30 pointys!!!!!
Please tell me the answer, 30 points avaliable,

Hence, the length of the curve defined by the parametric equations x = 4sin^2(t) and y = 4cos^2(t) over the interval 0 ≤ t ≤ 5π is 20π units.

To find the distance traveled by the particle, we need to calculate the length of the curve defined by the parametric equations x = 4sin^2(t) and y = 4cos^2(t) over the given time interval 0 ≤ t ≤ 5π.

We can use the arc length formula to calculate the length of the curve. The arc length formula for a parametric curve defined by x = f(t) and y = g(t) is given by:

L = ∫[a, b] √[f'(t)^2 + g'(t)^2] dt

where f'(t) and g'(t) are the derivatives of f(t) and g(t) with respect to t.

Let's start by finding the derivatives of x and y with respect to t:

x = 4sin^2(t)

x' = d/dt(4sin^2(t))

= 8sin(t)cos(t)

= 4sin(2t)

y = 4cos^2(t)

y' = d/dt(4cos^2(t))

= -8cos(t)sin(t)

= -4sin(2t)

Now, let's calculate the length of the curve using the arc length formula:

L = ∫[0, 5π] √[x'(t)^2 + y'(t)^2] dt

= ∫[0, 5π] √[16sin^2(2t) + 16sin^2(2t)] dt

= ∫[0, 5π] √[32sin^2(2t)] dt

= ∫[0, 5π] √[32sin^2(2t)] dt

= ∫[0, 5π] 4√[2sin^2(2t)] dt

= 4∫[0, 5π] √[2sin^2(2t)] dt

= 4∫[0, 5π] √[2(1 - cos^2(2t))] dt

= 4∫[0, 5π] √[2(1 - (1 - 2sin^2(t))^2)] dt

= 4∫[0, 5π] √[2(2sin^4(t))] dt

= 4∫[0, 5π] √[8sin^4(t)] dt

= 4∫[0, 5π] 2sin^2(t) dt

= 8∫[0, 5π] sin^2(t) dt

We can use the trigonometric identity sin^2(t) = (1 - cos(2t))/2 to simplify the integral further:

L = 8∫[0, 5π] sin^2(t) dt

= 8∫[0, 5π] (1 - cos(2t))/2 dt

= 4∫[0, 5π] (1 - cos(2t)) dt

= 4∫[0, 5π] dt - 4∫[0, 5π] cos(2t) dt

The integral of dt over the interval [0, 5π] is simply the length of the interval, which is 5π - 0 = 5π. The integral of cos(2t) over the same interval is zero since the cosine function is periodic with period π.

Therefore, the length of the curve is given by:

L = 4(5π) - 4(0)

= 20π

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Answer:

total price before sale = 20

total sale price = 5-original price

=5-20

= 15

mr randall bought 4 t-shirt =15×4=60

mr randall paid 60$ for the t-shirt

Answer:

x = - 2, x = 12

Step-by-step explanation:

Given

x² - 9x = x + 24 ( subtract x + 24 from both sides )

x² - 10x - 24 = 0 ← in standard form

(x + 2)(x - 12) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 2 = 0 ⇒ x = - 2

x - 12 = 0 ⇒ x = 12

Answer:

-5+8x

Step-by-step explanation:

Answer:

Increase of 9.3%

Step-by-step explanation:

Hence, the percent increase in salary did Jerry receive is \(4.68\)%.

What is the percentage?

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".

Here given that,

Jerry’s monthly salary was $\(3,050\) last year. This year his monthly salary is $\(3,350\).

His monthly salary in last year is $ \(3050\).

In this year his salary is $\(3350\).

So, the increase in salary is

\(3350-3050=300\)

And the overal salary is $ \(3050+3350=6400\)

So, the percent increase in salary did Jerry receive is

\(\frac{300}{6400}\) × \(100=4.68\)

Hence, the percent increase in salary did Jerry receive is \(4.68\)%.

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Answer:

g(x) = -2^x

Step-by-step explanation:

This is an exponential function from the looks of the value for x as it decreases toward the negative value (increase in the absolute value calculation) y is constricted from its vigorous values down to the fractional components of an equation of the standard appearance of y = C^x. For this, try some values. x = 1 y = 2, x = 2 y = 4. There is a trend that is established through these points, and g(x) = -2^x seems to follow this trend as it is reflect across the x-axis. Enter points, it is confirmed and a graphing calculator further substantiates. Try for yourself!

Answer:

A) added to both sides of the equation

Step-by-step explanation:

Answer:

please be more specific in your answer.

Step-by-step explanation:

i know that if it has a positive y coordinate then it would mean it has to be above the x-axis but thats all i can tell you from this.

Answer:

64-degree ( Alternate angles)

Answer:

y = 64 degree

Step-by-step explanation:

angle y =  64 degree (being alternate interior angles)

y° = 64°

B. save time in arriving at solutions to problems. Heuristics are problem-solving shortcuts or rules of thumb that can save time when arriving at solutions to problems.

The use of heuristics rather than algorithms is most likely to save time in arriving at solutions to problems. Heuristics are mental shortcuts or rules of thumb that are used to solve problems quickly and efficiently, whereas algorithms are step-by-step procedures for solving problems. While algorithms may produce more accurate solutions, they can be time-consuming to apply. Heuristics, on the other hand, allow for quick problem-solving and are often based on our language skills and prior knowledge. However, relying too heavily on heuristics can sometimes lead to errors or the overconfidence phenomenon.

They might not always yield the most accurate solutions compared to algorithms, which are step-by-step, systematic procedures that guarantee a correct solution. Heuristics usually rely less on language skills and do not necessarily minimize the overconfidence phenomenon.

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Heuristics are mental shortcuts or rules of thumb that people use to make decisions or solve problems quickly and efficiently.

B. save time in arriving at solutions to problems . The correct answer is:

Heuristics are mental shortcuts or rules of thumb that people use to make decisions or solve problems quickly and efficiently. They are simple and often automatic strategies that allow individuals to make judgments or decisions without engaging in a lengthy, deliberate, and systematic process of problem-solving.

One of the main advantages of using heuristics is that they can save time in arriving at solutions to problems. Instead of analyzing all available information and considering all possible alternatives, heuristics allow individuals to quickly narrow down options and make decisions based on simplified cognitive processes. This can be especially useful in situations where time is limited or when individuals need to make decisions on the spot.

However, heuristics are not always foolproof and can sometimes lead to errors or biases in decision-making.

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Hello! My name is Chris and I’ll be helping you with this problem.

Date: 9/28/20 Time:  9:19 am CST

Answer:

-12

Explanation:

(-3)4 = -12

I hope this helped answer your question! Have a great rest of your day!

Furthermore,

Chris  

Answer:

-12

Step-by-step explanation:

( -3 ) 4

= -12

hope this helps

Therefore, the probability that 0 < X < 1 is approximately 0.3413, or 34.13%.

If a random variable X is distributed normally with zero mean and unit standard deviation (X ~ N(0, 1)), the probability that 0 < X < 1 can be calculated using the standard normal distribution table or a statistical software.

In this case, we need to find the area under the normal curve between 0 and 1 standard deviations from the mean. Since the standard deviation is 1, we are interested in finding the probability that the value of X falls between 0 and 1.

Using the standard normal distribution table, we can look up the cumulative probability associated with 1 standard deviation from the mean, which is approximately 0.8413. Similarly, we can look up the cumulative probability associated with 0 standard deviations from the mean, which is 0.5.

To find the probability that 0 < X < 1, we subtract the probability associated with 0 from the probability associated with 1:

P(0 < X < 1) = P(X < 1) - P(X < 0) = 0.8413 - 0.5 = 0.3413

Therefore, the probability that 0 < X < 1 is approximately 0.3413, or 34.13%.

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The optimal values of these parameters are:

a. β₁ = 0

b. β₂ = 0

The linear regression y = β1 + β2x + e, assuming that the sum of squared errors (SSE) takes the following form:

SSE = 382 + 681 + 382 + 18β1β2

Derive the partial derivatives of SSE with respect to β1 and β2 and solve the optimal values of these parameters.

Given that SSE = 382 + 681 + 382 + 18β1β2 ∂SSE/∂β1 = 0 ∂SSE/∂β2 = 0

Now, we need to find the partial derivative of SSE with respect to β1.

∂SSE/∂β1 = 0 + 0 + 0 + 18β2 ⇒ 18β2 = 0 ⇒ β2 = 0

Therefore, we obtain the optimal value of β2 as 0.

Now, we need to find the partial derivative of SSE with respect to β2. ∂SSE/∂β2 = 0 + 0 + 0 + 18β1 ⇒ 18β1 = 0 ⇒ β1 = 0

Therefore, we obtain the optimal value of β1 as 0. Hence, the partial derivative of SSE with respect to β1 is 18β2 and the partial derivative of SSE with respect to β2 is 18β1.

Thus, the optimal values of β1 and β2 are 0 and 0, respectively.

Therefore, the answers are: a. β₁ = 0 b. β₂ = 0

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The missing value for x is [15w/15].

What is the value of x?

In similar polygons, corresponding sides are proportional, which means they have the same ratio. In this case, we have two polygons that are similar. We are given the value of one side, 15, and the variable w for the corresponding side in the other polygon. To find the missing value of x, we can set up a proportion based on the ratio of the corresponding sides: 15/15 = w/x. Solving this proportion, we can cross-multiply to get 15x = 15w. Dividing both sides by 15, we find that x = 15w/15, which simplifies to x = w.

Similar polygons are figures that have the same shape but may differ in size. When polygons are similar, their corresponding sides are in proportion to each other. This means that the ratio of the lengths of corresponding sides is constant. In the given question, we have two similar polygons with a missing value for x. By setting up a proportion based on the corresponding sides, we can solve for x. In this case, the ratio of the lengths of the given side 15 and the corresponding side w is equal to the ratio of x and the corresponding side. Solving the proportion, we find that x is equal to 15w/15, which simplifies to x = w.

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The polar form of the complex number 4 + 4i is:

4√2 cis(π/4)

To determine the polar form of the complex number 4 + 4i, we need to find the magnitude (r) and the argument (θ).

Magnitude (r):

The magnitude of a complex number is calculated using the formula:

r = √(a^2 + b^2)

where a is the real part and b is the imaginary part of the complex number.

In this case, a = 4 and b = 4, so the magnitude is:

r = √(4^2 + 4^2) = √(16 + 16) = √32 = 4√2

Argument (θ):

The argument of a complex number is calculated using the formula:

θ = arctan(b/a)

In this case, a = 4 and b = 4, so the argument is:

θ = arctan(4/4) = arctan(1) = π/4 (in radians)

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The probability that one or more of them tests negative is 0.85.

Probability is defined as the likeliness of an event to occur. The probability of any event to occur ranges from 0 to 1, and the sum of all the probabilities of all the events happening is 1.

If Marv and Patricia will each take a Covid test, then the events that will occur are : both will test positive, both will test negative, or one of them will test negative.

Based on the given information, the probability that both will test positive is 0.15. Subtract it from 1 to and get the probability that one or more of them tests negative.

probability = 1 - 0.15

probability = 0.85

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The Graham's scan algorithm calculates the convex hull of a set of points, which guarantees a convex polygon. The smallest area polygon containing the set of points is not the convex hull of those points.

To demonstrate that the smallest area polygon containing a set of points may not always be a convex hull of that set of points, let's consider a simple example.

Suppose we have a set of four points: A, B, C, and D, arranged in a square formation, as shown below:

A----B

|    |

|    |

D----C

If we calculate the convex hull of these points using the Graham's scan algorithm, we would obtain the convex hull as ABCDA, which is a square.

However, if we look for the smallest area polygon that contains these points, we can find a non-convex polygon. By connecting the points A, B, C, and D in that order, we form a quadrilateral:

A----B

\  /

 \/

 /\

/  \

D----C

This quadrilateral, ABCD, is not convex, as it has an internal angle greater than 180 degrees.

Hence, in this example, the smallest area polygon containing the set of points is not the convex hull of those points. This example demonstrates that the smallest area polygon can be non-convex, and the convex hull may not always provide the minimum area solution.

The Graham's scan algorithm specifically calculates the convex hull of a set of points, which guarantees a convex polygon. However, to find the smallest area polygon, other techniques or algorithms need to be applied.

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It will take 12 seconds until Hudson and Knox have run the same number of feet.

Let's denote the time it takes until Hudson and Knox have run the same number of feet as "t" (in seconds).

The distance Hudson runs can be calculated by multiplying his speed (8.8 feet/second) by the time "t". Thus, the distance Hudson covers is 8.8t feet.

Knox, on the other hand, had a head start of 30 feet. So the distance Knox covers can be calculated by multiplying his speed (6.3 feet/second) by the time "t" and adding the head start of 30 feet. Thus, the distance Knox covers is 6.3t + 30 feet.

To find the time when both runners have covered the same distance, we set their distances equal to each other:

8.8t = 6.3t + 30

Simplifying the equation:

2.5t = 30

Dividing both sides by 2.5:

t = 30 / 2.5

t = 12

Therefore, it will take 12 seconds until Hudson and Knox have run the same number of feet.

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The equation of the hyperbola is: (x^2/16) - (y^2/16) = 1

To begin, we need to use the definition of a hyperbola to find its equation. The standard form of the equation for a hyperbola centered at the origin is:

(x^2/a^2) - (y^2/b^2) = 1

Where a is the distance from the origin to the vertex along the x-axis, and b is the distance from the origin to the vertex along the y-axis.

We know that the foci of this hyperbola are at (4,0) and (-4,0). The distance between the foci is given by 2c, where c is the distance from the center of the hyperbola to a focus. So we have:

2c = 8
c = 4

We also know that the difference in the lengths of the focal radii from any point on the hyperbola is always ±6. This means that for any point (x,y) on the hyperbola, we have:

|sqrt((x-4)^2 + y^2) - sqrt((x+4)^2 + y^2)| = 6

Simplifying this equation, we get:

sqrt((x-4)^2 + y^2) - sqrt((x+4)^2 + y^2) = 6  (since the difference is always positive)
Squaring both sides, we get:

(x-4)^2 + y^2 - 2sqrt((x-4)^2 + y^2)*(x+4)^2 + y^2 = 36
Simplifying, we get:

(x-4)^2 + y^2 - (x+4)^2 - y^2 = 36
(x-4)^2 - (x+4)^2 = 36

Now we can simplify further using the difference of squares:

[(x-4)+(x+4)][(x-4)-(x+4)] = 36
2x*(-8) = 36
x = -9

Substituting x = -9 into our equation, we get:

(-9-4)^2 + y^2 - (-9+4)^2 - y^2 = 36
(-13)^2 - 5^2 = 36
144 = 36

This is a contradiction, so our original assumption that the hyperbola is centered at the origin must be incorrect. Instead, we know that the center of the hyperbola is at (0,0). This means that a = c = 4.

Substituting these values into the standard form of the equation for a hyperbola, we get:

(x^2/16) - (y^2/16) = 1

we can find the equation of the hyperbola. Since the foci are at (4, 0) and (-4, 0), the distance between them is 2a = 8, so a = 4. The difference in the lengths of the focal radii is 2b = ±6, so b = 3.

Now, we can use the formula for the equation of a hyperbola with horizontal major axis:

(x^2 / a^2) - (y^2 / b^2) = 1

Substitute the values of a and b:

(x^2 / 4^2) - (y^2 / 3^2) = 1

The equation of the hyperbola is:

(x^2 / 16) - (y^2 / 9) = 1

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The solution to the given system of equations is x = 7 and y = 1.

To solve the system of equations graphed on the coordinate axes, we need to find the point where the two lines intersect. This point represents the solution to the system of equations.

The given system of equations is:

y = x - 6

y = -x + 8

To find the point of intersection, we can set the two equations equal to each other and solve for x:

x - 6 = -x + 8

Combining like terms:

2x = 14

Dividing both sides by 2:

x = 7

Now, we substitute the value of x back into either of the original equations to find the corresponding y-coordinate. Let's use equation 1:

y = 7 - 6

y = 1

Therefore, the point of intersection is (7, 1). This point represents the solution to the system of equations.

Visually, on the coordinate axes, the lines y = x - 6 and y = -x + 8 intersect at the point (7, 1). This point is where the two lines cross each other.

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The type of triangle that is given by the dimensions, is a Scalene triangle.

A scalene triangle has sides that are all measured differently. In such a triangle, no side will be the same length as any other side. Every interior angle in a scalene triangle is also unique.

The lengths of two of the three sides in an isosceles triangle are equal. The angles that are opposite the equal sides are therefore equal. An isosceles triangle, in other words, has two equal sides and two equal angles.

Each side of an equilateral triangle has an equal length. Each internal angle in this scenario will be 60 degrees in length. An equilateral triangle is often referred to as an equiangular triangle since its angles are equal.

The dimensions here are not the same which means that all the sides are not equal and so this is a scalene triangle.

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Answer:

He added $25 to the contents of his wallet.

Step-by-step explanation:

Here's what's happening, symbolically:

$6 + m = $31, or m = $25.  Jake added $25 (represented by m) to the contents of his wallet.

The measure of the angle which makes the slides with the ground is equals to 51.1 degrees approximately.

Length of the slide = 14.5 feet

Distance from the end of the slide to the base of the ladder = 11.7 feet

To determine the measure of the angle that the slide makes with the ground, we can use trigonometry.

Use the tangent function to find the angle.

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Let us denote the angle we want to find as θ.

In a right triangle formed by the slide, the ground, and the ladder,

The slide is the opposite side and the distance from the end of the slide to the base of the ladder is the adjacent side.

Using the tangent function,

⇒tan(θ) = opposite / adjacent

⇒ tan(θ) = 14.5 / 11.7

To find the measure of the angle θ,

Take the inverse tangent (arctan) of both sides we get,

⇒ θ = arctan(14.5 / 11.7)

Using trigonometric calculator, the approximate value of θ is

⇒ θ ≈ 51.1 degrees

Therefore, the measure of the angle that the slide makes with the ground is approximately 51.1 degrees.

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The volume of the figure is approximately 1591.63 cm³.

We have,

To find the volume of the figure with a semicircle on top of a cone, we can break it down into two parts: the volume of the cone and the volume of the semicircle.

The volume of the Cone:

The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone.

Given that the diameter of the cone is 14 cm, the radius (r) is half of the diameter, which is 7 cm.

The height (h) of the cone is 17 cm.

Plugging the values into the formula, we have:

V_cone = (1/3)π(7 cm)²(17 cm)

V_cone = (1/3)π(49 cm²)(17 cm)

V_cone = (1/3)π(833 cm³)

V_cone ≈ 872.67 cm³ (rounded to two decimal places)

The volume of the Semicircle:

The formula for the volume of a sphere is V = (2/3)πr³, where r is the radius of the sphere. In this case, since we have a semicircle, the radius is half of the diameter of the base.

Given that the diameter of the cone is 14 cm, the radius (r) of the semicircle is half of that, which is 7 cm.

Plugging the value into the formula, we have:

V_semicircle = (2/3)π(7 cm)³

V_semicircle = (2/3)π(343 cm³)

V_semicircle ≈ 718.96 cm³ (rounded to two decimal places)

Total Volume:

To find the total volume, we add the volume of the cone and the volume of the semicircle:

V_total = V_cone + V_semicircle

V_total ≈ 872.67 cm³ + 718.96 cm³

V_total ≈ 1591.63 cm³ (rounded to two decimal places)

Therefore,

The volume of the figure is approximately 1591.63 cm³.

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The number of terms that must be added together to achieve this series s is 5000.

Given that,

The remainder of the following convergent series must be less than 10 Superscript negative 5 in magnitude.

We have to calculate the number of terms that must be added together to achieve this.

Given

S= summation n=0 to infinity (-1)ⁿaₙ.

An upper bound for the error of the series Rₙ is aₙ₊₁ so

Rₙ=|s-sₙ|≤aₙ₊₁

In this case we have

1/2n+1≤1/10⁴

10⁴≤2n+1

(10⁴-1)/2≤n

(10⁴-1)/2 is approximately 5000.

Therefore, the number of terms that must be added together to achieve this series s is 5000.

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Synthetic materials can play a significant role in preventing injuries and saving lives in several ways:

Protective Gear: Synthetic materials like Kevlar, high-density polyethylene, and synthetic fibers are used in the production of personal protective equipment (PPE). These materials provide excellent strength, durability, and resistance to impact, puncture, and abrasion. They are used in helmets, bulletproof vests, gloves, and other gear to protect individuals from physical injuries, impacts, and projectiles.

Medical Applications: Synthetic materials have revolutionized the medical field. Biocompatible materials like silicone, polymers, and metals are used in the production of prosthetics, implants, and medical devices. These materials help restore mobility and function to individuals with disabilities or injuries, enhancing their quality of life and saving lives in some cases.

Protective Barriers: Synthetic materials are used to create protective barriers in various industries and applications. For example, flame-resistant synthetic fabrics are used in firefighting suits to shield firefighters from heat and flames. Chemical-resistant materials are used in protective suits for workers handling hazardous substances. These barriers prevent direct contact with harmful elements, reducing the risk of injuries and saving lives.

Vehicle Safety: Synthetic materials are extensively used in the automotive industry to improve vehicle safety. High-strength synthetic composites, such as carbon fiber-reinforced polymers, are used to manufacture car frames and components. These materials offer excellent strength-to-weight ratios, enhancing the structural integrity of vehicles and reducing the risk of severe injuries in accidents.

Impact Absorption: Synthetic foams, such as expanded polystyrene (EPS) and polyurethane foam, are used in safety equipment and padding. These materials absorb and dissipate energy upon impact, reducing the severity of injuries. They are used in helmets, sports equipment, car seats, and mattresses, among others, to cushion impacts and protect individuals from harm.

Overall, synthetic materials provide customizable properties that can be engineered to enhance safety, durability, and protection. Their use in various applications can significantly reduce the risk of injuries and save lives in diverse scenarios.

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Both increasing the sample size (i) and using a lower confidence level (iii) would tend to decrease the width of a confidence interval. The answer is: E.

Increasing the sample size provides more data points, which leads to a more precise estimate of the population parameter. With a larger sample size, the variability within the sample is reduced, resulting in a narrower confidence interval.

Using a lower confidence level means being less confident in the estimation and allowing for a greater margin of error. A lower confidence level requires a smaller interval width to accommodate the increased uncertainty, resulting in a narrower confidence interval.

On the other hand, using a higher confidence level (ii) would tend to increase the width of a confidence interval. A higher confidence level indicates a greater degree of confidence in the estimation, requiring a wider interval to capture the range of possible values for the population parameter.

Hence, the correct option is: E. I and III only.

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Using Lagrange multipliers, the largest possible volume of a rectangular box can be found with a given diagonal length l.

Let's denote the dimensions of the rectangular box as length (L), width (W), and height (H). The volume (V) of the box is given by V = LWH. The constraint equation is the Pythagorean theorem: L² + W² + H² = l², where l is the given diagonal length.

To find the largest possible volume, we can set up the following optimization problem: maximize the volume function V = LWH subject to the constraint equation L² + W² + H² = l².

Using Lagrange multipliers, we introduce a new variable λ (lambda) and set up the Lagrangian function:

L = V + λ(L² + W² + H² - l²).

Next, we take partial derivatives of L with respect to L, W, H, and λ, and set them equal to zero to find critical points. Solving these equations simultaneously, we obtain the values of L, W, H, and λ.

By analyzing these critical points, we can determine whether they correspond to a maximum or minimum volume. The critical point that maximizes the volume will give us the largest possible volume of the rectangular box with a diagonal length l.

By utilizing Lagrange multipliers, we can optimize the volume function while satisfying the constraint equation, enabling us to determine the dimensions of the rectangular box that yield the maximum volume for a given diagonal length.

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