mad sounds ugh PLEASE HELP ME THIS IS URGENT I WILL GIVE BRAINLIEST sorry ik the Pic is dark my camera is broken ​

Answer:

see below

Step-by-step explanation:

I am not able to draw graphs or diagrams. However, I can provide a description of how to graph Aaron's data on a set of axes.

Note.The horizontal axis could be marked with a scale of every hour, while the vertical axis could be marked in increments of 2°F to fit all the data.

The graph of the exponential function f(x) = [tex]5^{x}[/tex] is an upward-sloping curve that starts at (0, 1) and gets steeper and steeper as x increases. It never touches or crosses the x-axis, but it gets arbitrarily close to it as x approaches negative infinity.

As we may raise 5 to any power, all real numbers (-∞,∞ ) are the function's domain (positive, negative, or zero).

Range: The range of the function is all positive real numbers (0, ∞), since the exponential function always gives a positive result.

Asymptote: The graph of the function approaches the x-axis as x approaches negative infinity. Therefore, the x-axis is the horizontal asymptote.

Y-intercept: The y-intercept of the function is (0, 1), since[tex]5^{0}[/tex] = 1.

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The situation representing independent events are ,

Drawing an ace and spade from a deck of 52 cards.

Choosing a destination by 3 randomly selected travelers.

Representing the dependent and the independent events,

Drawing an ace and then drawing a spade from a standard deck of 52 cards without replacement.

The probability of drawing a spade is not affected by whether or not an ace was drawn first,

This is independent event .

Training as a long-distance runner and winning a marathon.

The probability of winning a marathon depends on various factors such as training, skill, and the competition.

These factors are not independent.

so these events are dependent.

The destinations of 3 randomly selected travelers at an airport.

The choice of destination for each traveler is independent of the choices made by the other travelers.

so these events are independent.

Flipping a coin and getting heads twice in a row.

The probability of getting heads on the second flip depends on the outcome of the first flip.

If the first flip resulted in tails, then the probability of getting heads on the second flip is 1/2.

However, if the first flip resulted in heads, then the probability of getting heads on the second flip is still 1/2.

This events are dependent.

Therefore, the independent events are drawing an ace and spade from deck of cards and selecting destination of 3 randomly selected travelers.

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Answer:

-8/5f - 15

or

-1.6f - 15

Step-by-step explanation:

-2.3f + 0.7f - 12 - 3 =

-8/5f - 15

or

-1.6f - 15

Answer:

[tex] \sf \: -1.6f - 15 [/tex]

Step-by-step explanation:

Now we have to,

→ Simplify the given expression.

The expression is,

→ -2.3f + 0.7f - 12 - 3

Let's simplify the expression,

→ -2.3f + 0.7f - 12 - 3

→ (-2.3f + 0.7f) - 12 - 3

→ (-1.6f) + (-12 - 3)

→ -1.6f + (-15)

→ -1.6f - 15

Hence, the answer is -1.6f - 15.

Answer:

The temperature will be the same in two days

Step-by-step explanation:

We can set up two equations to represent the temperatures in Boston (B) and San Francisco (SF) after a certain number of days (d):

B = -1 + 3d

SF = 6 - (d/2)

To find the number of days when the two temperatures will be the same, we can set the equations equal to each other and solve for d:

-1 + 3d = 6 - (d/2)

Multiplying both sides by 2 to eliminate the fraction:

-2 + 6d = 12 - d

Combining like terms:

7d = 14

Dividing both sides by 7:

d = 2

Therefore, the two temperatures will be the same after 2 days.

(1) The probability of getting tails and a 3 is the probability of following the path T-3, which is 1/2 x 1/4 = 1/8.

(ii) The probability of getting heads and an even number is the probability of following the path H-2 or H-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.

(iii) The probability of not getting heads and an even number is the probability of following the paths T-2 or T-4, which is 1/2 x 1/2 + 1/2 x 1/2 = 1/2.

For the given equation of function, the production is equal when x = 1/3 and x = 3.

A relationship between a group of inputs and one output each is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function. y = f is how functions are typically represented (x).

The equation of the functions are:

A(x) = 3x²

B(x) = 8x + 3

The production when the functions are equal is given by:

3x² = 8x + 3

3x² - 8x - 3 = 0

3x² - 9x + x - 3 = 0

3x (x - 3) + 1 (x - 3) = 0

(3x + 1) (x - 3) = 0

x = 1/3 and x = 3

Thus, the production is equal when x = 1/3 and x = 3.

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Answer:

a. $76.00

b. y = 23x - 76

Step-by-step explanation:

a. If Anton earned $23 every week for twelve weeks, his total revenue would be 23 (12) = 276 dollars. He only had $200 after paying his parents back, though, so that means if you take away the $200 leftover money he got (after paying his parents back) from the $276 he earned over the 12 weeks, you'll get 276 - 200 = he borrowed $76.00 from his parents.

b. Anton is earning $23 every week, so the money, y, that he'll have after x weeks is y = 23x. But since he borrowed $76 from his parents, that much money should be subtracted from his money-making equation, leaving us with y = 23x - 76. Hope this was helpful!

Answer:

Step-by-step explanation:

2 triangles on either side = 4*3/2 = 6 * 2 = 12

rectangle on the back = 4*8 = 32

rectangle on top = 5*8 = 40

rectangle on bottom = 3*8 = 24

12 + 32 + 40 + 24 = 108 cm

Hope this is correct

Tell me if im wrong

Hence, after 12 minutes, there are 936 liters of water in the pond overall as the rate of water addition is constant at 28 liters per minute.

To answer mathematical issues involving proportional relationships between two or more quantities, one uses the unitary method. It is a technique for determining the value of one unit depending on the value of another, in other words. In several disciplines, including mathematics, physics, economics, and engineering, the unitary technique is frequently utilized. Finding the value of a single unit of a given quantity and using that value to calculate the value of a desired number of units is what this process entails.

given

The initial amount of water plus the water that has been added over time equals the total amount of water in the pond at any one time. We can apply the following formula because the rate of water addition is constant at 28 liters per minute:

W = 600 + 28T

where W is the pond's total volume of water (measured in liters), and T is the passing of time (in minutes).

T = 12 can be used in the equation to determine how much water was in the pond overall after 12 minutes:

W = 600 + 28T

W = 600 + 28(12) (12)

W = 600 + 336

W = 936

Hence, after 12 minutes, there are 936 liters of water in the pond overall as the rate of water addition is constant at 28 liters per minute.

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The complete question is:-   Writing and evaluating a function that models a real-world...

Owners of a recreation area are filling a small pond with water. They are adding water at a rate of 28 liters per minute. There are 600 liters in the pond to start. Let W represent the total amount of water in the pond (in liters), and let 7' represent the total number of minutes that water has been added. Write an equation relating W to T. Then use this equation to find the total amount of water after 12 minutes.

The radius and height that maximize the volume of the cone with a slant height of 12 are both 6√2.

To maximize the volume of a cone with a slant height of 12, we need to find the dimensions (radius and height) that give the maximum volume. Use this formula to get the volume of a cone:

[tex]V = 1/3 * \pi * r^2 * h[/tex]

where r and h represents radius ad height respectively.

From the problem statement, we know that the slant height s of the cone is 12. We can use the Pythagorean theorem to relate the height h, radius r, and slant height s:

[tex]s^2 = r^2 + h^2[/tex]

Substituting s = 12, we get:

[tex]12^2 = r^2 + h^2[/tex]

[tex]144 = r^2 + h^2[/tex]

We want to maximize the volume V, so we need to express it in terms of just one variable. We can use the equation relating r, h, and s:

[tex]s^2 = r^2 + h^2[/tex]

Solving for [tex]h^2[/tex], we get:

[tex]h^2 = s^2 - r^2[/tex]

Substituting s = 12, we get:

[tex]h^2 = 144 - r^2[/tex]

Now we can express the volume V as a function of just one variable, r:

[tex]V = 1/3 * \pi * r^2 * h[/tex]

[tex]V = 1/3 * \pi * r^2 * \sqrt{(144 - r^2)[/tex]

To find the maximum volume, we need to take the derivative of V with respect to r, set it equal to zero, and solve for r:

[tex]dV/dr = 1/3 * \pi * (2r * \sqrt{(144 - r^2)} - 2r^3 / \sqrt{(144 - r^2)})[/tex]

[tex]dV/dr = \pi r / 3 * (2\sqrt{(144 - r^2)} - 2r^2 / \sqrt{(144 - r^2)})[/tex]

Setting dV/dr = 0, we get:

[tex]2\sqrt{(144 - r^2)} - 2r^2 / \sqrt{(144 - r^2)} = 0[/tex]

[tex]2(144 - r^2) - 2r^2 = 0[/tex]

[tex]288 - 4r^2 = 0[/tex]

[tex]r^2 = 72[/tex]

So the radius that maximizes the volume is [tex]r = \sqrt{(72)} = 6\sqrt{(2)[/tex]. To find the height h, we can use the equation [tex]h^2 = 144 - r^2[/tex]:

[tex]h^2 = 144 - (6\sqrt{(2)})^2 = 72[/tex]

[tex]h = \sqrt{(72)} = 6\sqrt{(2)[/tex]

Therefore, the dimensions (radius and height) that maximize the volume of the cone with a slant height of 12 are [tex]r = 6\sqrt{(2)[/tex] and [tex]h = 6\sqrt{(2)[/tex].

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The complete question is:

What are the dimensions (radius and height) of all right circular cones with a slant height of 12 that maximize the cone's volume? The cone's slant height is the distance from the outer edge of the base to the vertex.

Answer: 75

Step-by-step explanation:

7+5=75

Answer:

it is typically 22 to 7

Step-by-step explanation:

if you divide 22 by 7 you get 3.142857143...

Answer:

  22/7

Step-by-step explanation:

You want to know a ratio commonly used as an alternative to 3.14 for an approximation to pi.

The value 3.14 as an approximation of pi is obtained by simply truncating the value to 2 decimal places. That makes the approximation lower than the actual value by about 0.0507%.

Rational approximations can be found from the expression of pi as a continued fraction. Truncating that fraction at various points gives the approximations ...

  22/7 . . . . high by about 0.0402%

  333/106 . . . . low by about 0.0026%

  355/113 . . . . . high by about 0.0000085%

22/7 is the most commonly used rational approximation of pi instead of 3.14.

__

Additional comment

The continued fraction is non-repeating. It starts ...

  [tex]3+\cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\cfrac{1}{292+\cfrac{1}{1+\dots}}}}}[/tex]

The values of the lengths of the sides and segments obtained using the equivalent ratios of the sides of similar triangles are;

1. AA

2. AA

3. 200 ft

4. 21 feet

5. 37.5 meters

6. 4.2 meters

7. 6 feet

Similar triangles are triangles that posses similar shape, but may have varied sizes.

1. Two angles in triangle ORS are congruent to two angles in triangle VUT, therefore, the triangles are similar by the Angle-Angle, AA similarity postulate

2. Whereby the angle ∠ABC is congruent to the angle ∠ADF, and the angle ∠ABC and ∠ADGF are corresponding angles, we get by the conversed of the corresponding angles theorem;

BC ║ DF

However, angle A is congruent to itself, by reflexive property, therefore;

Triangle ABC is similar to triangle ADF, according to the AA similarity postulate.

3. Using the proportional sides relationship between similar triangles or the definition of the tangent of an angle, we get;

50/12.5 = h/50

h = 50 × 50/12.5 = 200

The height of the building is 200 ft

4. Using similar triangles relationship, we get;

7/2 = h/6

h = (7/2) × 6 = 21

The height of the taller flagpole is 21 ft

5. Using the equivalent ratio of the sides of similar triangles, we get;

x/25 = 12/8

x = 25 × (12/8) = 37.5

The distance is 37.5 meters

6. The equivalent ratio of the sides of similar triangles indicates;

h/7 = 9/15

h = 7 × (9/15) = 4.2

The height of the brace is 4.2 meters

7. The ratios of the sides of similar triangles indicates that we get;

136/34 = h/(1 1/2)

h = 1 1/2 × (136/34) = 6

The height of the man is 6 feet

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Answer:

The answer would be

Step-by-step explanation:

A. develop a point estimate of the population mean number of units sold per month.

I must say, I do hope this helps so good luck dude

Answer: "A 265 pizzas"

Step-by-step explanation:

STEP 1: Divide the number of pizzas by the number of days (5565/21).

This should give you the answer 265.

The two digits are 3 and 2.

A number is a numerical unit of measurement and labeling in mathematics.

The natural numbers 1, 2, 3, 4, and so on are the first examples.

Number words are a linguistic way to express numbers.

Most commonly, specific numbers can be represented using symbols known as numerals; for instance, the numeral "5" stands in for the number five.

Basic numerals are frequently grouped in a numeral system, which is an ordered manner to represent any number, because only a small number of symbols can be learned.

The most widely used number system is the Hindu-Arabic number system, which enables the depiction of any number using a combination of 10 basic numerical symbols, or "digits."

According to our question-

3+2

=5

Hence, The two digits are 3 and 2.

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Anna should pay £3666.67 in 2 and a half years from now to meet her obligation. It means simple interest is £3666.67.

To solve this problem, we can use the formula for simple interest:

I = Prt

where I is the interest, P is the principal, r is the interest rate, and t is the time in years.

Given that Anna borrowed 20000 at 5% for half a year, we can calculate the interest as:

I = Prt = 20000 x 0.05 x 0.5 = 500

So, the total amount Anna needs to pay at maturity is 20000 + 500 = 20500.

Anna plans to pay 8000 on maturity, 2000 in 10 months, and 5000 in 16 months. Therefore, the remaining amount she needs to pay is:

20500 - 8000 - 2000 - 5000 = 5500

To find out how much she needs to pay in 2 and a half years from now, we need to first calculate the time in years:

2 and a half years = 2.5 years

The total time elapsed from the time of borrowing is:

0.5 year (at borrowing) + 2.5 years = 3 years

Using the formula for simple interest, we can calculate the principal amount that will accrue an interest of 5500 in 3 years at 5% per annum:

I = Prt

5500 = P x 0.05 x 3

P = 3666.67

Therefore, Anna should pay £3666.67 in 2 and a half years.

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The answers for the following questions and fill uos are given below respectively.

Study of correlations between triangles' sides and angles is the focus of the mathematical field of trigonometry.

It solves mathematical problems, especially those involving geometric shapes and motion, by using the properties of triangle angles and sides.

We can use the given side lengths of the right triangle KJL to determine the trigonometric ratios:

sin K = opposite/hypotenuse = KJ/LK = 9/10

cos L = adjacent/hypotenuse = KJ/LK = 9/10

To express these trig ratios as fractions in simplest terms, we can simplify each fraction by dividing both the numerator and denominator by their greatest common factor, which is 1 in this case. Therefore:

sin K = 9/10

cos L = 9/10

Since sin K is the ratio of the opposite side (KJ) to the hypotenuse (LK), it corresponds to angle K. Similarly, cos L is the ratio of the adjacent side (KJ) to the hypotenuse (LK), it corresponds to angle L.

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The ratio of the opposing side (KJ) to the hypotenuse is called sin K, which is equivalent to angle K. (LK). Cos L is the ratio of the neighboring side (KJ) to the hypotenuse and is related to angle L. (LK).

The area of mathematics known as trigonometry focuses on the analysis of relationships between shapes' sides and angles.

It uses the characteristics of triangle angles and sides to answer mathematical problems, particularly those involving geometric shapes and motion.

The trigonometric ratios can be calculated using the side lengths of the right triangle KJL as given:

sin K = opposite/hypotenuse = KJ/LK = 9/10

cos L = adjacent/hypotenuse = KJ/LK = 9/10

We can simplify each fraction by dividing the numerator as well as the denominator by their greatest common factor, which is 1 in this instance, to express these trig ratios as fractions in the simplest possible terms.

sin K = 9/10

cos L = 9/10

Sin K correlates to angle K because it is the ratio of the opposite side (KJ) to the hypotenuse (LK). Similar to cos L, which correlates to angle L, cos L is the ratio of the adjacent side (KJ) to the hypotenuse (LK).

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Answer:

Hope this helps

Step-by-step explanation:

Sum of series is donated by the for formula below

[tex] \sf \: Sn = \frac{n}{2} [2a + (n - 1)d][/tex]

Example 3,6,9,12,...

Let's say we're finding the sum of the 20 series

n = 20, a = 3, d = 6-3 = 3

Where,

n means number of terms

a means the first term

d means the constant value between terms

S20 = 20/2[2(3) + (20-1)d]

S20 = 10[6+19(3)]

S20 = 60 + 570

S20 = 630

Voila!!

Surface area of the cylinder in terms of pi, given a diameter of 3.4 inches and a height of 3 inches, is [tex]28.36\pi[/tex] square inches.

To find the surface area of a cylinder in terms of pi, you need to add the areas of the top and bottom circles (which are identical) and the area of the curved lateral surface.

The formula for the area of a circle is [tex]A = \pi r^2[/tex], where r is the radius of the circle. Since the diameter of the circle is given as 3.4 inches, we can find the radius by dividing the diameter by 2:

r = 3.4 / 2 = 1.7 inches

Using this radius, the area of one circle is:

A = [tex]\pi (1.7)^2[/tex] = [tex]9.08\pi[/tex] square inches

Since the cylinder has two identical circles, the total area of both circles is:

2A = 2([tex]9.08\pi[/tex]) = [tex]18.16\pi[/tex] square inches

The lateral surface area of a cylinder can be found by multiplying the circumference of the base circle by the height of the cylinder. The formula for the circumference of a circle is C = 2πr, so the circumference of our circle is:

C =[tex]2\pi (1.7)[/tex]= [tex]3.4\pi[/tex] inches

The height of the cylinder is given as 3 inches, so the lateral surface area is:

A = Ch = ([tex]3.4\pi[/tex])(3) = [tex]10.2\pi[/tex] square inches

To find the total surface area of the cylinder, we add the areas of the two circles and the lateral surface:

Total surface area = 2A + A = [tex]18.16\pi + 10.2\pi[/tex] = [tex]28.36\pi[/tex] square inches

Therefore, the surface area of the cylinder in terms of pi, given a diameter of 3.4 inches and a height of 3 inches, is [tex]28.36\pi[/tex] square inches.

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Answer:

b = 136°

Step-by-step explanation:

b + 44 = 180°

180° - 44° = 136°

The surface area of the storage container is 1536 square feet. To find the surface area of a storage container with dimensions 24 feet long, 18 feet wide, and 8 feet high, we need to calculate the area of each of its six faces and then add them together.

The front and back face each have an area of 24 feet x 8 feet = 192 square feet.

The top and bottom faces each have an area of 18 feet x 24 feet = 432 square feet.

The left and right faces each have an area of 8 feet x 18 feet = 144 square feet.

Adding these areas together, we get:

Surface area of the container = 2(192 sq. ft.) + 2(432 sq. ft.) + 2(144 sq. ft.)

The surface area of the container = 384 sq. ft. + 864 sq. ft. + 288 sq. ft.

The surface area of the container = 1536 square feet

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This is a classic river crossing puzzle. To safely get all three across take the goat across, leave the goat, take the lion across, take the cabbage across,  return with the empty boat, Take the goat across to reunite with the cabbage and lion

Here's one possible solution:

By following these steps, you will have successfully transported all three items across the river without violating the rules.

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For the number of small aircraft that arrive during a 75-min period, the expected value if 10 and standard deviation is 3.17.

The expected value of the number of small aircraft that arrive during a 75-minute period is ⇒ E[X] = μ = 8t.

We need to convert the time period to hours,

So, t = 75/60 = 1.25 hours.

So, the expected number of small aircraft arrivals during a 75-minute period is ⇒ E[X] = μ = 8(1.25) = 10,

To find the standard deviation, we know that for a Poisson distribution, the standard deviation(σₓ) is the square root of mean.

So, σₓ = √μ = √8t,

Substituting t = 1.25,

We get,

⇒ σₓ = √8(1.25) = √10 ≈ 3.17,

Therefore, the expected value for number of aircraft is 10, and standard deviation is 3.17.

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The given question is incomplete, the complete question is

Suppose small aircraft arrive at a certain airport according to a Poisson process with rate α =8 per hour, so that the number of arrivals during a time period of t hours is a Poisson rv with parameter μ = 8t.

What are the expected value and standard deviation of the number of small aircraft that arrive during a 75-min period?

An equation which is made true by the opposite angles theorem include the following: D. 3y − 15 = 85 + y.

In Mathematics, a parallelogram simply refers to a four-sided geometrical figure (shape) and it can be defined as a type of quadrilateral and two-dimensional geometrical figure that is composed of two (2) equal and parallel opposite sides.

According to the opposite angles theorem, the opposite angles of a parallelogram are always congruent or equal and as such, we have the following:

(3y − 15)° = (85 + y)°

Next, we would determine the value of y as follows;

(3y − 15)° = (85 + y)°

3y - y = 85 + 15

2y = 100

y = 100/2

y = 50.

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Answer:

Step-by-step explanation:

the answer is [3y-15=85+y]

The simplified expression is  [tex]\frac{1-4i\sqrt3}{7}[/tex].

What is simplification?

Simplifying an expression is just another way to say solving a math problem. When you simplify an expression, you're basically trying to write it in the simplest way possible. At the end, there shouldn't be any more adding, subtracting, multiplying, or dividing left to do. For example, take this expression: 4 + 6 + 5.

Here the given expression is

=> [tex]\frac{2-i\sqrt3}{2+i\sqrt3}[/tex]

Multiply and divide by [tex]2-i\sqrt3[/tex] then

=> [tex]\frac{2-i\sqrt3}{2+i\sqrt3}\times\frac{2-i\sqrt3}{2-i\sqrt3}[/tex]

=> [tex]\frac{(2-i\sqrt3)^2}{2^2-(i\sqrt3)^2}[/tex]

=> [tex]\frac{4+i^2\sqrt3^2-4i\sqrt3}{4-i^2\sqrt3^2}[/tex]

=> [tex]\frac{4-3-4i\sqrt3}{4+3}[/tex]

=> [tex]\frac{1-4i\sqrt3}{7}[/tex]

Hence the simplified expression is  [tex]\frac{1-4i\sqrt3}{7}[/tex].

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Answer:

[tex]\textsf{(C)} \quad -\dfrac{1}{4}[/tex]

Step-by-step explanation:

To differentiate an equation that contains a mixture of x and y terms, use implicit differentiation.

To find dy/dx for x² + 3y² = 8 + 2xy, differentiate each term with respect to x.

Begin by placing d/dx in front of each term of the equation:

[tex]\dfrac{\text{d}}{\text{d}x}x^2+\dfrac{\text{d}}{\text{d}x}3y^2=\dfrac{\text{d}}{\text{d}x}8+\dfrac{\text{d}}{\text{d}x}2xy[/tex]

Differentiate the terms in x only (and constant terms):

[tex]\implies 2x+\dfrac{\text{d}}{\text{d}x}3y^2=0+\dfrac{\text{d}}{\text{d}x}2xy[/tex]

Use the chain rule to differentiate terms in y only.

In practice, this means differentiate with respect to y, and place dy/dx at the end:

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=0+\dfrac{\text{d}}{\text{d}x}2xy[/tex]

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=\dfrac{\text{d}}{\text{d}x}2xy[/tex]

Use the product rule to differentiate the term in x and y.

[tex]\textsf{Let}\;u(x)=2x \implies u'(x)=2[/tex]

[tex]\textsf{Let}\;v(x)=y \implies v'(x)= 1\dfrac{\text{d}y}{\text{d}x}[/tex]

Therefore:

[tex]\implies \dfrac{\text{d}}{\text{d}x}[u(x) \cdot v(x)]=u(x) \cdot v'(x)+v(x) \cdot u'(x)[/tex]

[tex]\implies \dfrac{\text{d}}{\text{d}x}2xy=2x \cdot 1 \dfrac{\text{d}y}{\text{d}x}+y\cdot 2[/tex]

[tex]\implies \dfrac{\text{d}}{\text{d}x}2xy=2x \dfrac{\text{d}y}{\text{d}x}+2y[/tex]

So the final differentiated equation is:

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=2x \dfrac{\text{d}y}{\text{d}x}+2y[/tex]

Rearrange the resulting equation to make dy/dx the subject:

[tex]\implies 2x+6y\dfrac{\text{d}y}{\text{d}x}=2x \dfrac{\text{d}y}{\text{d}x}+2y[/tex]

[tex]\implies 6y\dfrac{\text{d}y}{\text{d}x}-2x \dfrac{\text{d}y}{\text{d}x}=-2x+2y[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}(-2x+6y)=-2x+2y[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-2x+2y}{-2x+6y}[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{2(-x+y)}{2(-x+3y)}[/tex]

[tex]\implies \dfrac{\text{d}y}{\text{d}x}=\dfrac{-x+y}{-x+3y}[/tex]

To find d²y/dx², differentiate again using the quotient rule and implicit differentiation.

[tex]\textsf{Let}\;u(x)=-x+y \implies u'(x)=-1+1\dfrac{\text{d}y}{\text{d}x}[/tex]

[tex]\textsf{Let}\;v(x)=-x+3y \implies v'(x)=-1+3\dfrac{\text{d}y}{\text{d}x}[/tex]

Therefore:

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\text{d}}{\text{d}x}\left[\dfrac{u(x)}{v(x)}\right]=\dfrac{v(x)\cdot u'(x)-u(x) \cdot v'(x)}{[v(x)]^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\text{d}}{\text{d}x}\left[\dfrac{-x+y}{-x+3y}\right]=\dfrac{(-x+3y) \cdot \left(-1+1\dfrac{\text{d}y}{\text{d}x}\right)-(-x+y) \cdot \left(-1+3\dfrac{\text{d}y}{\text{d}x}\right)}{(-x+3y)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{(3y-x)\left(\dfrac{\text{d}y}{\text{d}x}-1\right)-(y-x)\left(3\dfrac{\text{d}y}{\text{d}x}-1\right)}{(-x+3y)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{\left(3y\dfrac{\text{d}y}{\text{d}x}-3y-x\dfrac{\text{d}y}{\text{d}x}+x\right)-\left(3y\dfrac{\text{d}y}{\text{d}x}-y-3x\dfrac{\text{d}y}{\text{d}x}+x\right)}{(-x+3y)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2y+2x\dfrac{\text{d}y}{\text{d}x}}{(-x+3y)^2}[/tex]

Substitute in dy/dx:

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2y+2x\left(\dfrac{-x+y}{-x+3y}\right)}{(-x+3y)^2}[/tex]

To find d²y/dx² at (2, 2), substitute x = 2 and y = 2 into the second derivative:

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-2(2)+2(2)\left(\dfrac{-2+2}{-2+3(2)}\right)}{(-2+3(2))^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-4+4\left(\dfrac{0}{4}\right)}{(4)^2}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=\dfrac{-4}{16}[/tex]

[tex]\implies \dfrac{\text{d}^2y}{\text{d}x^2}=-\dfrac{1}{4}[/tex]