Some friends went out for a meal. The restaurant added a 10% service charge to the cost of the meal. The total bill was £126.50 including the service charge. What was the cost of the meal? Give your answer in pounds (£). Receipt Cost of the meal: £ Service charge: +10% Total: £126.50​

Using linear negative association, According to the  all four parts correct options are D ;A ;D ;D respectively

The slope of a line expresses a great deal about the linear relationship between two variables. If the slope is negative, there is a negative linear relationship, which means that as one variable increases, the other variable decreases. If the slope is zero, one increases while the other remains constant.

The first answer to the question is option D

The second answer to the question must be option A  

Option D must be chosen for the third question.

Option D  must be selected for Question 4.

Solution:

1.

square of 3 is 9

3 to the power of negative 2 is 1/ 9

cube of 3 is 27

3 to the negative power 3 is 1/27

2.

cylinder  volume =πr²h

Given value

pi =3.14

r=5

h=10

Volume=3.14×5²×10

cylinder volume =785m³

3.

When a point is rotated 90 degrees anticlockwise about the origin, it becomes the point (x,y) (-y,x).

The coordinates of Point N are (4, 3)

N' will be the new coordinates (-3, 4)

As a result, the y-coordinate of N' is 4.

4.

Option D must be selected for Question 4.

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The car must drive at a constant speed of approximately 112.89 km/hr to cover the same distance in 2 hours 35 minutes.

The formula for time is: time = distance / speed

where "distance" is the distance traveled by an object, and "speed" is the rate at which the object is moving.This formula can be used to calculate the time taken by an object to travel a certain distance at a constant speed, or to calculate the speed or distance if the other two variables are known.

The formula for speed is: speed = distance / time

where "distance" is the distance traveled by an object and "time" is the duration of travel.

This formula can be used to calculate the speed of an object if the distance it has traveled and the time it took to travel that distance are known. It can also be used to calculate the distance traveled by an object if its speed and the time it traveled at that speed are known.

In the given question,

Let's first calculate the distance traveled in 2 hours 45 minutes (2.75 hours) at an average speed of 106 km/hr.

distance = speed × time

distance = 106 × 2.75

distance = 291.5 km

Now, we need to find at which constant speed the car must drive to cover the same distance in 2 hours 35 minutes (2.5833 hours). Let's call this speed "x".

distance = speed × time

291.5 = x × 2.5833

x = 291.5 / 2.5833

x ≈ 112.89 km/hr

Therefore, the car must drive at a constant speed of approximately 112.89 km/hr to cover the same distance in 2 hours 35 minutes.

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Answer:

Percentage of the circle that lies outside the triangle = 15%

Step-by-step explanation:

Given,

The area of the intersection of circle and triangle is 45% of the area of the union.

The area of the triangle outside of the circle is 40% of the area of their union.

Required to find,

The percentage of the circle lies outside the triangle

Let us take 'C'  as the area of the circle and 'T' as the area of the triangle.

The union of the area of the circle and triangle = C∪T

Let CUT be  A

and the percentage of the circle that lies outside the triangle be 'x'

The intersection of the area of the circle and triangle = C∩T

Area of the triangle outside the circle = T - C

Area of the circle outside the triangle = C - T

Given,

C∩T = 45% of CUT = 45% of A

T - C = 40% of CUT = 40% of A

We know that,

The union of the area of the circle and triangle = Area of The intersection of the area of the circle and triangle + Area of the triangle outside the circle + Area of the circle outside the triangle

(CUT) = (C∩T) + (T - C) + (C - T)

[tex]A = 45\%A + 40\%A + x\%A[/tex]

[tex]A = (85+x)\% A[/tex]

[tex]85 +x = 100[/tex]

∴ [tex]x = 15[/tex]

Percentage of the circle that lies outside the triangle = 15%

The volume of the solid obtained by rotating the region bounded by the curves y = x and y = √x about the line x = 6 is (128π/15) - (13π/3), or approximately 3.013 cubic units.

To find the volume of the solid obtained by rotating the region bounded by the curves y = x and y = √x about the line x = 6, we can use the method of cylindrical shells.

First, we need to determine the limits of integration. Since the curves intersect at (0,0) and (1,1), we can integrate with respect to y from 0 to 1.

The radius of each cylindrical shell is the distance from the line x = 6 to the curve y = x or y = √x. We can express this distance as r = 6 - x or r = 6 - y^2, depending on which curve we are using.

The height of each cylindrical shell is the difference between the two curves at the given y-value. This is given by h = y - √x for y = x, and h = y^2 - x for y = √x.

Therefore, the volume of the solid is:

V = ∫(2πrh) dy from 0 to 1

Substituting r and h, we get:

V = ∫(2π(6 - x)(y - √x)) dy from 0 to 1 (for y = x)

V = ∫(2π(6 - y^2)(y^2 - x)) dy from 0 to 1 (for y = √x)

Evaluating these integrals using u-substitution and simplifying, we get:

V = (128π/15) - (13π/3)

Therefore, the volume of the solid is (128π/15) - (13π/3), or approximately 3.013 cubic units

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_____The given question is incomplete, the complete question is given below:

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified in y = x, y = sqrt(x) ; about x = 6

Answer:

look at the explanation

Step-by-step explanation:

okay so here if AC is equals to DC and BC is equals to CE then AB is equals to DE as well hence, angle A would be equals to angle D.

The second reason that prove that angle A is a angle D is that angle A and angle D are alternate angles

I am also in ninth grade so do recheck yourself

The triangle ABC with the given dimensions:AB = 10, BC = 6, CA = 90 are constructed.

The dimension of the triangle ABC are:

AB = 10

BC = 6

CA = 90

Thus, the triangle with the given dimensions are constructed.

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The length of the side of the hexagon is 12 cm and option d is the correct answer.

A regular polygon is a closed shape made up of straight line segments with sides and angles that are all of the same length. For instance, a regular hexagon is a polygon having six equal-length sides and six equal-sized angles. Regular polygons have a variety of intriguing characteristics. For instance, their diagonals (lines connecting non-adjacent vertices) all intersect at a single point, and their centre of symmetry is located at the centre of the polygon's circumscribed circle (the circle that passes through all of the polygon's vertices).

Given that, regular hexagon is inscribed into a circle.

The radius of a circle enclosing a regular hexagon is the same as the length of the hexagon's sides.

Hence, the length of the side of the hexagon is 12 cm and option d is the correct answer.

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The gradient of the curve x³ + 3xy²- y³ = 1 at the point with coordinates (1, 3) is -29/9

The gradient of a curve at a point is the value of its derivative at that point.

To find the gradient of the curve x³ + 3xy²- y³ = 1 at the point with coordinates (1, 3). we differentiate implicitly with respect to x.

So, we have that

x³ + 3xy²- y³ = 1

d(x³ + 3xy²- y³)/dx = d1/dx

dx³/dx + d3xy²/dx - dy³/dx = d1/dx

dx³/dx + 3y²dx/dx + 3xdy²/dx - dy³/dx = d1/dx

2x² + 3y² × 1 + 3x × dy²/dy × dy/dx - dy³/dy × dy/dx = d1/dx

2x² + 3y² × 1 + 3x × 2y × dy/dx - 3y² × dy/dx = 0

2x² + 3y² + 6xydy/dx - 3y²dy/dx = 0

6xydy/dx - 3y²dy/dx = -(2x² + 3y²)

Factorizing out dy/dx, we have that

6xydy/dx - 3y²dy/dx = -(2x² + 3y²)

(6xy - 3y²)dy/dx = -(2x² + 3y²)

dy/dx = -(2x² + 3y²)/(6xy - 3y²)

At (1,3) dy/dx =  -(2x² + 3y²)/(6xy - 3y²)

=  -(2(1)² + 3(3)²)/(6(1)(3) - 3(3)²)

=  -(2(1) + 3(9))/(6(1)(3) - 3(9))

=  -(2 + 27)/(18 - 27)

= -29/-9

= 29/9

So, dy/dx = 29/9

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The gravitational force that the moon produces on the Earth is approximately 1.99x10²⁰ N.

Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers, according to Newton's rule of universal gravitation.

We can use Newton's law of gravitation to solve this problem:

[tex]$F = G \frac{m_1 m_2}{r^2}$[/tex]

where F is the gravitational force, G is the gravitational constant, [tex]$m_1$[/tex] and [tex]$m_2$[/tex] are the masses of the two objects, and r is the distance between their centers.

Plugging in the given values, we get:

[tex]$F = (6.674\times10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2) \frac{(7.34\times10^{22} \text{ kg})(5.97\times10^{24} \text{ kg})}{(3.88\times10^8 \text{ m})^2}$[/tex]

Simplifying the expression, we get:

[tex]$F \approx 1.99\times10^{20} \text{ N}$[/tex]

Therefore, the gravitational force that the moon produces on the Earth is approximately 1.99x10²⁰ N.

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Answer: To find the distance of a chord from the center of a circle, we need to use the following formula:

Distance from center = sqrt(r^2 - (c/2)^2)

Where r is the radius of the circle and c is the length of the chord.

In this case, the radius of the circle is 3.7cm and the length of the chord is 7cm.

So, substituting these values in the formula, we get:

Distance from center = sqrt(3.7^2 - (7/2)^2)

= sqrt(13.69 - 12.25)

= sqrt(1.44)

= 1.2 cm

Therefore, the distance of the chord from the center of the circle is 1.2 cm.

Step-by-step explanation:

Answer: $2,324

Step-by-step explanation:

Let his salary be x.

He spent 1/4 of his salary, or x/4, and 1/7 of his salary, or x/7.

His salary was x. He spent x/4 and x/7, so we subtract those two amounts from his salary.

x - x/4 - x/7 is the amount he still has. He still has $1411. We equate the two and have an equation.

x - x/4 - x/7 = 1411

x/1 - x/4 - x/7 = 1411

We need to combine the three fractions on the left side, so we need to use a common denominator. The least common multiple of 1, 4, and 7 is 28, so 28 is the LCD.

28x/28 - 7x/28- 4x/28= 1411

17x/28= 1411

Multiply both sides by 28.

17x = 39,508

Divide both sides by 17.

x = 2,324

Check:

1/4 of his salary is  2,324/4 = 581

1/7 of his salary is  2,324/7 = 332

Now we subtract 581 and 332 from 2,324

2324 - 581 - 332 = 1411 which is what the problem stated.

Our answer $2,324 is correct.

Answer:

Yep

Step-by-step explanation:

Yep. Checking this, all of these are correct. Range on parabolas and absolutes will always go to infinity. Nice work

The aquarium has six rectangular faces, four of which are identical (two sides and two ends), and two of which are identical to each other but different from the others (top and bottom).

a) We can use the formula for the volume of a rectangular prism, which is given by V = lwh, where l is the length, w is the width, and h is the height. In this case, we have  [tex]V = 12 ft^3 and h = 1.5 ft[/tex] . We want to express y as a function of x, so we need to eliminate w from the equation.

We can rearrange the equation for the volume to get  [tex]w = V/(lh)[/tex] , and substitute in h [tex]= 1.5 ft[/tex] :

[tex]w = V/(1.5lx)[/tex]

Now we can substitute y for w to get:

[tex]y = V/(1.5lx)[/tex]

b)  To find the total surface area, we need to find the area of each face and add them up.

The area of one of the identical sides or ends is lw, so the total area of these four faces is:

[tex]4lw = 4xy[/tex]

The area of the top and bottom faces is lx, so the total area of these two faces is:

[tex]2lx[/tex]

Therefore, the total surface area S is given by:

[tex]S = 4xy + 2lx[/tex]

We can express y in terms of x using the equation from part a):

[tex]y = V/(1.5lx)[/tex]

Substituting this into the expression for S, we get:

[tex]S = 4x(V/(1.5lx)) + 2lx[/tex]

Simplifying, we get:

[tex]S = (8/3)V/x + 2lx[/tex]

So the total surface area S is a function of x, and we can use this equation to find the value of S for any given value of x.

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Answer:

y = -3/4 + 10 1/4

Step-by-step explanation:

y = mx + b  

We are given the slope -3/4.

y = -3/4x + b  To find the b we will use the point (-3,8)  We will use -3 for x and 8 for y and then solve for b

8 = -3/4 (-3) + b

8 = -9/4 + b  Add 9/4 to both sides

8 + 9/4 = -9/4 + 9/4 + b

41/4 = b or 10 1/4 = b

Helping in the name of Jesus.

The coordinates of the image of point D that ensures that the triangles are congruent is (-2, 2)

Given the triangle ABC and the incomplete triangle DEF

From the triangle transformation of points B and C, we can see that

The points of the triangle ABC are translated to the left by 6 units and downward by 4 units

Mathematically, this can be represented as

(x, y) = (x - 6, y - 4)

Given that

A = (4, 6)

So, we have

D = (4 - 6, 6 - 4)

Evaluate the difference

D = (-2, 2)

Hence, the position is (-2, 2)

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Answer:

14,4,10

Step-by-step explanation:

cardinality of the given set n(A'∩B') is 19

In a Venn diagram, circles are used to represent connections between objects or small groups of objects. Circles that overlap have some properties, but circles that do not overlap do not have properties. Venn diagrams are very useful for showing how two concepts are related and different graphically.

This Venn diagram illustrates the relationship between the subsequent set of integers.

Whereas the other set comprises the numbers in the 5x table from 1 to 25, the first set only contains even numbers from 1 to 25.

The intersection component demonstrates that 10 and 20 are both multiples of 5 from 1 to 25 and even integers.

Complement of A =14+5+12+7=38

Complement  of B=7+4+3+9+12+7=42

n(A'∩B')=19

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The area of the composite figure is 159.9 cm²

From the question, we are to determine the area of the given composite figure

In the given diagram, the area of the composite figure = Area of triangle + Area of rectangle

First, we will calculate the area of the triangle

Area of triangle = 1/2 × base × height

Thus,

Area of the triangle = 1/2 × 16.4 × 5.5

Area of the triangle = 45.1 cm²

Calculating the area of the rectangle

Area of rectangle = Length × Width

Thus,  

Area of the rectangle = 16.4 × 7

Area of the rectangle = 114.8 cm²

Therefore,

The area of the composite figure = 45.1 cm² + 114.8 cm²

The area of the composite figure = 159.9 cm²

Hence, the area is 159.9 cm²

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Answer:

I'd be happy to help!

a. From the picture of the window, we can identify the following quadrilaterals:

Rectangle: ABCD (all angles are right angles and opposite sides are parallel and congruent)

Parallelogram: EFGH (opposite sides are parallel and congruent)

Trapezoid: BCGH (at least one pair of opposite sides are parallel)

b. To identify the parallelograms in the picture, we would need to know the following properties of parallelograms:

Opposite sides are parallel and congruent

Opposite angles are congruent

Diagonals bisect each other

Using these properties, we can identify the following parallelograms in the picture:

Parallelogram EFGH: Opposite sides EF and GH are parallel and congruent, and opposite sides EG and FH are also parallel and congruent. Additionally, angles E and G are congruent, and angles F and H are congruent.

Rectangle ABCD: Opposite sides AB and CD are parallel and congruent, and opposite sides AD and BC are also parallel and congruent. Additionally, angles A and C are congruent, and angles B and D are congruent. The diagonals AC and BD bisect each other, meaning that they intersect at their midpoints.

Step-by-step explanation:

Answer:

One possible function that models the ocean tide is:

h(t) = A sin(ωt + φ) + B

where:

h(t) represents the height of the tide (in meters) at time t (in hours)

A is the amplitude of the tide (in meters)

ω is the angular frequency of the tide (in radians per hour)

φ is the phase shift of the tide (in radians)

B is the mean sea level (in meters)

This function is a sinusoidal function, which is a common type of function used to model periodic phenomena. The sine function has a natural connection to circles and periodic motion, making it a good choice for modeling the regular rise and fall of ocean tides.

The amplitude A represents the maximum height of the tide above the mean sea level, while B represents the mean sea level. The angular frequency ω determines the rate at which the tide oscillates, with one full cycle (i.e., a high tide and a low tide) occurring every 12 hours. The phase shift φ determines the starting point of the tide cycle, with a value of zero indicating that the tide is at its highest point at time t=0.

To determine specific values for A, ω, φ, and B, we would need to gather data on the tide height at various times and locations. However, typical values for these parameters might be:

1. A = 2 meters (representing a relatively large tidal range)

2. ω = π/6 radians per hour (corresponding to a 12-hour period)

3. φ = 0 radians (assuming that high tide occurs at t=0)

4. B = 0 meters (assuming a mean sea level of zero)

Using these values, we can write the equation for the tide as:

h(t) = 2 sin(π/6 t)

We can evaluate this equation for various values of t to get the height of the tide at different times. For example, at t=0 (the start of the cycle), we have:

h(0) = 2 sin(0) = 0

indicating that the tide is at its lowest point. At t=6 (halfway through the cycle), we have:

h(6) = 2 sin(π/2) = 2

indicating that the tide is at its highest point. We can also graph the function to visualize the rise and fall of the tide over time:

Tide Graph

Overall, this function provides a simple and effective way to model the ocean tide using trigonometric functions.

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The reason why total energy eigen-functions cannot be written in form x(x)y(y)z(z) c(x, y, z) is that the energy eigenfunctions depend on all three spatial coordinates, x, y, and z, simultaneously.

In other words, the probability density of finding the electron at any point in space is not just a product of separate functions that only depend on one of the coordinates at a time.

Instead, the probability density of finding the electron is determined by the complex wave function, which depends on all three spatial coordinates at the same time.

The wave function gives us information about the energy and the behavior of the electron in the system, and it cannot be factorized into separate functions that depend on only one coordinate at a time.

Therefore, we cannot write the total energy eigenfunction as a product of separate functions of x, y, and z. The total energy eigenfunction is a complex function that depends on all three coordinates simultaneously, and it cannot be separated into individual functions that depend only on one coordinate at a time.

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Answer:

E.  x² + (y - 3)² = 16

Step-by-step explanation:

The equation of a circle in the standard (x, y) coordinate plane with center (h, k) and radius r is given by:

[tex]\boxed{(x - h)^2 + (y - k)^2 = r^2}[/tex]

To find the equation of the circle with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0, we need to first write the equation of the second circle in the standard form.

We can complete the square for y to rewrite this equation in standard form. To do this move the constant to the right side of the equation:

[tex]\implies x^2 + y^2 - 6y + 4 = 0[/tex]

[tex]\implies x^2 + y^2 - 6y = -4[/tex]

Add the square of half the coefficient of the term in y to both sides of the equation:

[tex]\implies x^2 + y^2 - 6y +\left(\dfrac{-6}{2}\right)^2= -4+\left(\dfrac{-6}{2}\right)^2[/tex]

[tex]\implies x^2 + y^2 - 6y +9= -4+9[/tex]

[tex]\implies x^2 + y^2 - 6y +9=5[/tex]

Factor the perfect square trinomial in y:

[tex]\implies x^2+(y-3)^2=5[/tex]

[tex]\implies (x-0)^2 + (y-3)^2=5[/tex]

So the center of this circle is (0, 3) and its radius is √5 units.

Since the new circle has the same center, its center is also (0, 3).

We know its radius is 4 units, so we can write the equation of the new circle as:

[tex]\implies (x - 0)^2 + (y - 3)^2 = 4^2[/tex]

[tex]\implies x^2 + (y - 3)^2 = 16[/tex]

Therefore, the equation of the circle in the standard (x, y) coordinate plane with a radius of 4 units and the same center as the circle determined by x² + y² - 6y + 4 = 0 is x² + (y - 3)² = 16.

To find:-

Answer:-

The given equation of the circle is ,

[tex]\implies x^2+y^2-6y + 4 = 0 \\[/tex]

Firstly complete the square for y in LHS of the equation as ,

[tex]\implies x^2 + y^2 -2(3)y + 4 = 0 \\[/tex]

Add and subtract 3² ,

[tex]\implies x^2 +\{ y^2 - 2(3)(y) + 3^2 \} -3^2 + 4 = 0 \\[/tex]

The term inside the curly brackets is in the form of a²-2ab+b² , which is the whole square of "a-b" . So we may rewrite it as ,

[tex]\implies x^2 + (y-3)^2 -9 + 4 = 0 \\[/tex]

[tex]\implies x^2 + (y-3)^2 - 5 = 0 \\[/tex]

[tex]\implies x^2 + (y-3)^2 = 5\\[/tex]

can be further rewritten as,

[tex]\implies (x-0)^2 + (y-3)^2 = \sqrt5^2\\[/tex]

now recall the standard equation of circle which is ,

[tex]\implies (x-h)^2 + (y-k)^2 = r^2 \\[/tex]

where,

So on comparing to the standard form, we have;

[tex]\implies \rm{Centre} = (0,3)\\[/tex]

Now we are given that the radius of second circle is 4units . On substituting the respective values, again in the standard equation of circle, we get;

[tex]\implies (x-h)^2 + (y-k)^2 = r^2 \\[/tex]

[tex]\implies (x-0)^2 + (y-3)^2 = 4^2 \\[/tex]

[tex]\implies \underline{\underline{\red{ x^2 + (y-3)^2 = 16}}}\\[/tex]

and we are done!

Answer:

(15+x)÷4 = 80-x

by criss cross we'll get:

15+x = 4(80-x)

15+x = 320-4x

x+4x=320-15

5x = 305

x = 61

Answer:

35/8

Step-by-step explanation:

A mixed fraction in the form [tex]a \dfrac{b}{c}[/tex] can be converted to an improper fraction using the following calculation:

[tex]a \dfrac{b}{c} = \dfrac{(a \times b) + b}{c}[/tex]

Here we have the improper fraction [tex]4 \dfrac{3}{8}[/tex]

Using the technique described
[tex]4 \dfrac{3}{8} = \dfrac{4 \times 8 + 3}{8} = \dfrac{32+ 3}{8} = \dfrac{35}{8}[/tex]

Ans: 35/8

Answer:

To calculate Justin's monthly installment, we need to use the formula for calculating the monthly payment for a loan:

P = (r * A) / (1 - (1 + r)^(-n))

where P is the monthly payment, r is the monthly interest rate, A is the loan amount, and n is the total number of payments (in months).

In this case, A = 18500, r = 0.06/12 = 0.005 (since the annual interest rate is 6%, we divide by 12 to get the monthly rate), and n = 4*12 = 48 (since the loan is for 4 years, or 48 months).

Plugging in these values, we get:

P = (0.005 * 18500) / (1 - (1 + 0.005)^(-48))

P ≈ $432.85

Therefore, Justin's monthly installment would be approximately $432.85

The accurate comparison shown by Histogram is that the mean is likely greater than the median because the data is skewed to the right. So, the correct option is A).

The histogram shows a right-skewed distribution, with a tail extending to the right of the peak. This indicates that there are a few cities with very high values that are pulling the mean to the right. In a right-skewed distribution, the mean is always greater than the median. This is because the mean is sensitive to extreme values and the median is not.

Therefore, option A is the accurate comparison. Option B is incorrect because the data is not skewed to the left. Option C is incorrect because the median is always less than the mean in a right-skewed distribution. Option D is also incorrect because the median is always less than the mean in a left-skewed distribution. The correct answer is A).

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_____The given question is incomplete, the complete question is given below:

3 4 5 8 9 10

The histogram below shows the average number of days per year in 117 Oklahoma cities where the high temperature was greater than 90 degrees Fahrenheit.

Which is an accurate comparison?

A  The mean is likely greater than the median because the data is skewed to the right.

B The mean is likely greater than the median because the data is skewed to the left.

C The median is likely greater than the mean because the data is skewed to the right.

D The median is likely greater than the mean because the data is skewed to the left.

Answer:

Step-by-step explanation:

The three inequalities that x and y must satisfy because of the package's requirements for protein, fat, and carbohydrate are:

Protein: 1x + 1y ≥ 8 (at least 8 units of protein)

Carbohydrates: 2x + 1y ≥ 11 (at least 11 units of carbohydrates)

Fat: 1x + 1y ≤ 10 (no more than 10 units of fat)

The inequalities that x and y must satisfy because they cannot be negative are:

x ≥ 0

y ≥ 0

Step-by-step explanation:

To satisfy the requirements for protein, fat, and carbohydrates, the following three inequalities must be satisfied:

1. 1x + 1y ≥ 8 (At least 8 units of protein)

2. 2x + 1y ≥ 11 (At least 11 units of carbohydrates)

3. 1x + 1y ≤ 10 (No more than 10 units of fat)

To ensure that x and y are non-negative, the following inequalities must be satisfied:

x ≥ 0y ≥ 0

Therefore, the complete set of inequalities for x and y are:

x + y ≥ 82x + y ≥ 11x + y ≤ 10x ≥ 0y ≥ 0

Answer:

(y - 2)² = -12(x - 5)

Step-by-step explanation:

A parabola is a locus of points, which are equidistant from the focus and directrix;

Generic cartesian equation of a parabola:

y² = 4ax, where the:

Focus, S, is: (a, 0)

Directrix, d, is: x = -a

a > 0

Put simply, a is the horinzontal difference between the directrix and the vertex or between the vertex and focus;

Always a good idea to do a quick drawing of the graph;

We are the told the focus, F, is: (2, 2) and directrix, d, is: x = 8;

First thing to note, the vertex, or turning point will be in line with the focus vertically, i.e. they will share the same y-coordinate;

Horizonatally, it will be halfway between the focus and the directrix, i.e. halfway between 8 and 2;

Therefore, the vertex will be will be (5, 2);

We can also work out a:

a = 8 - 5 = 5 - 2

a = 3

Substituting this value of a into the generic cartesian equation:

y² = 4(3)x

y² = 12x

The focus and directrix will be:

S: (3, 0)

d: x = -3

Next thing to note, a parabola curves away from the directrix;

In this case, the directrix is x = 8, so the vertex will be the right-most point on the parabola, it will curve off to the left and the focus will also be to the left;

What we want to do is compare with y² = 12x;

This parabola, has a vertex (0, 0), which is the left-most point that curves off to the right and a focus also to the right;

Since we know the formula of this parabola, if we figure out how to transform it into the one in the question, we can find out it's equation;

What we should recognise first is that the parabola in the question is reflected in the y-axis, compared to y² = 12x;

So we apply the transformation that corresponds to this, i.e. use the f(-x) rule:

y² = 12(-x)

y² = -12x

Now the two graphs will have the same shape and orientation;

The focus and directrix will also be affected:

S: (-3, 0)

d: x = 3

Now, the only remaining difference would be the coordinates of the focus and directrix of the two graphs;

The focus of the graph in the question is 5 units to the right and 2 units upwards compared to the focus of y² = -12x;

The directrix is 5 units to the right of that of y² = -12x;

So we apply a translation transformation of 5 units right and 2 units up, like so:

(y - 2)² = -12(x - 5)

Replace y with (y - 2) to translate up 2 units;

Replace x with (x - 5) to translate 5 units right.

We know have a parabola with focus, (2, 2), directrix, x = 8 and vertex, (5, 2), i.e. the parabola in the question;

Hence, the equation of the parabola in the question is:

(y - 2)² = -12(x - 5)

It might seem a bit long and complicated to begin with, but can be done very quickly if you can get used to it.

Yes, that is correct. The expression 0.25t2–4t+28 shows that the value of a first-edition deck of Wandering Wizards will decrease initially when it is first released, then increase as the deck becomes more rare. The maximum value of the deck is 28, which is when t=0 (when the game is first released).