2 Tom Milk his cow and got 2 litres of Milk He gave Kevin 350 ml and Sold 500m to Bob. How much milk is left?​

The critical points of [tex]f(x,y)[/tex] are: (0,0), (1,1), (-1,-1), [tex](1/\sqrt2,-1/\sqrt2)[/tex], [tex](-1/\sqrt2,1/\sqrt2), (i/\sqrt2,-i/\sqrt2)[/tex], and [tex](-i/\sqrt2,i/\sqrt2)[/tex]. The points (1,1) and (-1,-1) are local maxima, while the remaining critical points are saddle points

To find the critical points of the function [tex]f(x,y)[/tex], we need to find where its partial derivatives with respect to x and y are equal to zero:

∂f/∂x = 4x³ - 4y = 0

∂f/∂y = 4y³ - 4x = 0

From the first equation, we get y = x³, and substituting into the second equation, we get:

[tex]4x - 4x^9 = 0[/tex]

Simplifying this equation, we get:

[tex]x(1 - x^8) = 0[/tex]

So the critical points occur at x = 0, x = ±1, and [tex]x = (^+_-i)/\sqrt2[/tex].

To determine the nature of these critical points, we need to look at the second partial derivatives of [tex]f(x,y)[/tex]:

∂²f/∂x² = 12x²

∂²f/∂y² = 12y²

∂²f/ = -4

At (0,0), we have ∂²f/∂x² = ∂²f/∂y² = 0 and ∂²f/∂x ∂y = -4, so this is a saddle point.

At (1,1), we have ∂²f/∂x² = ∂²f/∂y² = 12, and ∂²f/∂x ∂y = -4, so this is a local maximum.

At (-1,-1), we have ∂²f/∂x² = ∂²f/∂y² = 12, and ∂²f/∂x ∂y = -4, so this is also a local maximum.

At , we have ∂²f/∂x² = 6, ∂²f/∂y² = 6, and ∂²f/∂x ∂y = -4, so these are saddle points.

At [tex](i/\sqrt2,-i/\sqrt2)[/tex] and [tex](-i/\sqrt2,i/\sqrt2)[/tex], we have ∂²f/∂x² = -6, ∂²f/∂y² = -6, and ∂²f/∂x ∂y = -4, so these are also saddle points.

Therefore, the critical points of [tex]f(x,y)[/tex] are: [tex](0,0), (1,1), (-1,-1), (1/\sqrt2,-1/\sqrt2), (-1/\sqrt2,1/\sqrt2), (i/\sqrt2,-i/\sqrt2)[/tex], and [tex](-i/\sqrt2,i/\sqrt2)[/tex]. The points (1,1) and (-1,-1) are local maxima, while the remaining critical points are saddle points

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Answer:

Step-by-step explanation:

The absolute maximum is f(-2) = 10/3, and absolute minimum is f(0) = 8.

First, let's find the derivative of the function:

f(x) = 8+x/2-x

f'(x) = (1/2) - 1 = -1/2

Next, we need to find the critical points of the function on the given domain.

In this case, the derivative is always defined and is never zero. Therefore, there are no critical points on the given domain.

Next, we check the endpoints of the domain, x = -2 and x = 0:

f(-2) = 8 + (-2)/(2-(-2)) = 10/3

f(0) = 8 + 0/(2-0) = 8

Since the function is continuous on the closed interval [-2, 0],

The extreme value theorem tells us that the function must have both an absolute maximum and an absolute minimum on the interval.

Therefore, the absolute maximum occurs at x = -2 and is f(-2) = 10/3, and the absolute minimum occurs at x = 0 and is f(0) = 8.

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The probability of selecting a red bead is 3/20 when the probability of selecting a white bead, replacing it, and then selecting a red bead.

We need to find the probability of selecting a red bead when first a white bead is selected and then it is replaced and then selected a red bead. The formula to find the probability is,

P(A) = f / N

Where,

f = number of outcomes

N = total number of outcomes

Given data:

Total number of beads =  10

Number of blacks beads = 2

Number of white beads = 3

Number of red beads = 5

The probability of selecting a white bead is given as,

P(A) = f / N

P(W) = 3/10

When the bead is replaced, the probability of selecting a red bead is  P(R) = 5/10

The probability of selecting a white bead and then a red bead is the product of the probabilities of each event:

P(white and red) = P(white) × P(red)

= (3/10) × (5/10)

= 3/20

Therefore, the probability of selecting a white bead, replacing it, and then selecting a red bead is 3/20.

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the solution of equation problem is estimated distance of the gold medal winning discus throw in 1980 is approximately 65.03 meters. The answer is option C.

An equation is a statement that says two things are equal. It can contain variables, which can take on different values. Equations are used to solve problems and model real-world situations by expressing relationships between variables.

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("="). For illustration, 2x - 5 = 13.

Here,

5 and 13 are expressions for 2x.

These two expressions are joined together by the sign "="

To estimate the distance of the gold medal winning discus throw in 1980 using the line of best fit, we need to first calculate the value of x for the year 1980

x = 1980 - 1920 = 60

Now, we can substitute x=60 into the equation of the line of best fit to find the estimated distance:

y = 0.34x + 44.63

y = 0.34(60) + 44.63

y = 20.4 + 44.63

y ≈ 65.03

Therefore, the estimated distance of the gold medal winning discus throw in 1980 is approximately 65.03 meters. The answer is option C.

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The value of x in the solution to this system of equations 5x - 4y = 27 and y = 2x + 3 is -13.

To find the value of x in this system of equations, we can use substitution method to find the its solution. Start by isolating x in one of the equations and then substituting that value into the other equation.

Let's start by isolating x in the second equation:

y = 2x + 3

Subtracting 3 from both sides:

y - 3 = 2x

Dividing both sides by 2:

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

Now we can substitute this expression for x into the first equation:

5x - 4y = 27

5((1/2)y - (3/2)) - 4y = 27

Simplifying:

(5/2)y - 15/2 - 4y = 27

Combining like terms:

-(3/2)y = 69/2

Dividing by -(3/2):

y = -23

Now we can substitute this value of y back into the expression we found for x:

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

x = (1/2)(-23) - (3/2)

x = -13

Therefore, the solution to this system of equations is x = -13, y = -23.

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The driver was driving at a speed of about 14 mi/h at the time of the skid. option is C. 14 mi/h

Using the formula s = √(30fd), where f is the coefficient of friction (0.7) and d is the length of the skid marks in feet (9), we can estimate the speed at the time of the skid:

s = √(30 × 0.7 × 9)

s ≈ 14.53 mi/h

Rounding to the nearest mi/h, the driver was driving at approximately 15 mi/h at the time of the skid. However, none of the given options match this result. The closest option is C. 14 mi/h, so I would choose that as the best available answer.

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5 or more Xs

To complete the activities based on the given data:

Compare the sizes: By looking at the shooting star lengths, we can determine the number of Xs that would appear on the line plot. The shooting star lengths "10" and "8" appear more than 5 times, so they would be placed in the "More than 5 Xs" box. The shooting star lengths "6" and "4" appear fewer than 5 times, so they would be placed in the "Fewer than 5 Xs" box.

Complete the line plot: Using the given set of data, we can create a line plot to represent the lengths of shooting stars. We mark each measurement on the number line and place an X above the corresponding value.

The line plot would have an X above the number 10, 8, 6, and 4, each representing the occurrence of shooting stars with those lengths.

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Answer:69x-44

Step-by-step explanation:

69-44=67

To withdraw $3,200 quarterly at an interest rate of 4.5% for 18 years, the account balance needs to be approximately $178,311. This is calculated using the formula for the present value of an annuity, where the payment, interest rate, time period, and compounding frequency are considered.

To find the account balance needed, we need to use the present value of an annuity formula.

Convert the annual interest rate to a quarterly rate: 4.5% / 4 = 1.125%

Convert the number of years to the number of quarters: 18 years * 4 quarters per year = 72 quarters

Calculate the present value of the annuity using the formula:

PV = PMT * (1 - (1 + r)⁻ⁿ) / r

where PV is the present value, PMT is the regular withdrawal amount, r is the quarterly interest rate, and n is the number of quarters.

Plugging in the values, we get

PV = 3200 * (1 - (1 + 0.01125)⁻⁷²) / 0.01125

= 3200 * (1 - 0.2717) / 0.01125

= 178,311.11

Round the answer to the nearest dollar: $178,311

Therefore, the account needs to hold $178,311 to make regular withdrawals of $3200 per quarter for 18 years at a quarterly interest rate of 4.5%.

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The relative frequency of people who strongly agree with the statement "I feel respected by those I work for" is 0.312, or 31.2%.

This means that out of the 500 employees surveyed, 156 strongly agreed with the statement. To find the relative frequency, you simply divide the number of people who strongly agree by the total number of people surveyed (156/500).

This result suggests that the majority of employees feel respected by their employers, which is a positive sign for the workplace culture.

However, it's important to note that there are still a significant number of employees who either disagree or feel neutral about this statement, indicating that there may be room for improvement in terms of fostering a more respectful and supportive work environment.

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Answer: Your answer is 9y(y^3 - 4) (y^3 + 4

(The fours are not being subtracted with the exponent 3. They are separate)

Using mathematical operations, each box of a dozen donuts should cost $3.40.

The basic mathematical operations used to determine the cost of a dozen donuts include multiplication and addition.

Firstly, the total cost of 12 donuts is computed by multiplication, while the total cost of the donuts per box (including the cost of the box) is obtained by addition.

1 dozen = 12 donuts

The cost unit of a donut = $0.27

The total cost of donuts = $3.24 ($0.27 x 12)

The cost per square foot of cardboard = $0.16

The total cost of a dozen donuts and the box = $3.40 ($3.24 + $0.16)

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The slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.

To find the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees,:

Step 1: Determine the angle of elevation. Since the angle of depression is 32 degrees, the angle of elevation is also 32 degrees, because they are alternate angles.

Step 2: Use the tangent function to find the slope. The tangent of an angle in a right triangle is equal to the ratio of the side opposite the angle (rise) to the side adjacent to the angle (run). In this case, the tangent of the angle of elevation (32 degrees) is equal to the slope of the line.

Step 3: Calculate the tangent of 32 degrees. Using a calculator or a trigonometric table, you can find that tan(32°) ≈ 0.625.

So, the slope of the line on a coordinate plane that makes an angle of depression of 32 degrees is approximately 0.625.

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The linear optimization technique for allocating constrained resources among different products is linear programming. (option d).

In the context of allocating constrained resources, linear optimization aims to maximize the output of a system while minimizing the input required to produce that output. This is achieved by formulating the problem as a set of linear equations or inequalities, which represent the constraints on the resources.

The linear equations or inequalities define the relationship between the input and output variables, which are typically expressed as linear functions. These functions represent the production possibilities of each product or activity and the available resources, such as labor, materials, and equipment.

The goal of linear optimization is to find the optimal values of the input and output variables that satisfy the constraints and maximize the objective function. The objective function is a linear function that represents the measure of performance or profitability of the system.

Hence the correct option is (d).

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To prove that ABCD is a parallelogram, we need to show that opposite sides are parallel.

Since AE and CF are perpendicular to DB, we know that DB is the transversal that creates four right angles at the intersections.

Using the given information, we know that:

   AE = CF (given)

   AE || CF (since they are perpendicular to DB, they are parallel to each other)

   DE = FB (given)

   ∠AED = ∠CFB = 90° (since AE and CF are perpendicular to DB)

Now we can prove that AB || CD:

Thus, by the converse of the corresponding angles theorem, we can conclude that AB || CD.

Similarly, we can prove that AD || BC:

Thus, by the converse of the corresponding angles theorem, we can conclude that AD || BC.

Since we have shown that opposite sides are parallel, we can conclude that ABCD is a parallelogram.

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To find the Laplace transform of f(t), we use the formula:

L{f(t)} = ∫[0,∞) [tex]e^(-st)[/tex] f(t) dt

where L{f(t)} denotes the Laplace transform of f(t) and u(t) is the unit step function.

Using the linearity of the Laplace transform, we can find the Laplace transform of each term separately and add them up.

L{6u(t-2)} = [tex]6e^(-2s)[/tex] / s (applying the time-shift property)

L{3u(t-5)} = [tex]3e^(-5s)[/tex] / s (applying the time-shift property)

L{-4u(t-6)} = -[tex]4e^(-6s[/tex]) / s (applying the time-shift property)

Therefore, the Laplace transform of f(t) is:

F(s) = L{f(t)} = 6[tex]e^(-2s)[/tex] / s + [tex]3e^(-5s)[/tex] / s - [tex]4e^(-6s)[/tex]/ s

= [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s[/tex]

Hence, the Laplace transform of f(t) is F(s) = [tex](6e^(-2s) + 3e^(-5s) - 4e^(-6s)) / s.[/tex]

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Step-by-step explanation:

you can solve cos(u) by

cos(u) = adjecent / hypotenes...general formula of cos

cos(u) = √44 / 12

cos(u) = 2√11 / 12 ..... √44 = √4×11 = 2√11

cos(u) = √11 / 6

u = cos^-1 ( √11 / 6 ) ..... divided both aide by cos ( multiple by cos invers )

u = 56.442 .... so we get it's angle

Answer:

[tex]cos(U)=\frac{\sqrt{11} }{6}[/tex]

Step-by-step explanation:

In a right triangle, the cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse. Therefore, we have:

cos(U) = adjacent/hypotenuse = TU/SU

We are given that TU = sqrt(44) and SU = 12, so:

cos(U) = sqrt(44)/12

To simplify this expression, we can first factor 44 into 4 * 11, since 4 is a perfect square and a factor of 44:

cos(U) = sqrt(4 * 11) / 12

cos(U) = (sqrt (4) * sqrt (11)) / 12

cos (U) = (2 * sqrt (11)) / 12

Simplifying the fraction by dividing both the numerator and denominator by 2, we get:

cos(U) = sqrt(11)/6

Therefore, the exact value of cos(U) in simplest radical form is sqrt(11)/6

Furthermore, if you want another way to write the answer, dividing by 6 is the same as multiplying by 1/6 so you can do cos (U) = 1/6 * sqrt (11)

Although the other individual was correct that you use inverse trig (cos ^ -1) to find the measure of U, getting an exact answer requires us to leave it in simplest radical form since the number is so large and at best will yield an approximation if you don't keep it in simplest radical form.

Approximately 680 out of the 850 entering students will probably need to take a refresher mathematics course which is calculated using simplified fraction.

We are given that about 8 out of 10 people entering a community college need to take a refresher mathematics course. We need to find out how many of the 850 entering students will probably need this course.

Step 1: Determine the proportion of students who need the refresher course.
The proportion is 8 out of 10, which can be written as a fraction: 8/10.

Step 2: Simplify the fraction.
Divide both the numerator (8) and the denominator (10) by their greatest common divisor, which is 2:
8 ÷ 2 = 4
10 ÷ 2 = 5
So, the simplified fraction is 4/5.

Step 3: Calculate the number of students who need the refresher course.
To find the number of students who probably need the course, multiply the total number of entering students (850) by the simplified fraction (4/5):
850 * (4/5) = (850 * 4) / 5 = 3400 / 5 = 680

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Answer:

3. 254.34 mm^2

4. 615.44 cm^2

5. 314 in^2

6. 7.065 in^2

7. 3.14 cm^2

8. 1.76625 ft^2

Step-by-step explanation:

AREA FORMULA: π * r^2

This question is asking to use 3.14 or 22/7 for x.

The following steps will use 3.14.

3. r = 9 mm (r^2 = 81 mm)

A = 81 * 3.14 = 254.34 mm^2

4. r = 14 cm (r^2 = 196 cm)

A = 196 * 3.14 = 615.44 cm^2

5. r = 10 in (r^2 = 100 in)

A = 100 * 3.14 = 314 in^2

Questions 6-8 show the diameter of the circle.

Divide by 2 to find the radius, then plug that into the area formula

6. r = 1.5 in (r^2 = 2.25 in)

A = 2.25 * 3.14 = 7.065 in^2

7. r = 1 cm (r^2 = 1 cm)

A = 1 * 3.14 = 3.14 cm^2

8. r = 0.75 ft (r^2 = 0.5625 ft)

A = 0.5625 * 3.14 = 1.76625 ft^2

Using basic mathematical procedures, we can determine that the total cost of the 8 bags of popcorn and 5 drinks is $53.

An operation, in mathematics, is a mathematical function that transforms zero or more input values into a precisely defined output value.

The quantity of operands affects the operation's arity.

The rules that specify the order in which we should carry out the operations required to solve an equation are referred to as the order of operations.

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. (from left to right).

The whole price is thus:

Popcorn costs $3.50 for one bag.

One beverage costs $5.

Total cost for 5 beverages and 8 bags of popcorn:

(8 × 3.50) + (5 × 5) 28 + 25 $53

Therefore ,Using basic mathematical procedures, we can determine that the total cost of the 8 bags of popcorn and 5 drinks is $53.

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The appropriate response is provided below:

Zahra is going to the cinema with a few of her friends. A bag of popcorn costs $3.50 and a drink costs $5 at the theatre. How much would it cost if they purchased 5 beverages and 8 bags of popcorn?

Answer:

= 112.5π sq. units is the area of pi

Step-by-step explanation:

In your case it's 15π

So that becomes:

2πr=15π

Now dividing the equation on both sides by π,the result is:

2r=15

That means 2 times radius(r) is 15

r=15/2

r computes out to be 7.5

Now r=7.5

So the area of circle(AoC) i.e. πr^2

AoC=3.14*(7.5)^2

AoC=3.14*(7.5)*(7.5)

AoC=176.625

Note: Don't forget to multiply the result to respective unit square for e.g. if the circumference was 15π in cm then the Area would compute out as 176.625 cm^2

It cost Trudy $200 to get the loan for $500. The correct answer is C) $200.

Trudy has taken a loan of $500, and the cost of the loan is $10 for every $100 borrowed. Therefore, the cost of borrowing $500 will be:

Cost of borrowing $500 = ($10/$100) * $500 = $50

In addition to the above cost, there is a one-time fee of $150 to be paid. So, the total cost of the loan will be:

Total cost of the loan = Cost of borrowing + one-time fee

= $50 + $150

= $200

Hence, it cost Trudy $200 to get the loan for $500.

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The sum of all edges of the regular quadrilateral prism is 108 cm.

To find the sum of all edges of a regular quadrilateral prism with height h = 11 cm and base edges b = 8 cm, follow these steps:

1. Determine the number of base edges: A quadrilateral has 4 edges, so there are 4 base edges for the top and 4 for the bottom, totaling 8 base edges.
2. Determine the number of height edges: There are 4 vertical edges connecting the top and bottom bases.
3. Add the number of base and height edges: 8 base edges + 4 height edges = 12 edges in total.
4. Calculate the sum of all edge lengths: (8 base edges (8 cm)) + (4 height edges (11 cm)) = 64 cm + 44 cm = 108 cm.

So, the sum of all edges of the regular quadrilateral prism is 108 cm.

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The solution to the quadratic equation using quadratic formula is: -1 or -1/2

The general form of expression of a quadratic equation is:

ax² + bx + c = 0

The quadratic formula for solving quadratic functions is:

x = [-b ± √(b² - 4ac)]/2a

If we have a quadratic equation as: 5x² + 6x + 1 = 0.

Using quadratic formula, we have:

x = [-6 ± √(6² - 4(5*6))]/2*5

x = -1 or -1/2

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The given problem involves evaluating the surface integral of the vector field F(X, y, 2) over the top half of a sphere x^2 + y^2 + z^2 = 1, oriented upwards, using the Divergence Theorem.

The Divergence Theorem states that the flux of a vector field F through a closed surface S is equal to the triple integral of the divergence of F over the region enclosed by S.

In this problem, the given vector field F(X, y, z) is F(X, y, 2) = 3z^2xi + (y^3 + tan(2)J + (3x^2z + 1y^2)k.

The surface S is the top half of the sphere x^2 + y^2 + z^2 = 1, oriented upwards. This means that z is positive on S, and the normal vector points in the positive z-direction.

To use the Divergence Theorem, we need to find the divergence of F. The divergence of F is given by div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z, where ∂Fx/∂x, ∂Fy/∂y, and ∂Fz/∂z are the partial derivatives of F with respect to x, y, and z, respectively.

Taking the partial derivatives of F with respect to x, y, and z, we get:

∂Fx/∂x = 6xz

∂Fy/∂y = 3y^2 + 2y

∂Fz/∂z = 0

So, the divergence of F is: div(F) = 6xz + 3y^2 + 2y

Now, we can apply the Divergence Theorem, which states that the surface integral of F over S is equal to the triple integral of the divergence of F over the region enclosed by S.

The triple integral of the divergence of F over the region enclosed by S can be written as: ∫∫∫ div(F) dV, where dV is the volume element.

Since the given problem asks for the surface integral of F over S, we only need to consider the part of the triple integral that involves the surface S.

The surface integral of F over S can be written as: ∫∫ F · dS, where dS is the outward-pointing normal vector on S and · represents the dot product.

The dot product F · dS can be expressed as: Fx * dSx + Fy * dSy + Fz * dSz, where Fx, Fy, and Fz are the components of F, and dSx, dSy, and dSz are the components of the outward-pointing normal vector on S.

Since the normal vector on S points in the positive z-direction, we have dSx = 0, dSy = 0, and dSz = 1.

Substituting the components of F and the components of dS into the expression for the dot product, we get: Fx * dSx + Fy * dSy + Fz * dSz = (3z^2x)(0) + (y^3 + tan(2)J + (3x^2z +

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The frequency table shows that 152 female students chose website design out of a total of 362 female students, which is about 0.421 or 42%.

The frequency table shows that out of the total 362 female students attending the technology summer camp, 152 chose website design, which is a design class. This means that the percentage of female students who chose a design class is 152/362 = 0.4202 or about 42%.

However, the camp counselor says that about 68% of female students chose a design class. It is unclear where the counselor obtained this information from as it is not reflected in the frequency table. It is possible that the counselor gathered this information from a different survey or observation.

On the other hand, the camp director's statement is more accurate as it is based on the frequency table.

The frequency table shows that 152 female students chose website design out of a total of 362 female students, which is about 0.421 or 42%. It is important to rely on data and accurate information when making statements or drawing conclusions.

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The area of the squares are;

1.  9x²ft². Option D

2.  6x² - 7x - 3 in². Option C

The formula for calculating the area of a square is expressed as;

A = a²

Such that the parameters of the formula are;

From the information given, we have that;

Area = (3x)²

Find the square of the expression, we have that;

Area = 9x²ft²

2. Substitute the values, we have that;

Area = (2x -3)(3x + 1)

expand the bracket, we have;

Area = 6x² + 2x - 9x - 3

collect the like terms

Area = 6x² - 7x - 3

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For the piecewise-defined function, f(-4) = 42.

The given function is a piecewise-defined function, which means that it is defined differently depending on the value of x. In this case, we have two different formulas for the function depending on whether x is less than or greater than 3. For values of x less than 3, the function is given by f(x) = 22 - 5x, while for values of x greater than 3, the function is given by f(x) = 2x + 5.

To find f(-4), we need to determine which part of the function applies to the value of x = -4. Since -4 is less than 3, we use the first part of the function, which gives us f(-4) = 22 - 5(-4) = 22 + 20 = 42. This means that if x is equal to -4, the function f(x) evaluates to 42.

Piecewise-defined functions can be useful in modeling real-world problems where the relationship between variables changes depending on certain conditions or constraints. By defining the function differently depending on the value of x, we can more accurately capture the behavior of the system being modeled.

In this case, the function could be used to model a situation where the value of a variable has different relationships to other variables depending on whether it is less than or greater than a certain threshold value.

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