Suppose you are the building rectangular puppy kennel for your new puppy with 25 feet of fence. The side of the kennel next to your house does not need a fence.this side is 9 feet long. Find the dimensions of the kennel.

The critical point of the function g is (0,0), the value of D(0,0) is -6272, and the critical point is a saddle point.

(a) To find the critical point of g, we need to find the partial derivatives of g with respect to x and y, and set them equal to zero:

[tex]∂g/∂x = -8xe^(-4x^2+7y^2+14√8y) = 0[/tex]

[tex]∂g/∂y = 14ye^(-4x^2+7y^2+14√8y) + 14√8e^(-4x^2+7y^2+14√8y) = 0[/tex]

From the first equation, we get x = 0. Substituting this value into the second equation, we get:

[tex]14ye^(7y^2+14√8y) + 14√8e^(7y^2+14√8y) = 0[/tex]

Dividing both sides by [tex]14e^(7y^2+14√8y)[/tex], we get:

y + √8 = 0

Thus, the critical point of g is (0, -√8).

(b) To find the value of D(a,b) from the Second Partials test, we need to compute the second-order partial derivatives of g with respect to x and y:

[tex]∂^2g/∂x^2 = 32x^2e^(-4x^2+7y^2+14√8y) - 8e^(-4x^2+7y^2+14√8y)[/tex]

[tex]∂^2g/∂y^2 = 98y^2e^(-4x^2+7y^2+14√8y) + 196√8ye^(-4x^2+7y^2+14√8y) + 686e^(-4x^2+7y^2+14√8y)[/tex]

[tex]∂^2g/∂x∂y = -112xye^(-4x^2+7y^2+14√8y) - 196√8xe^(-4x^2+7y^2+14√8y)[/tex]

At the critical point (0, -√8), we have:

[tex]∂^2g/∂x^2 = -8[/tex]

[tex]∂^2g/∂y^2 = 686[/tex]

[tex]∂^2g/∂x∂y = 0[/tex]

Therefore, D(0, -√8) =[tex](∂^2g/∂x^2)(∂^2g/∂y^2) - (∂^2g/∂x∂y)^2[/tex] = (-8)(686) - [tex](0)^2[/tex] = -5488.

(c) Since D(0, -√8) is negative, and [tex]∂^2g/∂x^2[/tex] is negative at the critical point, the Second Partials test tells us that (0, -√8) is a saddle point.

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The function f(2) from the piecewise function has a value of 0

From the question, we have the following parameters that can be used in our computation:

The piecewise function

The x value of 2 belongs to the domain x > 2

So, we have

f(x) = -x + 2

Substitute the known values in the above equation, so, we have the following representation

f(2) = -2 + 2

Evaluate

f(2) = 0

Hence, the value iss 0

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Possible rational zeros of the polynomial is

±1, ±1/2, ±1/4, ±2, ±5, ±5/2, ±5/4, ±10.

Polynomials are particular type of algebraic expressions which consists of variables and coefficients. Various arithmetic operations such as addition, subtraction, multiplication, and also positive integer exponents for polynomial expressions can be performed on polynomial but not division by variable. An example of a polynomial is x-12 . Here it has two  terms:  x and -12.

Given polynomial p(x)= 4x⁵-2x³+10

The polynomial can be rewritten as p(x)= 4x⁵+ 0x⁴-2x³+0x²+0x+10

Comparing the polynomial with

p(x) = aₙ xⁿ + aₙ₋₁xⁿ⁻¹+ ---------- + a₁ x + a₀ we get the  leading coefficient

aₙ= 4 and the constant term a₀= 10

So the possible zeros are=  ±( factors of a₀)/ (factors of aₙ)

Factors of 10:

1, 2, 5, 10

Factors of 4:

1, 2, 4

Hence, Possible rational zeros:

±1, ±1/2, ±1/4, ±2, ±5, ±5/2, ±5/4, ±10

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Answer:  (6, -2)

Step-by-step explanation:

         First, we will graph these equations. See attached. One has a y-intercept of -5 and then moves 2 units right for every unit up (we get this from the slope of 1/2). The other has a y-intercept of 4, and moves right one unit for every unit down (we get this from the slope of -1).

         The point of intersection is the solution, this is the point at which both graphed lines cross each other. Our solution is:

                         (6, -2)     x = 6, y = -2

Answer:

the area is 1000

Step-by-step explanation:

This one is for the boys with the booming system

Top down, AC with the cooler system

As a result, the range of the population's mean height for male Swedes is between 70.331 and 73.669 inches.

The range of a function or relation in mathematics is the collection of all feasible output values (also known as the dependent variable). It is the collection of all possible values that such a function can have. For instance, since its function can output anything non-negative real integer, f(x) = x2 would have a range of all non-negative real numbers. The range is frequently stated as a collection of numbers or as a circle, and it may be either finite or infinite.

given

For the population mean height of male Swedes, the following formula provides a 95% confidence interval:

where n is the sample size, x is the sample mean, y is the standard deviation of height, n* is the z-score corresponding to a 95% confidence level, which is 1.96, and z* is the z-score.

By entering the specified values, we obtain:

CI = 72 ± 1.96 * (3/√48)

CI = (70.331, 73.669) (70.331, 73.669)

As a result, the range of the population's mean height for male Swedes is between 70.331 and 73.669 inches.

By multiplying the critical value (1.96), by the standard error, one may determine the error bound:

EBM = 1.96 * (3/√48) ≈ 1.382

The error bound is therefore roughly 1.382 inches.

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Answer:

To find the slope of a line parallel to 3x - y = -1, we need to first rearrange this equation into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Starting with 3x - y = -1, we can add y to both sides to get: 3x = y - 1 Then, we can add 1 to both sides to get: 3x + 1 = y This is now in slope-intercept form, with y = 3x + 1, so we can see that the slope of the line is 3. Since we want to find the slope of a line parallel to this one, we know that it must have the same slope of 3. Therefore, the answer is: Slope = 3.

So, the corresponding point for g(x) on the graph of f(x) is (-3/2, F (-3/2)).

In mathematics, a function is a relation between a set of inputs (the function's domain) and a set of possible outputs (the function's range) with the property that each input is associated with exactly one output.

Functions are often described using a formula or an equation, such as [tex]f(x) = x^2,[/tex] where "f" is the name of the function, "x" is the input variable, and "[tex]x^2[/tex]" is the output. The value of the function for a specific input value of "x" can be found by substituting that value into the equation.

Functions are used in many areas of mathematics, science, engineering, and economics to model real-world phenomena, analyze data, and solve problems. They are also important in computer science, where they are used to write algorithms and design software.

if we know the point (x, y) on the graph of a function f(x), we can find the corresponding point for the function g(x) = F(-1/2x) by substituting -1/2x for x in the expression for f(x) and simplifying. That is, we can find the point (u, v) on the graph of g(x) such that u = -1/2x and v = F(-1/2x), where (x, y) is a point on the graph of f(x).

For example, if f(x) = x^2 and the point (3, 9) is on the graph of f(x), then we can find the corresponding point for g(x) = F(-1/2x) by substituting -1/2x for x in the expression for f(x) and simplifying:

[tex]u = -1/2x = -1/2(3) = -3/2\\v = F(-1/2x) = F(-1/2(3)) = F(-3/2)[/tex]

So, the corresponding point for g(x) on the graph of f(x) is (-3/2, F(-3/2)). However, without knowing the specific definition of fix,

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0.2525 is the probability that a simple random sample.

What is the simple definition of probability?

A probability is a number that expresses the possibility or likelihood that a specific event will take place. Probabilities can be stated as proportions with a range of 0 to 1, or as percentages with a range of 0% to 100%. Probability is a measure of how likely or likely-possible something is to happen.

Since the distribution of the life expectancies of a certain protozoan is normal, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = life expectancies of the certain protozoan.

µ = mean

σ = standard deviation

n = number of samples

From the information given,

µ = 46 days

σ = 10.5 days

n = 49

The probability that a simple random sample of 49 protozoa will have a mean life expectancy of 47 or more days is expressed as

P(x ≥ 47) = 1 - P(x < 47)

For x = 47

z = (47 - 46)/(10.5/√49) = 0.67

Looking at the normal distribution table, the probability corresponding to the z score is 0.0.75

P(x ≥ 47) = 1 - 0.75 = 0.25

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The measure of angle A is 59°

A cyclic quadrilateral is a quadrilateral which has all its four vertices lying on a circle. It is also sometimes called inscribed quadrilateral. The circle which consists of all the vertices of any polygon on its circumference is known as the circumcircle or circumscribed circle.

The sum of opposite angle in a cyclic quadrilateral is 180. This means, 121+<A = 180°

therefore,

= angle A = 180-121

= 59°

therefore the measure of angle A is 59°

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Answer: 5

Step-by-step explanation:

If the area of a square equals 25 square inches, then the side length of the square can be found by taking the square root of the area. The formula for the area of a square is A = s^2, where A is the area and s is the length of a side of the square.

So, in this case, we have:

A = 25 sq inches

s = sqrt(A) = sqrt(25 sq inches) = 5 inches

Therefore, the side of the square is 5 inches.

Answer:

Step-by-step explanation:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

The solution to the set of equations can be found by solving the system:

3x + 3y = 12

x + y = 4

We can simplify the second equation by solving for y:

y = 4 - x

Substituting this expression for y into the first equation, we get:

3x + 3(4 - x) = 12

3x + 12 - 3x = 12

12 = 12

This is a true statement, which means that the system is consistent and has infinitely many solutions. Therefore, the correct answer is:

There are infinitely many solutions.

Answer:

Step-by-step explanation:

To find:-

Answer:-

We are here given that there are two linear equations, namely,

[tex]\begin{cases} 3x+3y = 12 \\ x+y = 4 \end{cases}[/tex]

These can be rewritten as ,

[tex]\begin{cases} 3x+3y - 12 =0\\ x+y -4=0 \end{cases}[/tex]

Before we precede we must know that,

Conditions for solvability :-

If there are two linear equations namely,

[tex]\begin{cases} a_x + b_1y+c_1 = 0 \\ a_2x+b_2y+c_2=0\end{cases}[/tex]

Then ,

Case 1 :-

If we have,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} } \\[/tex]

Then , the lines are coincident and there are infinitely many solutions .

Case 2 :-

If we have,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2} } \\[/tex]

Then, the linear equations are inconsistent and have no solutions , thus the lines are parallel .

[tex]\rule{200}2[/tex]

So here with respect to angle standard form of pair linear equations, we have;

Hence here we have,

[tex]\longrightarrow \dfrac{a_1}{a_2} = \dfrac{3}{1} \\[/tex]

[tex]\longrightarrow \dfrac{b_1}{b_2}=\dfrac{3}{1} \\[/tex]

[tex]\longrightarrow \dfrac{c_1}{c_2}=\dfrac{-12}{-4}=\dfrac{3}{1} \\[/tex]

Therefore we can clearly see that,

[tex]\longrightarrow \boxed{ \dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2} =\boxed{\dfrac{3}{1}}} \\[/tex]

Hence there are infinitely many solutions and the lines are coincident .

The desalination plant can produce [tex]9.84*10^6[/tex] gallons of water in four day in scientific notation.

We must divide the daily production rate by the number of days in order to get the total volume of water that a desalination plant can generate in four days:

4. days at a rate of [tex]2.46*10^6[/tex] gallons equals [tex]9.84*10^6[/tex] gallons.

Hence, in four days, the desalination plant can produce [tex]9.84 * 10^6[/tex] gallons of water.

Large or small numbers can be conveniently represented using scientific notation, especially when working with measurements in science and engineering. A number is written in scientific notation as a coefficient multiplied by 10 and raised to a power of some exponent. For example, [tex]2.46*10^6[/tex] denotes 2,460,000, which is 2.46 multiplied by 10 to the power of 6.

The solution in this case is [tex]9.84*10^6[/tex], or 9,840,000, which is 9.84 multiplied by 10 to the power of 6. Large numbers can be written and compared more easily using this format, and scientific notation rules can be used to conduct computations with them.

In conclusion, the desalination plant has a four-day capacity of [tex]9.84 * 10^6[/tex] gallons of water production.

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a. Your friend's mistake is that the reflection of EFG to its image E'F'G' is not a reflection across the x-axis.

b. The correct description of the reflection is a reflection across the line y = x.

The correct description of the reflection are:

a. Your friend's mistake is that the reflection of EFG to its image E'F'G' is not a reflection across the x-axis.

b. The correct description of the reflection is a reflection across the line y = x.

This is because the line of reflection is the perpendicular bisector of the segment connecting each point to its image. In this case, the segment connecting E to E' has a slope of 1 (since it is a diagonal line), so the perpendicular bisector of this segment must have a slope of -1.

The line with slope -1 passing through the midpoint of this segment (which is the point on the line y = x) is the line of reflection. Therefore, the reflection of EFG to its image E'F' is a reflection across the line y = x, not the x-axis.

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Each student will read 2 1/2 pages aloud. The answer is option B, 2 1/2.

What is arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term.

For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a common difference of 3, since each term after the first is found by adding 3 to the preceding term.

Ms. Avila's reading class has 18 students and they are going to read aloud from a book that has 45 pages. To determine the total number of pages that each student will read, we need to divide the total number of pages by the number of students.

So, 45 divided by 18 equals 2 1/2 pages per student.

This means that each student will read 2 1/2 pages aloud.

Therefore, the answer to the question is option B, 2 1/2.

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Answer:

The triangles are similar by AA since vertical angles are congruent.

[tex] \frac{8}{12} = \frac{x}{7} [/tex]

[tex]12x = 56[/tex]

[tex]x = \frac{14}{3} = 4 \frac{2}{3} [/tex]

Answer:

Step-by-step explanation:

Here is a step-by-step explanation of the solution and how it leads to the maximum revenue:

Let's start with the current revenue generated by the TV cable company, which is the product of the number of subscribers and the monthly fee:

Current revenue = Number of subscribers * Monthly fee

Current revenue = 6400 * $28

Current revenue = $179,200

To increase the revenue, the TV cable company can decrease the monthly fee by $0.50, which will result in an increase in the number of subscribers by 160 for each $0.50 decrease. Let's calculate the additional revenue generated by each $0.50 decrease in the monthly fee:

Additional revenue per decrease in monthly fee = Increase in subscribers * Decrease in monthly fee

Additional revenue per decrease in monthly fee = 160 * $0.50

Additional revenue per decrease in monthly fee = $80

The TV cable company can continue to decrease the monthly fee by $0.50 until the revenue stops increasing. Let's calculate the number of $0.50 decreases in monthly fee required to reach the maximum revenue:

Number of $0.50 decreases in monthly fee = (Total revenue - Current revenue) / Additional revenue per decrease in monthly fee

Number of $0.50 decreases in monthly fee = ($20,000,000 - $179,200) / $80

Number of $0.50 decreases in monthly fee = 248,525

The corresponding decrease in monthly fee is:

Decrease in monthly fee = Number of $0.50 decreases in monthly fee * Decrease in monthly fee

Decrease in monthly fee = 248,525 * $0.50

Decrease in monthly fee = $124,262.50

The corresponding increase in subscribers is:

Increase in subscribers = Number of $0.50 decreases in monthly fee * Increase in subscribers

Increase in subscribers = 248,525 * 160

Increase in subscribers = 39,764,000

The total number of subscribers at the maximum revenue point is:

Total subscribers = 6400 + Increase in subscribers

Total subscribers = 6400 + 39,764,000

Total subscribers = 39,770,400

The monthly fee at the maximum revenue point is:

Monthly fee = Current revenue / Total subscribers

Monthly fee = $179,200 / 39,770,400

Monthly fee = $0.0045

The maximum revenue is:

Maximum revenue = Total subscribers * Monthly fee

Maximum revenue = 39,770,400 * $0.0045

Maximum revenue = $178,966.80

Therefore, the TV cable company can generate maximum revenue of $178,966.80 per month by decreasing the monthly fee by $0.50 for each 160 additional subscribers.

Answer:

[tex](x+1)^{2} +(y+2)^{2} =20[/tex]

Step-by-step explanation:

Using the mid point formula to find the center of the circle:

midpoint = [tex](\frac{x_{1} +x_{2} }{2} ,\frac{y_{1} +y_{2} }{2} )[/tex]

Midpoint = [tex](\frac{3+-5}{2} ,\frac{0+-4}{2} )[/tex]

Midpoint = [tex](-1,-2)[/tex]

The midpoint is the same as the centre of the circle

Find the distance(the diameter of the circle) between those two points to find the radius:

Distance formula = [tex]\sqrt{(y_{2}-y_{1} )^{2} +(x_{2} -x_{1} ) ^{2} }[/tex]

Distance formula = [tex]\sqrt{(-4-0)^{2} +(-5-3)^{2} }[/tex]

Distance formula = [tex]\sqrt{16+64}[/tex]

Distance formula = [tex]\sqrt{80}[/tex]

So,the diameter is [tex]\sqrt{80}[/tex] and to find the radius we need to divide the diameter by two

Radius = [tex]\frac{\sqrt{80} }{2}[/tex]

Radius = [tex]2\sqrt{5}[/tex]

the equation of circle:

Radius = [tex]2\sqrt{5}[/tex]

Center = (-1,-2)

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

[tex](x- -1)^{2} +(y- - 2)^{2} =(2\sqrt{5} )^{2}[/tex]

[tex](x+1)^{2} +(y+2)^{2} =20[/tex]

Answer:

Step-by-step explanation:

I am assuming the triangles are similar.

You can make a proportion between the two triangles and solve for x but you must match the sides properly.

RS ≈ MN

ST ≈ NO

Making a proportion between the sides:

[tex]\frac{x}{3}[/tex] = [tex]\frac{6}{4}[/tex]  

cross multiplyand then divide

6 · 3 = 18

18 ÷ 4 = 4.5

x = 4.5

If you check the ratios - they are all the same.

For example  6 ÷ 4 = 1.5

7.5 ÷ 5 = 1.5

4.5 ÷ 3 = 1.5

this checks out

Answer: 165 degrees

Step-by-step explanation:

Let's start off and name the angle between the hour and minute hand x, representing the number of degrees between the hands. Next, let's look at the minute hand first. After exactly an hour, the minute hand does not change its position(since after one hour, it comes back to where it was previously). As for the hour hand however, its position differs by exactly "1 hour tick", which is equivalent to 30 degrees (360 / 12 hours). What that implies is that x would have to equal x + 30, which is clearly impossible. However, we can think about the problem in a different way: by looking at the acute angle only between the hands, that would indeed be possible. Assuming that x + 30 is greater than 180(since if it wasn't, that wouldn't be possible), we can write the following equation:

x = 360 - (x+30)

, with the right hand side coming from the fact that the obtuse angle of x + 30 must have an acute counterpart, of which both add up to 360.

Then, we can simplify to:

x = 330 - x

or:

2x = 330

x = 165.

This gives us our only answer of 165 degrees

we can conclude that the expression [tex]2x^2 + 3x -1[/tex] is the quadratic expression among the given options.

A quadratic expression is an algebraic expression in which the highest power of the variable is 2. The general form of a quadratic expression is:

[tex]ax^2 + bx + c[/tex]

where a, b, and c are constants and x is the variable. The coefficient a must be non-zero for the expression to be considered quadratic.

In the given options, we can see that only one expression has a degree of 2, which is the second term of the quadratic expression.

So, we can conclude that the expression [tex]2x^2 + 3x -1[/tex] is the quadratic expression among the given options.

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The domain of F(d) is: {d ∈ R | d ≥ 0}

The domain is a mathematical concept that refers to the set of all possible input values (independent variables) for which a function is defined. In other words, the domain of a function is the range of values that can be used as input to the function.

Here we have

A large passenger airplane is making the 9,537-mile trip from New York to Singapore. During this flight, the plane will use 47,685 gallons of fuel or 5 gallons per mile of flight.

The function F(d) represents the amount of fuel, in gallons, consumed by the airplane on this trip after d miles of flight.

Therefore, we can express F(d) as:

F(d) = 5d

The domain of F(d) represents the set of all valid values of d for which the function is defined.

In this case, d represents the distance flown by airplane, and it must be a non-negative number since the plane cannot fly a negative distance.

Therefore,

The domain of F(d) is: {d ∈ R | d ≥ 0}

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Answer:

Leonhard Euler was a Swiss mathematician who lived from 1707-1783. He made significant contributions to many areas of mathematics, including number theory, calculus, and geometry.

One of Euler's significant contributions to the field of sequences and series is his work on the summation of infinite series. In particular, he developed a method for finding the sum of an infinite geometric series, which has the form:

a + ar + ar^2 + ar^3 + ...

where a is the first term, r is the common ratio, and the series continues infinitely.

Euler's method for finding the sum of this series is as follows:

If |r| < 1, then the sum of the series is:

a / (1 - r)

For example, let's consider the infinite geometric series:

2 + 4 + 8 + 16 + ...

In this series, a = 2 and r = 2. Since |r| < 1, we can use Euler's formula to find the sum:

sum = a / (1 - r)

= 2 / (1 - 2)

= -2

Therefore, the sum of the infinite geometric series 2 + 4 + 8 + 16 + ... is -2.

Euler's formula is just one example of his many contributions to mathematics. His work on sequences and series laid the foundation for many important concepts in calculus, and his ideas continue to influence the field of mathematics today.

Answer: We can simplify the expression as follows:

sin theta / (1 + sin theta) - sin theta / (1 - sin theta)

= [(sin theta)(1 - sin theta) - (sin theta)(1 + sin theta)] / [(1 + sin theta)(1 - sin theta)]

= [(sin theta - sin^2 theta) - (sin theta + sin^2 theta)] / [1 - sin^2 theta]

= -2 sin theta / cos^2 theta

= -2 tan theta

So the simplified expression is -2 tan theta, with no quotients.

Step-by-step explanation:

The Two figures are (Not similar) because 2/5 ( does not equal) 3/6.

What is similar rectangle?

If two rectangles are similar, their corresponding sides are proportionately equal.

Now comparing the length and width of the two rectangles then.

=> 2/5 = 3/6

Which is not equally proportional.

Hence the both rectangles are similar because 2/5 does not equal 3/6 .

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Answer:

10.5

Step-by-step explanation:

250% = 2.5

What number is 250% of 4.2?
We Take

2.5 x 4.2 = 10.5

So, 10.5 is 250% of 4.2

In a rectangle, there are 2 pairs of congruent sides. Therefore, the area of a rectangle can be found using:

[tex]\text{A}=\text{l} \times \text{w}[/tex]

We know that the length is 18 and the width is 12, so we can multiply 18 in for l, and 12 in for w.

[tex]\text{A}=\text{18} \times \text{12}[/tex]

Multiply the numbers together

[tex]\text{A}=216[/tex]

So, the area is 216 inches.

To find the area of a rectangular placemat, you multiply the length by the width. In this case, the length is 18 inches and the width is 12 inches. Therefore, the area of the tablecloth is:

Area = Length × Width

Area = 18 inches × 12 inches

Area = 216 square inches

So, the correct answer is option a. 216.

Answer: Given a 2x2 matrix A:

A = [a11 a12]

   [a21 a22]

The minor of any element a_ij is the determinant of the 1x1 matrix obtained by deleting the i-th row and j-th column from A. In other words, the minor of a_ij is given by:

M_ij = det(A_ij)

where A_ij is the matrix obtained by deleting the i-th row and j-th column from A.

For example, the minor of a11 is given by:

M_11 = det([a22])

    = a22

Similarly, we can find the minors of the other elements:

M_12 = det([a21])

    = a21

M_21 = det([a12])

    = a12

M_22 = det([a11])

    = a11

Therefore, the minors of the matrix A are:

M = [a22  a21]

   [a12  a11]

Step-by-step explanation: