fraction or piece of a total. The word "percent" means "per hundred."

If 4% of the students bring their lunch to school every day, we can find the number of students who brought their lunch on Thursday by multiplying the total number of students by the percentage that brought their lunch:

Number of students who brought their lunch = (4/100) x 300

Number of students who brought their lunch = 12

Therefore, On Thursday, 12 students brought their lunch at school.

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For Gasoline Expense, the account number may be something like 4540 (4 for expenses, 5 for operating expenses, and 40 for the specific account number). For Water Expense, the account number may be something like 4550 (4 for expenses, 5 for utility expenses, and 50 for the specific account number

How to assign account number?

To add new accounts to the chart of accounts, Mr. Ross should first determine the account type and the appropriate category. In this case, Gasoline Expense would be classified as an operating expense account, while Water Expense would be classified as a utility expense account.

Once the account types have been identified, Mr. Ross should then determine the account number for each account. The account numbering system may vary depending on the accounting system in use, but generally, account numbers are structured hierarchically based on the account category and type.

For example, if the account numbering system uses a four-digit code, the first digit may represent the account category (e.g., 1 for assets, 2 for liabilities, 3 for equity, 4 for revenues, and 5 for expenses), the second digit may represent the specific account type (e.g., 4 for operating expenses and 5 for utility expenses), and the last two digits may be assigned sequentially to each account within that category and type.

So, for Gasoline Expense, the account number may be something like 4540 (4 for expenses, 5 for operating expenses, and 40 for the specific account number). For Water Expense, the account number may be something like 4550 (4 for expenses, 5 for utility expenses, and 50 for the specific account number).

However, the specific account numbering system used may vary depending on the accounting system in use, so Mr. Ross should consult with his accountant or accounting software provider for guidance on how to properly create and assign account numbers in his specific system.

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Answer: 37.5

Step-by-step explanation:

75 - 180 = 105

105 degrees = the obtuse angle, bottom triangle.

75/2= 37.5 (since both sides of the bottom triangle are equal angles)

Answer:

Total cost for groceries = ($3.75, $1.09,

$4.50, $3.25, $2.58, $4.71, $5.19, $0.89, and $5.34. add them all). = $ 31.3

the amount she paid= $ 35

balance =$ 3.7

therefore she have enough money

The area of this triangle is 30 m².

Area is a surface measure, that is, it is the amount of space that a geometric figure occupies on a flat surface.

To calculate the area of a triangle, we can use the formula:

Area = (base x height) / 2

In the case of the given triangle, we can choose the measure of 5m as the base and the measure of 12m as the height, since the height forms a right angle with the base and is perpendicular to it.

So, we have:

Area = (b*h)/2

Area = (5m * 12m) / 2

Area = 30m²

Answer:

19782m

Step-by-step explanation:

1  revolution = circumference

circumference = π * diameter

π = 3.1416

Then

circumference = 3.1416 * 63

= 197.92m

1 revolution = 197.82m

100 revolutions = 100*197.82m

= 19782m

Answer:

19.8 km

Step-by-step explanation:

To find:-

Answer:-

We are here given that,

We can first find the circumference of the wheel using the formula,

[tex]:\implies \sf C = 2\pi r \\[/tex]

Here radius will be 63/2 as radius is half of diameter. So on substituting the respective values, we have;

[tex]:\implies \sf C = 2\times \dfrac{22}{7}\times \dfrac{63}{2} \ m \\[/tex]

[tex]:\implies \sf C = 198\ m \\[/tex]

Now in one revolution , the cycle will cover a distance of 198m . So in 100 revolutions it will cover,

[tex]:\implies \sf Distance= 198(100)m\\[/tex]

[tex]:\implies \sf Distance = 19800 m \\[/tex]

[tex]:\implies \sf Distance = 19.8 \ km\\[/tex]

Hence the bicycle would cover 19.8 km in 100 revolutions.

The appropriate inference from the data is (B) Since [tex]20%[/tex] of this group read below grade level, we cannot draw any generalizations about the population. Thus, option B is correct.

A representative sample is a subset of data, often drawn from a wider population, that can show qualities that are similar.

Because the data produced contains more manageable, smaller representations of the larger group, representative sampling aids in the analysis of bigger groups.

Although the sample may be representative of seventh-grade American students, it is not necessarily representative of all seventh-graders or all American children. Hence, without additional data or research, we cannot extrapolate the sample's results to the overall population.

Therefore, 20%20, percent of all American students in seventh grade were reading below grade level.

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If a 5-year swap contract can be viewed as a portfolio of 5 forward contracts with maturities of 1, 2, 3, 4 and 5 years. One important exception is option (a) the forward price is the same for the swap contract but not for the forward contracts

In a swap contract, the fixed rate is agreed upon at the beginning of the contract, and the floating rate is determined by a reference rate such as LIBOR. The swap contract's value is based on the difference between the fixed and floating rates at each settlement date. In contrast, forward contracts involve an agreement to buy or sell an asset at a specified price on a specific future date. The forward price is determined at the time the contract is entered into and is based on the spot price of the underlying asset, the time to maturity of the contract, and the cost of carry.

Therefore, the forward price will be different for each forward contract in the swap portfolio, whereas the forward price for the swap contract will be the same throughout the contract's life. The other options mentioned are not true for swap or forward contracts.

Therefore, the correct option is (a) the forward price is the same for the swap contract but not for the forward contracts

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When a quantity increases by 25%, the new quantity becomes 100% + 25% = 125% of the original quantity.

After the first year, the quantity has increased to 125% of its original value. After the second year, it increases an additional 25%, making the new value 125% + (25% of 125%) = 156.25% of the original value. Finally, after the third year, it increases by another 25%, making the new value 156.25% + (25% of 156.25%) = 195.3125% of the original value.

So the quantity has increased by 195.3125% - 100% = 95.3125% over the three-year period. Therefore, the combined percentage growth p over the three-year period is 95.3125%.

Answer:

Step-by-step explanation:

The center of the circle is the midpoint of the diameter. To find the midpoint, we use the midpoint formula:

Midpoint = [(x1 + x2)/2, (y1 + y2)/2]

where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the diameter.

Midpoint = [(3 + (-2))/2, (1 + 3)/2]

Midpoint = [1/2, 2]

So, the center of the circle is (1/2, 2).

The radius of the circle is half the length of the diameter. To find the length of the diameter, we use the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)

where (x1, y1) and (x2, y2) are the coordinates of the endpoints of the diameter.

Distance = sqrt((-2 - 3)^2 + (3 - 1)^2)

Distance = sqrt(25 + 4)

Distance = sqrt(29)

So, the length of the diameter is sqrt(29).

The radius of the circle is half of sqrt(29), which is sqrt(29)/2.

Therefore, the equation of the circle is:

(x - 1/2)^2 + (y - 2)^2 = (sqrt(29)/2)^2

Simplifying this equation, we get:

(x - 1/2)^2 + (y - 2)^2 = 29/4

So, the equation of the circle with a diameter whose endpoints are (3, 1) and (-2, 3) is (x - 1/2)^2 + (y - 2)^2 = 29/4.

Answer:

Step-by-step explanation:

To find the greatest profit, we need to determine the fare that will maximize revenue, while also considering the decrease in ridership due to the fare increase.

Let's assume the initial fare is $2, and the number of passengers is 800 per day. So, the initial revenue is:

$2 x 800 = $1600 per day

Now, let's say we increase the fare by 4 cents to $2.04. According to the problem, for each 4 cents increase in fare, there will be 5 fewer passengers. So, the number of passengers will decrease to:

800 - (5 x 4) = 780 passengers per day

The new revenue at this fare will be:

$2.04 x 780 = $1591.20 per day

By increasing the fare, the revenue decreased. This means that we may have increased the fare too much. Let's try another fare.

If we increase the fare by 2 cents to $2.02, the number of passengers will decrease by:

800 - (5 x 2) = 790 passengers per day

The new revenue at this fare will be:

$2.02 x 790 = $1595.80 per day

This is more revenue than the initial fare of $2 per person. Let's continue this process:

If we increase the fare by another 2 cents to $2.04, the number of passengers will decrease by:

790 - (5 x 2) = 780 passengers per day

The new revenue at this fare will be:

$2.04 x 780 = $1591.20 per day

This is less revenue than the $2.02 fare, so we can stop here.

Therefore, the greatest profit can be earned by charging $2.02 per person per day, and the maximum revenue will be:

$2.02 x 790 = $1595.80 per day

This is a bit less than the initial daily revenue of $1600, but it is the most revenue we can get by increasing the fare without causing a significant reduction in ridership.

Answer:

  $2205

Step-by-step explanation:

You want the greatest profit that can be earned by a commuter railway that has 800 passengers per day at a fare of $2, and 5 fewer for each 4¢ increase in the fare.

The number of riders (q) as a function of price (p) can be described by ...

  q = 800 -5(p -2)/0.04

  q = 1050 -125p . . . . . . . simplified

The daily revenue is the product of price and the number of riders who pay that price.

  r = pq

  r = p(1050 -125p)

  r = 125p(8.40 -p)

This function describes a parabola that opens downward. It has zeros at p=0 and p=8.40. The vertex of the parabola is on the line of symmetry, halfway between the zeros:

  pmax = (0 +8.40)/2 = 4.20

The maximum revenue is ...

  r(4.20) = 125·4.20(8.40 -4.20) = 125(4.20²) = 2205

The maximum revenue that can be earned is $2205.

__

Additional comment

The ridership at that fare is 125(4.20) = 525.

Profit is the difference between revenue and cost. Here, we have no information about the cost function, so we cannot predict the maximum profit. The question seems to assume that profit is equal to revenue.

Answer: 6234 × 32 = 199488.

Step by step:

To calculate 6234 × 32 without using a calculator, you can use the traditional multiplication method as follows:

   6234

x    32

-------

  12468   (2 x 6234)

+ 62340   (3 x 6234 with a zero added)

--------

 199488

Answer: Permutations that preserve distances are also known as isometries or distance-preserving transformations.

Step-by-step explanation:

Permutations that preserve distances refer to a type of mathematical transformation that preserves the distances between pairs of points in a geometric space. In other words, if you have a set of points arranged in a particular way and you apply a permutation that preserves distances, the resulting arrangement of points will have the same distances between each pair of points as the original arrangement. This type of permutation is important in geometry and can be used to study properties of geometric objects such as polyhedra, graphs, and other structures. Permutations that preserve distances are also known as isometries or distance-preserving transformations.

The area of enclosed region as a function of x is A(x) = 8x^2/3 + (200/3)x - 2x. Using a table, the estimated dimensions that produce maximum area are longer side x = 12 and shorter side y = 17.3 m. Using a graph, the estimated dimensions that produce maximum area are longer side 2x = 25 m and shorter side y = 16.7 m. The dimensions for an enclosed area of 336 square meters are smaller value of x: longer side y = 23.3 m, shorter side 2x = 15 x m and larger value of x: longer side 2x = 35 m, shorter side y = 10 x m.

The area of the enclosed region can be written as a function of x as follows.

A(x) = 2xy

Substituting y = (100 - 4x)/3, we get

A(x) = 2x(100 - 4x)/3 = (200x - 8x^2)/3 = 66.7x - 2.67x^2

Using a graphing utility, we can generate additional rows of the table as follows.

xy Area

2    92 368

3    122.7 367.1

4     128 512

5     116.7 583.3

6     100.7 604.2

7     80.0 560.0

8     65.3 522.7

9     56.0 504.0

10    51.3 513.0

11     51.0 561.0

12    54.7 656.7

13     62.0 806.2

14     72.7 1003.3

From the table, we can see that the maximum area occurs when x = 12 and y = 17.3.

The graph of the area function is shown.

From the graph, we can estimate that the dimensions that will produce a maximum area are x ≈ 12.5 and y ≈ 16.7.

From the graph, we can estimate that the smaller value of x for an area of 336 square meters is x ≈ 7.5 and the larger value of x is x ≈ 17.5. Substituting these values in the equation for y, we get:

For the smaller value of x:

y = (100 - 4(7.5))/3 ≈ 23.3

So, the required dimensions are longer side = 15 m and shorter side = 2(7.5) = 15 m.

For the larger value of x:

y = (100 - 4(17.5))/3 ≈ 10.0

So, the required dimensions are longer side = 2(17.5) = 35 m and shorter side = 10 m.

To find the required dimensions algebraically, we need to solve the equation A(x) = 336. We already have the expression for A(x) as:

A(x) = 8x(50 - 4x)

Substituting 336 for A(x), we get:

8x(50 - 4x) = 336

Dividing both sides by 8, we get:

x(50 - 4x) = 42

Expanding the left side, we get:

50x - 4x^2 = 42

Rearranging and simplifying, we get a quadratic equation:

4x^2 - 50x + 42 = 0

We can solve this quadratic equation using the quadratic formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 4, b = -50, and c = 42.

Substituting these values, we get:

x = (50 ± sqrt(50^2 - 4(4)(42))) / 2(4)

Simplifying, we get:

x = (50 ± sqrt(196)) / 8

x = (50 ± 14) / 8

So, the two possible values of x are:

x = 7 or x = 1.25

For x = 7, the corresponding dimensions are:

longer side 2x = 14 m

shorter side y = (50 - 4x) = 22 m

For x = 1.25, the corresponding dimensions are:

longer side 2x = 2.5 m

shorter side y = (50 - 4x) = 32.5 m

Therefore, the required dimensions for an enclosed area of 336 square meters are:

Smaller value of x: longer side 2x = 2.5 m, shorter side y = 32.5 m

Larger value of x: longer side 2x = 14 m, shorter side y = 22 m

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The toll calculation for driving on a certain stretch of a toll road is $5 except during rush hours when the toll is $7, depending on the time the driver uses the toll road.

To compute the toll for driving on the toll road during non-rush hours, simply add $5 to the total. During rush hour, however, the toll is $7.

To compute the toll for driving during rush hour, you must first determine when the driver intends to utilize the toll road. If the period is between 7 AM and 10 AM or 4 PM and 7 PM, the toll is $7.

For instance, if a vehicle expects to use the toll road at 8 a.m., the toll is $7. If the vehicle intends to use the toll road at 2 p.m., the toll is $5.

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Answer:

father age 45 son age 0 this is answer

Answer:

The answer is 2.45 litres in one week.

Answer:

2.45 liters

Step-by-step explanation:

1 days - 350 ml of Milk

We know that, there are 7 days per week.

Therefore,

To find the milliliters of milk that she drinks in a week,

multiply milliliters of milk she drinks for a day by 7 ( number of days in a week).

Let us find it now.

350 ml × 7 = 2450 ml

And now to convert your answer into litres divide it by 1000.

2450ml ÷ 1000 = 2.45 liters

so as we speak, the subfunctions are discontinued, the 1st goes close to 2 and who knows what happens it goes somewheres, the 2nd one makes it to 2.

we know that since the 2nd one makes to 2, to x = 2 that is, well, f(2) = kx, well, let's make f(2) for the 2nd one be equal to the 1st one then, if both they equate each other, that's where they meet, at x = 2.

[tex]f(x)= \begin{cases} k^2-24x,&x > 2\\\\ kx,&x\leqslant 2 \end{cases} \\\\[-0.35em] ~\dotfill\\\\ k^2-24x~~ = ~~kx\hspace{5em}\stackrel{\textit{now let's go to f(2)}}{k^2 - 24(2)~~ = ~~k(2)}\implies k^2-48=2k \\\\\\ k^2-2k-48=0\implies (k-8)(k+6)=0\implies \boxed{k= \begin{cases} 8\\ -6 \end{cases}}[/tex]

The number of fractions between 0 and 1 (inclusive) with a denominator of 15 can be found using the formula (n-1)/n, where n is the denominator.

So, to answer your question, we can use the formula and plug in 15 for the value of n:

(15-1)/15 = 14/15

Therefore, there are 14 fractions between 0 and 1 (inclusive) with a denominator of 15.

The angle congruent to angle FYE is angle EYD.

A line known as a bisector splits an angle or a line into two equally sized segments. A segment's midpoint is always contained in the segment's bisector. Based on the geometric shape that they bisect, there are two different sorts of bisectors. An angle is divided into equal angles by an angle bisector. The line segment is split into two equal halves by a line segment bisector. It travels through the line segment's centre.

Given that, YE bisects the angle FYD.

That is, the segment YE divides the angle FYD into two equal parts.

Thus, the angle congruent to angle FYE is angle EYD.

Learn more about angle bisector here:

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