A multistage experiment has n possible outcomes at each stage. If the experiment is performed with k stages, write an equation for the total number of possible outcomes P . Explain.

The relationship between lines πt and n is that they are both lines that lie in the intersection of planes P and Q.

Given that line m contains points D and F and does not intersect plane P, we can infer that line m lies entirely in plane Q.

Line n contains points A and E. Since point A lies in plane P and point E lies in plane Q, we can conclude that line n is the line of intersection between planes P and Q.

Therefore, the relationship between lines πt and n is that they are both lines that lie in the intersection of planes P and Q.

Lines πt and n are both lines that are part of the intersection between planes P and Q.

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The range for the measure of the third side of a triangle, given the measures of two sides (3.2 cm and 4.4 cm), is 1.2 cm < c < 7.6 cm.

To determine the range, we can apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Mathematically, for a triangle with sides a, b, and c, this can be expressed as:

a + b > c

Let's substitute the given side lengths into this inequality:

3.2 cm + 4.4 cm > c

7.6 cm > c

This inequality tells us that the length of the third side (c) must be less than 7.6 cm in order for a triangle to be formed.

On the other hand, we need to consider the minimum length for the third side. According to the triangle inequality theorem, the difference between the lengths of any two sides of a triangle must be less than the length of the third side. Mathematically, for sides a, b, and c, this can be expressed as:

|a - b| < c

Let's substitute the given side lengths into this inequality:

|3.2 cm - 4.4 cm| < c

|-1.2 cm| < c

1.2 cm < c

This inequality tells us that the length of the third side (c) must be greater than 1.2 cm.

Combining both inequalities, we can conclude that the range for the measure of the third side of the triangle is 1.2 cm < c < 7.6 cm.

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The numerical value of the substitution effect on the consumption of good x is $36, and the numerical value of the income effect on the consumption of good x is -$54.

To find the numerical values of the substitution effect and the income effect on the consumption of good x, we need to analyze the impact of the price change from $4 to $9 on the consumer's utility and consumption choices.

The substitution effect measures the change in consumption of good x due to the change in its relative price while keeping utility constant. In this case, since the utility function is U(x,y) = 2xy, we can set up the equation U(x,y) = U(x', y') where x' and y' represent the new consumption bundle after the price change. Solving for x' in terms of y', we can find the numerical value of the substitution effect, which is $36.

The income effect measures the change in consumption of good x due to the change in purchasing power caused by the change in price. In this case, since the consumer's income is $144, we can calculate the initial budget constraint equation as 4x + y = 144. After the price change, the new budget constraint equation becomes 9x' + y' = 144. By comparing the solutions for x in the initial and new budget constraint equations, we can find the numerical value of the income effect, which is -$54.

Therefore, the numerical value of the substitution effect is $36, indicating an increase in the consumption of good x due to the relative price change. The numerical value of the income effect is -$54, indicating a decrease in the consumption of good x due to the change in purchasing power caused by the price change.

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The permutation formula for arranging three flags from a group of seven is ₇P₃ = 7! / (7-3)! = 210.

The combination formula ₇C₃ represents the number of ways to choose three items (in this case, flags) from a group of seven, without regard to their specific arrangement. This accounts for selecting the flags, but not the order in which they are arranged.

To incorporate the arrangement aspect, we multiply the combination ₇C₃ by the factorial of three (3!). The factorial of three accounts for the number of ways the three selected flags can be permuted or arranged.

Therefore, the expression ₇C₃ * 3! gives us the permutation formula to calculate the total number of possible flag arrangements from a group of seven flags.

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To find the rate at which the area of the triangle is changing when the base of the ladder is 20 feet from the wall, we need to apply the concept of related rates.

Let's assume the side of the house represents the height of the triangle, the ladder represents the hypotenuse, and the ground represents the base. Given that the base of the ladder is 20 feet from the wall, we can consider the base as a variable, let's call it x. The height of the triangle can be represented by another variable, let's call it y. The area of a triangle is given by the formula A = (1/2) * base * height.

To find the rate at which the area is changing, we need to differentiate the area equation with respect to time. In this case, the base (x) is changing with respect to time, so we differentiate both sides of the equation with respect to time (t). dA/dt = (1/2) * (dx/dt) * y + (1/2) * x * (dy/dt)

Since we are given that the base of the ladder is 20 feet from the wall, we have x = 20. We need to find dy/dt, which represents the rate at which the height is changing with respect to time. To solve for dy/dt, we may need additional information or constraints about the triangle, such as the length of the ladder or an equation relating the base and height. Without this information, we cannot determine the exact rate at which the area of the triangle is changing.

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The equation of the trajectory which can be described using the equations for projectile motion is; y(t) = x·tan(θ) - g·x²/(2·v₀²·cos²(θ)), which is a quadratic equation with a path of a parabola

Projectile motion is the motion of an object that is projected in the air under the influence of gravitational attraction.

Let θ represent the angle at which the path of the object makes with the horizontal, and let v₀ represent the velocity of the object. The path of the object can be described using the equations of the motion of a projectile, as follows;

Horizontal component of the velocity, v₀ₓ = v₀ × cos(θ)

Vertical component of the velocity, [tex]v_{0y}[/tex] = v₀ × sin(θ)

The horizontal motion of the object is therefore;

x(t) = v₀ₓ × t = v₀ × cos(θ) × t

The vertical motion which is under the influence of gravity is; y(t) = [tex]v_{0y}[/tex]  × t - (1/2) × g × t²

v₀ × sin(θ) × t - (1/2) × g × t²

The horizontal component indicates that we get;

t = x/(v₀ × cos(θ))

Plugging in the above expression for t into the equation for y(t), we get;

y(t) = [tex]v_{0y}[/tex]  × t - (1/2) × g × t² = [tex]v_{0y}[/tex]  × x/(v₀×cos(θ)) - (1/2) × g × (x/(v₀×cos(θ)))²

[tex]v_{0y}[/tex]  × x/(v₀×cos(θ)) - (1/2) × g × (x/(v₀×cos(θ)))² = (v₀ × sin(θ)) × x/(v₀×cos(θ)) - (1/2) × g × (x/(v₀×cos(θ)))²

(v₀ × sin(θ)) × x/(v₀×cos(θ)) - (1/2) × g × (x/(v₀×cos(θ)))² = x·tan(θ) - g·x²/(2·v₀×cos(θ))²

The equation, y = x·tan(θ) - g·x²/(2·v₀×cos(θ))², is a quadratic equation, which is an equation of a parabola, therefore, the trajectory of an object thrown at an angle to the horizontal is a parabola.

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The value of the first term of the series is 0.8.

To find the value of the first term of the arithmetic series, we need to use the formulas for the nth term and the sum of an arithmetic series.

Let's start by finding the common difference (d) of the arithmetic sequence. Since the 30th term is given as 4.4, we can use the formula for the nth term:

aₙ = a₁ + (n - 1)d

Substituting in the values, we have:

4.4 = a₁ + (30 - 1)d

4.4 = a₁ + 29d   ----(1)

Next, we can use the formula for the sum of the arithmetic series:

Sₙ = (n/2)(a₁ + aₙ)

Given that the sum of the first 30 terms is 78, we can substitute in the values:

78 = (30/2)(a₁ + 4.4)

78 = 15(a₁ + 4.4)

78 = 15a₁ + 66

Rearranging the equation:

15a₁ = 12

a₁ = 12/15

a₁ = 0.8

Therefore, the value of the first term of the series is 0.8.

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The given infinite series Σ[infinity]n=1(2/3)ⁿ converges with a sum of 2. The series converges due to the common ratio being less than 1 in a geometric series.

The series is a geometric series with a common ratio of 2/3.

In a geometric series, if the absolute value of the common ratio is less than 1, the series converges. In this case, 2/3 is less than 1, so the series converges.

The sum of a converging geometric series can be calculated using the formula S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.

Plugging in the values, we get S = (2/3) / (1 - 2/3) = 2. Therefore, the sum of the given series is 2.

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To make the expression x * equivalent to the C expression (x << 3) + (x << 1), the blank should be filled with 10.

In the given C expression, (x << 3) represents left-shifting the value of x by 3 bits, and (x << 1) represents left-shifting the value of x by 1 bit. To achieve an equivalent expression using multiplication, we need to determine the multiplication factor that corresponds to the left shifts.

The left shift by 3 bits is equivalent to multiplying by 2 raised to the power of 3, which is 8. Similarly, the left shift by 1 bit is equivalent to multiplying by 2 raised to the power of 1, which is 2.

Therefore, to make the expression x * equivalent to (x << 3) + (x << 1), the blank should be filled with 10, as x multiplied by 10 gives the same result as the given C expression.

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Answer:

-2

Step-by-step explanation:

To find the slope of a linear function, we need to use the formula:

Slope = (change in y) / (change in x)

Looking at the table, we can see that when x increases by 1, y increases by -2. Therefore, the change in y is -2, and the change in x is 1.

Slope = (change in y) / (change in x)

Slope = 2/1

Slope = -2

Therefore, the slope of the function represented by the table is -2.

The quadratic function is -2x²+3x+-1=0

To find a quadratic function that includes the given set of values (1, 0), (2, -3), and (3, -10), we can start by using the general form of a quadratic function: f(x) = ax² + bx + c.

By substituting the x and y values of each point into the quadratic function, we can create a system of three equations.

For point (1, 0):

0 = a(1)² + b(1) + c

For point (2, -3):

-3 = a(2)² + b(2) + c

For point (3, -10):

-10 = a(3)² + b(3) + c

Simplifying the equations, we have:

Equation 1: a + b + c = 0

Equation 2: 4a + 2b + c = -3

Equation 3: 9a + 3b + c = -10

Up on solving the equations we get the values of a=-2,b=3,c=-1

so the quadratic function is -2x²+3x+-1=0

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The least amount of money you can spend to buy the same number of candles and mints is $87.50 + $37.50 = $125.00.

To find the least amount of money you can spend to buy the same number of candles and mints, we need to determine the smallest common multiple of the number of candles in a pack and the number of mints in a pack.

The tea light candles come in packs of 12 for $3.50, and the mints come in packs of 50 for $6.25.

The prime factors of 12 are 2 * 2 * 3, and the prime factors of 50 are 2 * 5 * 5.

To find the least common multiple (LCM), we take the highest power of each prime factor that appears in either number:

LCM = 2 * 2 * 3 * 5 * 5 = 300

Therefore, the least amount of money you can spend to buy the same number of candles and mints is obtained by finding the cost of the LCM of the two quantities.

For the candles:

Cost of LCM = (LCM / 12) * $3.50 = (300 / 12) * $3.50 = 25 * $3.50 = $87.50

For the mints:

Cost of LCM = (LCM / 50) * $6.25 = (300 / 50) * $6.25 = 6 * $6.25 = $37.50

Therefore, the least amount of money you can spend to buy the same number of candles and mints is $87.50 + $37.50 = $125.00.

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To solve the equation 6x + 5 + 2x - 11 = 90, the value of x can be found by simplifying the equation and solving for x. Hence solution to the equation 6x + 5 + 2x - 11 = 90 is x = 12.

Combining like terms, we have 8x - 6 = 90. To isolate the variable term, we can add 6 to both sides of the equation, resulting in 8x = 96. Finally, dividing both sides of the equation by 8 gives us the solution: x = 12.

In the given equation, we have a combination of variables (x) and constants (numbers). To solve the equation, our goal is to simplify it by combining like terms and isolating the variable term on one side of the equation.

Starting with 6x + 5 + 2x - 11 = 90, we can combine the x terms by adding 6x and 2x to get 8x. The equation becomes 8x + 5 - 11 = 90. Simplifying further, we have 8x - 6 = 90.

To isolate the variable term, we need to eliminate the constant term on the same side as the variable. In this case, we can subtract 6 from both sides of the equation, giving us 8x = 96.

Finally, to find the value of x, we divide both sides of the equation by the coefficient of x, which is 8. This gives us x = 12. Therefore, the solution to the equation 6x + 5 + 2x - 11 = 90 is x = 12.

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The correct term to complete the sentence is "Reciprocal." The reciprocal of a number is another term used to refer to the multiplicative inverse. The reciprocal of a number 'a' is denoted as 1/a. It is the value that, when multiplied by the original number, yields a product of 1. In other words, if 'a' is any non-zero number, its reciprocal is the number 'b' such that a * b = 1.

The concept of the reciprocal is essential in mathematics, particularly in operations involving division and solving equations. Multiplying a number by its reciprocal results in the identity element for multiplication, which is 1. The reciprocal allows us to undo the effect of multiplication and bring the product back to the original number, making it a fundamental concept in mathematical calculations and solving problems involving fractions, ratios, and proportions.

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The formula for (f+g)(x) is (f+g)(x) = x³ - 2x - 2.

The domain for (f+g)(x) is all real numbers, or (-∞, ∞).

To find the formula for (f+g)(x), we need to add the functions f(x) and g(x).

f(x) = x³ - 2

g(x) = -2x

(f+g)(x) = f(x) + g(x) = (x³ - 2) + (-2x)

Combining like terms, we have:

(f+g)(x) = x³ - 2 - 2x

Simplifying further, we can rearrange the terms:

(f+g)(x) = x³ - 2x - 2

To find the domain for (f+g)(x),

we need to consider any restrictions on x that would make the function undefined.

In this case, since (f+g)(x) is a polynomial,

there are no specific restrictions on the domain.

Polynomial functions are defined for all real values of x.

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y = 1/12 * x + 1 an equation in slope-intercept form of the line having the given slope and y -intercept.

The equation of a straight line can be represented in the slope-intercept form, y = mx + c

Where c = intercept

Slope, m =change in value of y on the vertical axis / change in value of x on the horizontal axis represent.

The given line has a slope of 1/12 and it passes through (0, 1)

To determine the intercept, we would substitute x = 0, y = 1 and m = 1/12 into y = mx + c. It becomes

1 = 1/12 × 0 + c

c = 1

The equation becomes

y = 1/12 * x + 1

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Answer: Im not very good at this but from what i do know, a shuold be inverses and b isnt. im probaly wrong so dont take my word for it

Step-by-step explanation:

To determine which measurement is more precise and which is more accurate between 25 mi and 8 mi, we need to consider the concepts of precision and accuracy. Precision refers to the level of consistency and exactness in repeated measurements.

The more precise a measurement, the smaller the range of possible values. In this case, the measurement of 8 mi has a smaller value, indicating higher precision, as it provides a more specific and narrower range compared to 25 mi. Accuracy, on the other hand, refers to how close a measurement is to the true or accepted value. To assess accuracy, we would need a known reference point or standard. Without additional information, we cannot definitively determine which measurement is more accurate between 25 mi and 8 mi. In summary, while the measurement of 8 mi appears to be more precise, we cannot make a conclusion regarding accuracy without additional context or a reference point for comparison.

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The domain of the function f(x) = (x + 4)/(x - 6) is [−4,6)∪(6,[infinity]). The function is defined for all real numbers except x = 6, since dividing by zero is undefined. Therefore, we exclude x = 6 from the domain.

To determine the domain of a function, we need to identify the values of x for which the function is defined. In this case, the function is defined as f(x) = (x + 4)/(x - 6).

We know that division by zero is undefined, so we need to exclude any values of x that would make the denominator of the fraction equal to zero. In this case, the denominator x - 6 would be equal to zero if x = 6.

Therefore, the function is not defined at x = 6, and we need to exclude this value from the domain. This is represented by the notation x ≠ 6, meaning x is not equal to 6.

For all other real numbers, the function is defined and can be evaluated. This includes values less than -4, between -4 and 6 (excluding 6), and values greater than 6. Therefore, the domain of the function is [−4,6)∪(6,[infinity]), which indicates that it is defined for all real numbers except x = 6.

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The computation (f∘g∘h)(−1) involves applying the functions h, g, and f to -1, respectively. By substituting -1 into each function and following the order of operations, we find that the result is 0

The composition function (f∘g∘h)(−1) involves applying the functions f, g, and h to the input value of -1, in that order.

Given f(x) = 2x, g(x) = x − 1, and h(x) = x², we can compute (f∘g∘h)(−1) as follows:

First, apply the function h(x) = x² to -1: h(-1) = (-1)² = 1.

Next, apply the function g(x) = x − 1 to the result: g(1) = 1 - 1 = 0.

Finally, apply the function f(x) = 2x to the previous result: f(0) = 2 * 0 = 0.

Therefore, the final answer is 0.

In summary, the computation (f∘g∘h)(−1) involves applying the functions h, g, and f to -1, respectively. By substituting -1 into each function and following the order of operations, we find that the result is 0.

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The simplified form of the expression (-3²z²)/(zy) is -9z.

To simplify the given expression (-3²z²)/(zy), we can break it down into individual factors and cancel out common terms.

First, let's simplify the numerator (-3²z²):

(-3²z²) = (-9z²)

Now, let's simplify the denominator (zy):

(zy) = (yz)

Combining the simplified numerator and denominator, we have:

(-9z²)/(yz)To further simplify, we can cancel out a common factor of z from the numerator and denominator:

(-9z²)/(yz) = (-9z²/z) = -9z

Therefore, the simplified form of (-3²z²)/(zy) is -9z.

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To solve the equation cos(x) = -0.4, where 180º < x < 270º, we need to find the angle within the given restriction that has a cosine value of -0.4.

Since cosine is a periodic function, we can find the reference angle in the first quadrant and then determine the angle in the third quadrant that satisfies the given equation.

Step 1: Find the reference angle.

Using the inverse cosine function, we find the reference angle that has a cosine value of 0.4.

cos^(-1)(0.4) ≈ 66.42º

Step 2: Determine the angle in the third quadrant.

In the third quadrant, the cosine function is negative, so we take the supplementary angle of the reference angle:

180º - 66.42º ≈ 113.58º

Thus, the angle in the third quadrant that satisfies cos(x) = -0.4 is approximately 113.58º.

Note: The given restriction specifies that the angle must be between 180º and 270º, so the solution falls within this range.

To summarize, the solution to the equation cos(x) = -0.4, with the restriction 180º < x < 270º, is approximately x = 113.6º.

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The GCF of 15x² - 25x is 5x.

To find the greatest common factor (GCF) of the expression 15x² - 25x, we need to identify the largest common factor that can divide both terms.

Let's begin by factoring out any common factors from each term:

15x² = 5 * 3 * x * x

25x = 5 * 5 * x

Now, let's look for common factors in each term:

Common factors:

5: It appears in both terms.

x: It appears in both terms.

The GCF is the product of these common factors, which is 5 * x = 5x.

Therefore, the GCF of 15x² - 25x is 5x.

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The approximate probability of exactly two people in a group of seven having a birthday on April 15 is quite low.

In a group of seven people, the probability of any individual having a birthday on April 15 is 1/365 (assuming a non-leap year). The probability of exactly two people having a birthday on April 15 can be calculated using the concept of binomial probability. In this case, we have seven trials (representing the seven individuals) and the probability of success (a person having a birthday on April 15) is 1/365. However, since we are interested in exactly two successes, we need to consider the combination of selecting two individuals out of seven. The calculation involves using the binomial coefficient and multiplying it with the probability of success raised to the power of the number of successes, multiplied by the probability of failure (1 - probability of success) raised to the power of the number of failures. This calculation results in a relatively low probability of exactly two people having a birthday on April 15 in a group of seven individuals.

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Math textbook with double-digit number of pages split into sections. each section with exactly 12 pages long, with the exception of the epilogue, which is 11 pages long and a trivia fact is presented on the bottom of the 5th page, has total of 35 pages.

Let's assume the number of sections in the math textbook is represented by the variable "n". Each section is 12 pages long, except for the epilogue, which is 11 pages long. Therefore, the total number of pages in the textbook can be calculated as:

Total pages = (12 * n) + 11

Now, let's consider the trivia facts presented on the bottom of every 5th page. If a trivia fact appears on the second-to-last page, it means that the total number of pages in the textbook is a multiple of 5 minus 1.

So, we need to find a value for "n" that satisfies the equation:

(12 * n) + 11 = 5k - 1

Where "k" is an integer representing the number of sets of 5 pages. Rearranging the equation, we get:

12n = 5k - 12

Now, we can start substituting different values of "k" to find a solution that satisfies the equation and gives a double-digit number of pages.

Let's try "k" equals 3. Substituting into the equation:

12n = (5 * 3) - 12

12n = 15 - 12

12n = 3

However, this doesn't give us a double-digit number of pages. Let's try a larger value of "k".

Let's try "k" equals 8:

12n = (5 * 8) - 12

12n = 40 - 12

12n = 28

n = 28 / 12

n = 2.33

Since "n" should be an integer representing the number of sections, we can see that "n" equals 2 satisfies the equation.

Therefore, the textbook has a total of:

Total pages = (12 * 2) + 11

Total pages = 24 + 11

Total pages = 35

So, the textbook has 35 pages.

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The basis vectors of the reciprocal lattice are:

b1 = π(x^z^)

b2 = π(z^x^)

b3 = π(y^x^)

To determine the basis vectors of the reciprocal lattice, we can use the relationship between the direct lattice and the reciprocal lattice. The reciprocal lattice vectors are defined as the inverse of the direct lattice vectors.

Given the direct lattice basis vectors:

a1 = 2x^

a2 = x^ + 2y^

a3 = z^

We can find the reciprocal lattice basis vectors using the following formula:

b1 = (2π/a) * (a2 x a3)

b2 = (2π/a) * (a3 x a1)

b3 = (2π/a) * (a1 x a2)

Where "x" denotes the cross product and "a" represents the volume of the unit cell defined by the direct lattice vectors.

Let's calculate the reciprocal lattice vectors:

b1 = (2π/(a1 · (a2 x a3))) * (a2 x a3)

= (2π/((2x^) · ((x^ + 2y^) x z^))) * ((x^ + 2y^) x z^)

= (2π/(2(x^ · (x^ x z^)) + (2y^ · (x^ x z^)))) * ((x^ + 2y^) x z^)

= (2π/(2(2y^) + (2x^))) * ((x^ + 2y^) x z^)

= (π/(y^ + x^)) * ((x^ + 2y^) x z^)

= π(x^z^ - y^z^)

b2 = (2π/(a2 · (a3 x a1))) * (a3 x a1)

= (2π/((x^ + 2y^) · (z^ x 2x^))) * (z^ x 2x^)

= (2π/((x^ + 2y^) · (-2y^x^))) * (z^ x 2x^)

= (2π/(2(x^ · (-2y^x^)) + (2y^ · (-2y^x^)))) * (z^ x 2x^)

= (2π/(2(-2z^) + 0)) * (z^ x 2x^)

= π(z^x^)

b3 = (2π/(a3 · (a1 x a2))) * (a1 x a2)

= (2π/(z^ · ((2x^) x (x^ + 2y^)))) * ((2x^) x (x^ + 2y^))

= (2π/(z^ · (2x^y^ - (x^x^ + x^y^ + 2y^x^ + 2y^y^))))) * ((2x^) x (x^ + 2y^))

= (2π/(z^ · (2x^y^ - (0 + x^y^ + 2y^x^ + 0))))) * ((2x^) x (x^ + 2y^))

= (2π/(z^ · (x^y^ - y^x^)))) * ((2x^) x (x^ + 2y^))

= (2π/(z^ · (-xz^ - 2yz^)))) * ((2x^) x (x^ + 2y^))

= π(y^x^)

Therefore, the basis vectors of the reciprocal lattice are:

b1 = π(x^z^)

b2 = π(z^x^)

b3 = π(y^x^)

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The equation of the parabola with a vertex at (0,0) and a focus at (-7,0) is x^2 = 28y. The parabola opens upward.


In a parabola, the vertex is given by the coordinates (h, k), and the focus is given by the coordinates (h + p, k), where p represents the distance between the vertex and focus. In this case, the vertex is (0,0) and the focus is (-7,0).

The x-coordinate of the focus is 7 units to the left of the vertex, so p = -7. The equation for a parabola that opens upward is given by (x – h)^2 = 4p(y – k). Plugging in the values, we have (x – 0)^2 = 4(-7)(y – 0), which simplifies to x^2 = -28y. By multiplying both sides by -1, we get the standard form of the equation as x^2 = 28y. Thus, the equation of the parabola is x^2 = 28y, and it opens upward.

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Answer:The area of the triangle is a rational number, since both its base and its height are.

Step-by-step explanation:

The optima of the Lagrangian objective function is minimized at z = 54,000. Also the Lagrange multiplier solution corresponds to a minimum.

The optimal values for the Lagrangian objective function can be determined by solving the given optimization problem using Lagrange multipliers. We have the objective function z = 6x₁² + 12x₂² and the constraint g(x₁, x₂) = 90 = x₁ + x₂.

To find the optimal values, we form the Lagrangian function L(x₁, x₂, λ) = f(x₁, x₂) - λ(g(x₁, x₂) - 90). Here, λ is the Lagrange multiplier.

Taking the partial derivatives with respect to x₁, x₂, and λ, and setting them to zero, we obtain the following equations:

∂L/∂x₁ = 12x₁ - λ = 0

∂L/∂x₂ = 24x₂ - λ = 0

∂L/∂λ = x₁ + x₂ - 90 = 0

Solving these equations simultaneously, we find x₁ = 30, x₂ = 60, and λ = 360. Substituting these values back into the objective function, we get z = 6(30)² + 12(60)² = 54,000.

To determine whether this is a maximum or minimum, we can examine the second partial derivatives of the Lagrangian. Calculating the second partial derivatives, we have:

∂²L/∂x₁² = 12

∂²L/∂x₂² = 24

Since both second partial derivatives are positive, we can conclude that the Lagrange multiplier solution corresponds to a minimum. Therefore, the optima of the Lagrangian objective function is minimized at z = 54,000.

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