Grading categories: creating equations, reasoning with equations, linear models


mr. bro opened a chick-fil-a on white plains road, and ms. kramer opened a popeye’s across the street. both had to borrow money to open their fast food franchises. after 500 customers, mr. bro was still $4000 in debt. by the time he had served 3000 customers, he was ahead $1000.



after 2000 customers, ms. kramer still owed $6000 to the bank. however, after 4500 customers, she was ahead by $1500.



which restaurant would you rather own? support your answer with mathematical evidence. you must show all work and math calculations. some things we have to see in your solution:


amount each had to borrow to open their restaurant


amount of customers they needed to break even (break even means the total profit is $0)


the solution to your system of equations showing when one restaurant would produce more total income than the other

Answer:

There are 16 total outcomes for tossing 4 quarters

This is because each coin flip has 2 possibilities, so if you flip the coin 4 times it will equal

2x2x2x2.

To find the revenue function, we need to multiply the price (p) by the quantity sold (x). The price function given is p = 5,300 - dollars, so we can substitute this into our revenue function as follows:

Revenue = p * x
Revenue = (5,300 - dollars) * x

We also know that production is limited to 100 units, so we need to take that into account when determining the revenue function. If x is greater than 100, then the revenue will be limited to 100 units sold. If x is less than or equal to 100, then the revenue will be based on the actual quantity sold.

To incorporate this constraint into our revenue function, we can use a piecewise function:

Revenue = { (5,300 - dollars) * 100           if x > 100
             (5,300 - dollars) * x             if x <= 100 }

Simplifying the piecewise function, we get:

Revenue = { 530,000 - (dollars * 100)        if x > 100
             (5,300 - dollars) * x             if x <= 100 }

Therefore, the revenue function for this monopoly is:

Revenue = { 530,000 - 100 * dollars        if x > 100
             (5,300 - dollars) * x             if x <= 100 }
Hi! To find the revenue function, we first need to determine the total revenue, which is the product of the price per unit (p) and the number of units sold (x). Given the demand function p = 5,300 - x dollars and the average cost function C = 3,010 + 2x dollars, we can find the revenue function as follows:

Revenue function, R(x) = p * x
R(x) = (5,300 - x) * x

By simplifying the equation, we get:
R(x) = 5,300x - x^2

So, the revenue function for this monopoly is R(x) = 5,300x - x^2.

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Answer:

so

Step-by-step explanation:

Let's call the length of each of the other two sides x. Since the triangle is isosceles, it has two sides of equal length. Therefore, the perimeter of the triangle can be expressed as 6 + x + x Simplifying this equation, we get 2x + 6 We know that the perimeter is 22 cm so we can set up an equation and solve for x. 22 = 2x + 6 Subtracting 6 from both sides, we get 16 = 2x Dividing both sides by 2, we get x=8

On the basic of square peg sydney smith written point the probability that it is im-possible to fit a square peg in a round hole is equals to the zero.

Probability is defined as the number of chances of occurrence of an event. It is the ratio of the number of favorable outcomes to the total outcomes in that sample space. Mathematical formula is written as, probability of an event, P(E) = (Number of favorable outcomes)/(Total possible outcomes). Now, we have specify that according to sydney smith written in on the conduct of the understanding about square peg that it is impossible to fit a square peg in a round hole. Let consider an event A of fit a square peg in a round hole. We have specify that it is impossible to fit a square peg in a round hole. So, the favourable possible outcomes for event A = 0. Therefore, the probability that to fit a square peg in a round hole, P(A) = 0

Hence, required probability value is zero.

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The function f(x) = x² + 12x - 3 has two distinct real roots and no rational or complex roots. The possible rational roots are ±1 and ±3, but they are not actual roots of the equation.

To find the number of complex zeros and possible rational zeros of the function f(x) = x² + 12x - 3 = 0, we can use the same methods as in the previous question.

According to the rational root theorem, any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term (-3) and q is a factor of the leading coefficient (1). The possible rational roots are therefore ±1, ±3.

To determine if there are any complex roots, we can use the quadratic formula. The solutions to the equation f(x) = 0 are given by

x = (-b ± √(b² - 4ac)) / 2a

Plugging in the coefficients of the polynomial, we get

x = (-12 ± √(12² - 4(1)(-3))) / 2(1)

x = (-12 ± √(156)) / 2

x = -6 ± 3√(13)

Since the discriminant (b² - 4ac) is positive, the square root is real, and the roots are two distinct real numbers. Therefore, there are no complex roots.

The possible rational roots we found earlier, ±1 and ±3, are not actual roots of the equation, since plugging them into the equation does not give us zero. Therefore, there are no rational roots either.

In conclusion, the equation f(x) = x² + 12x - 3 = 0 has two distinct real roots and no complex or rational roots.

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--The given question is incomplete, the complete question is given

" State the number of complex zeros and the possible rational zeros for the function f(x) = x² + 12x -3 = 0

show your work"--

The IQR is approximately 1 unit.

Yes, you can use the box plot to find the IQR (Interquartile Range).

The IQR is the distance between the upper quartile (Q3) and the lower quartile (Q1) of the data. In a box plot, Q1 is the bottom of the box, and Q3 is the top of the box. The IQR is the height of the box.

Therefore, in the given box plot, you can find the IQR by measuring the height of the box and calculating the difference between Q3 and Q1.

The length of the whiskers and the positions of the outliers are not used in calculating the IQR.

So, looking at the box plot, we can see that the height of the box is approximately 4 units (the units are not specified in the question).

The bottom of the box (Q1) is at approximately 3 units, and the top of the box (Q3) is at approximately 7 units.

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The slope of the line is 3/5 and the y-intercept is 4/7.

The equation of the line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.

In this form, the slope of the line tells us how steeply the line is rising or falling, while the y-intercept tells us where the line crosses the y-axis.

In the given equation y = 3/5x + 4/7, we can see that the coefficient of x, 3/5, is the slope of the line. This means that for every 1 unit increase in x, the line will increase by 3/5 units in y.

A positive slope means that the line is rising from left to right, while a negative slope means that the line is falling.

We can also see that the constant term, 4/7, is the y-intercept of the line. This tells us that the line crosses the y-axis at the point (0, 4/7). In other words, when x is 0, y is equal to 4/7.

So, to summarize, the slope of the line y = 3/5x + 4/7 is 3/5, which means the line rises 3 units for every 5 units it moves to the right.

The y-intercept is 4/7, which tells us the line crosses the y-axis at the point (0, 4/7).

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The surface area of Dante's tent is approximately 180 square feet.

To find the surface area of Dante's tent, we need to find the area of each of its faces and then add them up. Since the tent is in the shape of a triangular prism, it has three rectangular faces and two triangular faces.

First, let's find the area of each rectangular face. We know that the tent is 8 feet long and 4.3 feet tall. To find the width of each face, we need to use the Pythagorean theorem since the base of the tent is equilateral. So, we can use a triangle with sides of 5 feet (the base of the tent) and 4.3 feet (the height of the triangular face) to find the width of each rectangular face.

a^2 + b^2 = c^2
5^2 + 4.3^2 = c^2
25 + 18.49 = c^2
43.49 = c^2
c = √43.49
c ≈ 6.6 feet

Now that we know the width of each rectangular face, we can find the area of each face:

Area of rectangular face = length x width
Area of rectangular face = 8 x 6.6
Area of rectangular face ≈ 52.8 square feet

Next, let's find the area of each triangular face. Since the base of the tent is equilateral, each triangular face has a base of 5 feet and a height of 4.3 feet. The area of each triangular face is:

Area of triangular face = (base x height) / 2
Area of triangular face = (5 x 4.3) / 2
Area of triangular face ≈ 10.8 square feet

Now we can add up the areas of all five faces to find the total surface area of the tent:

Surface area of tent = 3 x area of rectangular face + 2 x area of triangular face
Surface area of tent = 3 x 52.8 + 2 x 10.8
Surface area of tent = 158.4 + 21.6
Surface area of tent ≈ 180 square feet

Therefore, the surface area of Dante's tent is approximately 180 square feet.

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Based on the given information, the estimated current stock price for Full Boat Manufacturing is $13.11. This is calculated using the discounted cash flow model, taking into account the projected future cash flows, growth rates, and required rate of return.

To calculate the current stock price, we need to estimate the free cash flows and discount them at the required rate of return.

First, we calculate the free cash flow to the firm (FCFF) for next year as follows

FCFF = Sales - Costs - Net Investment*(1-t)

= $115 million - $67 million - $12 million*(1-0.21)

= $31.02 million

Next, we calculate the expected growth rate in FCFF using the formula:

g = (FCFF Year 2 / FCFF Year 1) - 1

where FCFF Year 2 = FCFF Year 1 * (1 + g)

Using the given information, we get

g = (FCFF Year 2 / FCFF Year 1) - 1

= (FCFF Year 1 * (1 + 0.14) * (1 - 0.02) / FCFF Year 1) - 1

= 0.12

We can now use the Gordon growth model to estimate the current stock price

Current stock price = FCFF Year 1 * (1 + g) / (r - g)

where r is the required rate of return.

Substituting the values, we get

Current stock price = $31.02 million * (1 + 0.12) / (0.13 - 0.12)

= $72.13 million

Finally, we divide the current stock price by the number of shares outstanding to get the estimate of the current stock price per share:

Current stock price per share = $72.13 million / 5.5 million

= $13.11 per share

Therefore, the estimate of the current stock price is $13.11 per share.

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--The given question is incomplete, the complete question is given

"  Full Boat Manufacturing has projected sales of $115 million next year. Costs are expected to be $67 million and net investment is expected to be $12 million. Each of these values is expected to grow at 14 percent the following year, with the growth rate declining by 2 percent per year until the growth rate reaches 6 percent, where it is expected to remain indefinitely. There are 5.5million shares of stock outstanding and investors require a return of 13 percent on the company’s stock. The corporate tax rate is 21 percent.

What is your estimate of the current stock price?

Sophie will pay money equivalent to $100 for 15m at the same store.

The term "money" in mathematics refers to a form of payment, such as bills, coins, and demand deposits, that is used to purchase goods and services. Money is used to pay for the worth or price of an item or service.

A country's monetary system is referred to as its currency.

In the case of Gareth,

Money paid for 9 m of climbing [tex]=\$60[/tex]

Money paid per m of climbing [tex]=\$60\div9[/tex]

Thus, money paid by Sophie for 15 m of climbing [tex]= 15 \times (60\div9)[/tex]

[tex]\boxed{\bold{= \$100}}[/tex]

Hence Sophie will pay money equivalent to $100 for 15m at the same store.

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Answer:

$100

Step-by-step explanation:

We know that for 9m of rope, Gareth had to pay $60.

The question is asking us to find out how much Sophie will pay for 15m of rope.  To do this, we have to find out how much is paid per meter of rope.

[tex]60/9\\=6\frac{2}{3}[/tex]

For the sake of not using fractions, let's keep it as an improper fraction: 60/9

So, we can write an equation for the price of 15m of rope:

(60/9)·15

=100

So, Sophie will pay $100 for 15m of rope.

Hope this helps! :)

Triangle ABD is scalene and obtuse.

The triangle has three sides of unequal length, hence it is classified as an scalene triangle.

(it would be equilateral if all had the same length, and isosceles if two sides have the same length).

The triangle has one angle larger than 90º, hence it is classified as an obtuse triangle.

(acute with no angles of 90º or greater, right with one angle of exactly 90º).

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The minimum surface area of fabric needed for the sleeve is approximately 628.32 cm².

Based on the given model arm, a right circular cylinder would best represent the model for the sleeve.

To calculate the minimum surface area of fabric needed for the sleeve, we need to find the lateral surface area of the right circular cylinder.

The lateral surface area of a right circular cylinder is given by the formula:

Lateral surface area = 2πrh

where r is the radius of the cylinder, h is the height of the cylinder.

In this case, the height of the cylinder needs to be 20 cm (to cover the distance from the shoulder to the elbow), and the radius can vary depending on the desired fit. Let's assume a radius of 5 cm for the purposes of this calculation.

Plugging in the values, we get:

Lateral surface area = 2π(5 cm)(20 cm)

= 628.32 cm²

Therefore, the minimum surface area of fabric needed for the sleeve is approximately 628.32 cm².

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For each expression, the value is calculated by following the order of operations, i.e. first solving any multiplication or division, then addition or subtraction. The resulting values are: a + b - c = -3, a - b + c = -1, a + 2b - 2c = -5, (7b)/(b+c) = 3, ac + 2b - 2c = -10, and c(b-a) = 20.

To calculate a + b - c, we substitute a = -2, b = 3, and c = 4. So,

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

To calculate a - b + c, we substitute a = -2, b = 3, and c = 4. So,

a - b + c = -2 - 3 + 4 = -1

To calculate a + 2b - 2c, we substitute a = -2, b = 3, and c = 4. So,

a + 2b - 2c = -2 + 2(3) - 2(4) = -5

To calculate (7b) / (b + c), we substitute b = 3 and c = 4. So,

(7b) / (b + c) = (7(3)) / (3 + 4) = 21 / 7 = 3

To calculate ac + 2b - 2c, we substitute a = -2, b = 3, and c = 4. So,

ac + 2b - 2c = (-2)(4) + 2(3) - 2(4) = -10

To calculate c(b - a), we substitute a = -2, b = 3, and c = 4. So,

c(b - a) = 4(3 - (-2)) = 4(5) = 20

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The rate of change of a function as x approaches infinity is determined by the leading term in the function.

For f(x) = 2x - 10, the leading term is 2x.

For g(x) = 16x - 4, the leading term is 16x.

For h(x) = 3x^2 - 7x + 8, the leading term is 3x^2.

Since the coefficient of the leading term in h(x) is positive, and it has a higher degree than the leading terms of f(x) and g(x), h(x) has the greatest rate of change as x approaches infinity.

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If you select a marble without looking and then put it back if you do this 90 times, the probability of selecting an orange or a pink marble on any given trial is 1/2 or 0.5.

If you select a marble without looking and then put it back, the probability of selecting any particular marble on any given trial is the same for all marbles. Assuming there are only two possible outcomes - selecting an orange marble or a pink marble - the probability of selecting an orange or a pink marble on any given trial is 1/2 or 0.5.

Since the probability of selecting an orange or a pink marble on each trial is the same, the best prediction for the number of times you will select an orange or a pink marble out of 90 trials is an equal number of times for each color. Therefore, the best prediction for the number of times you will select an orange or a pink marble is 45 times each.

This is only a prediction based on probability, and the actual number of times you select an orange or a pink marble may differ from the prediction due to chance. However, over a large number of trials, the actual outcomes should approach the predicted probabilities.

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Answer:

The third choice is the correct answer.

Answer:

1)55,44

2)6,4

3)49,8

4)11,7

5)12,444

6)23,04

7)12,46

8)15,06

9)31,68

10)30,4

Step-by-step explanation:

Recheck them and make sure there are no mistakes

Check the picture below.

Answer:

C (12)

Step-by-step explanation:

2/5 of the packages weighed more than 1 2/5 pounds.

To find the fraction of packages that weighed more than 1 2/5 pounds, we need to count the number of packages that weighed more than 1 2/5 pounds and divide it by the total number of packages.

There are a total of 10 packages. Out of these, we can see that 4 packages weighed more than 1 2/5 pounds: 1 1/5, 1 3/5, 1 3/5, and 1 4/5.

Therefore, the fraction of packages that weighed more than 1 2/5 pounds is:

4/10

This can be simplified to:

2/5

Hence, only 2/5 of the package weighed more.

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In this case, the researchers have been following the development of a sample of 11-year-old children since 1932, which is a very long period of time, making it a classic example of a longitudinal study

The type of study that the researchers in Scotland are conducting is a longitudinal study. Longitudinal studies involve following a group of individuals over an extended period of time, often years or even decades, in order to observe changes or continuity in their development, behaviors, or other characteristics.Longitudinal studies are considered to be one of the most powerful research designs for understanding how individuals change over time. By following a group of people over an extended period, researchers can gain insight into how different factors, such as social, environmental, and biological, interact to shape development.

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The general solution to the given differential equation:

[tex]y = (9x + C)e^(-6x)[/tex]

To solve the given differential equation, dy + 6ydx = 9e^(-6x)dx, we can first rewrite it as a first-order linear differential equation:

[tex]dy/dx + 6y = 9e^(-6x)[/tex]

Now, we will find the integrating factor, which is e^(∫P(x)dx), where P(x) is the coefficient of y in the equation:

Integrating factor =[tex]e^(∫6dx) = e^(6x)[/tex]

Next, we multiply the entire differential equation by the integrating factor:

[tex]e^(6x)(dy/dx + 6y) = 9e^(-6x)e^(6x)[/tex]

This simplifies to:

[tex]e^(6x)(dy/dx) + 6e^(6x)y = 9[/tex]

Now, the left side of the equation is the derivative of the product of y and the integrating factor:
[tex]d/dx(y * e^(6x)) = 9[/tex]

To solve for y, integrate both sides with respect to x:

[tex]∫[d/dx(y * e^(6x))]dx = ∫9dx[/tex]

This results in:

[tex]y * e^(6x) = 9x + C[/tex]

Finally, we solve for y by dividing by e^(6x):

[tex]y = (9x + C)e^(-6x)[/tex]

And that is the general solution to the given differential equation.

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1. The ratio of the surface areas of the two similar solids is 3. 2. The ratio of the volumes of the two similar solids is 3.

Surface area is a measure of the total area of the surface of a three-dimensional object. It is the sum of the areas of all the faces, sides, and bases of the object. Surface area is expressed in square units, such as square meters (m²) or square feet (ft²).

The formula for calculating the surface area of a particular object depends on its shape. Some common shapes and their formulas for surface area include:

Rectangular prism: Surface area = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height.

Cube: Surface area = 6s², where s is the length of one side.

Sphere: Surface area = 4πr², where r is the radius.

Cylinder: Surface area = 2πr² + 2πrh, where r is the radius and h is the height.

Cone: Surface area = πr² + πrl, where r is the radius and l is the slant height.

1. To find the ratio of the surface areas of two similar solids, we need to use the fact that the ratio of the surface areas is equal to the square of the ratio of their corresponding side lengths. Since the solids are similar, their corresponding side lengths are in proportion.

Let's call the ratio of the corresponding side lengths "r". Then, we have:

Ratio of surface areas = r²

To find "r", we can use the fact that the ratio of the volumes of two similar solids is equal to the cube of the ratio of their corresponding side lengths. Therefore:

(Ratio of side lengths)³ = Ratio of volumes

Let's call the ratio of the corresponding side lengths "k". Then, we have:

k³ = 45/15 = 3

k = ∛3

Now, we can find the ratio of surface areas:

Ratio of surface areas = (corresponding side length ratio)²

Ratio of surface areas = (∛3)² = 3

Therefore, the ratio of the surface areas of the two similar solids is 3.

2. To find the ratio of the volumes of the two similar solids, we simply divide the larger volume by the smaller volume:

Ratio of volumes = 45/15 = 3

Therefore, the ratio of the volumes of the two similar solids is 3.

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Answer:

a) y=0.50x+15  

b) The graph of this equation form on a coordinate plane is a line.

c) Slope  =0.50 and y-intercept = 15

Step-by-step explanation:

Let x = Number of miles driven by car

Given: The cost of renting a car for a day is $0.50 per mile plus a $15 flat fee.

a) Total cost = 0.50x+15

If y =total cost of renting the car, then y=0.50x+15   (i)

b) Above equation is similar to y= mx+c  (ii) [m = slope , xc=y-intercept] which a linear equation .

So the graph of this equation form on a coordinate plane is a line.

c) Comparing (i) and (ii)

m=0.50 , c=15

Hope this helps :)

The expression that is equivalent to 1/4(8 - 6x + 12) is 2 - 3x/2 + 6

Algebraic expressions are simply defined as those mathematical expressions that are composed of terms, variables, their coefficients, their factors and constants.

These mathematical expressions are also comprised of arithmetic operations.

These operations are listed thus;

From the information given, we have that;

1/4(8 - 6x + 12)

expand the bracket, we have;

8 - 6x + 12/4

Divide in group, we have;

8/4 - 6x/4 + 12/4

Divide the values

2 - 3x/2 + 6

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The area of the painted face of the brick is given as follows:

108 in².

The area of a rectangle of length l and width w is given by the multiplication of dimensions, as follows:

A = lw.

The dimensions for this problem are given as follows:

Hence the area of the painted face of the brick is given as follows:

A = 12 x 9 = 108 in².

(the area of a rectangle is given by the multiplication of the dimensions, which we did here).

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The cost of points for this mortgage is $3,510 rounded to the nearest dollar if Julie pays 2.25 points.

Cost of house =  $225,000.

Mortgage amount =  $156,000

Fixed-rate = 4.5%

Bank offer rate = 4.25%

Points to pay = 2.25 points

If we assume that one point is equal to 1%, then 2.25 points are equal to 2.25% of the loan amount.

The cost of points for the mortgage can be calculated by the product of the loan amount by 2.25%.

Cost of points = 0.0225 × $156,000

Cost of points = $3,510

Therefore, we can conclude that the cost of points for this mortgage is $3,510 rounded to the nearest dollar.

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The value of the function f(x) at x = 2 is 48.

The question asks us to find the value of the function f(x) = 4[tex]x^3[/tex] at x = 2 and its derivative f'(x) at x = 2.

The value of the function f(x) at x = 2, we simply plug in x = 2 into the function and evaluate:

f(2) = 4[tex](2)^3[/tex] = 4(8) = 32

Therefore,

The value of the function f(x) at x = 2 is 32, which we can write as f(2) = 32.

To find the derivative of the function f(x) at x = 2, we first need to find the general formula for the derivative of f(x), which we can do using the power rule for derivatives.

The power rule states that

To find the derivative of f(x) at x = 2, we simply plug in x = 2 into the formula for f'(x) that we just found:

f'(2) = [tex]12(2)^2[/tex] = 48

Therefore,

The derivative of the function f(x) at x = 2 is 48, which we can write as f'(2) = 48To find the derivative of the function

f'(x) = 12[tex]x^2[/tex]

To find the value of f(2), we can simply plug in x = 2 into the original function:

f(2) = 4[tex](2)^3[/tex] = 32

To find the value of f'(2), we can plug in x = 2 into the derivative we just found:

f'(2) = 12[tex](2)^2[/tex] = 48

Therefore,

f(2) = 32 and f'(2) = 48.

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The true statement about Amy's weekly income is that her net income is less than $120 (option c).

To find out the true statement about Amy's weekly income, we need to calculate her gross income first.

Step 1: Multiply the hours worked per week by the hourly wage.
Amy works 15 hours a week and earns $8 an hour, so:
15 hours * $8/hour = $120

Now we have her gross income, which is $120.

Comparing the gross income to the statements given:

a) Her net income is more than $120: False, as net income is usually less than gross income due to deductions (taxes, social security, etc.).
b) Her gross income is less than $120: False, as we calculated her gross income to be exactly $120.
c) Her net income is less than $120: True since net income is generally less than the gross income due to deductions.
d) Her gross income is more than $120: False, as her gross income is exactly $120.

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The absolute maximum occurs at x = -2 and the absolute minimum occurs at x = 0 and x = 2 and  The absolute maximum is at x = -2 and the absolute minimum is at x = 0, 2.
a. The local minimum is at x=2 and there is no local maximum.
b. The local maximum is at x=1 and the local minimum is at x=-2 and x=2.
c. The absolute maximum is at x=0 and the absolute minimum is at x=2.
(Note: To find the absolute maximum and minimum, we need to evaluate the function at the critical points and endpoints of the interval. The critical points are x=0 and x=2, and the endpoints are x=-2 and x=2. The absolute maximum is the largest value among these, which is f(0)=0. The absolute minimum is the smallest value among these, which is f(2)=-4.)
Given the function f(x) = x²(x - 2) on the interval [-2, 2]:

A. To find the local minima and maxima, we need to take the first derivative and find its critical points.

f'(x) = 3x² - 4x
Solving for x, we get x = 0 and x = 4/3.

However, x = 4/3 is not within the interval [-2, 2], so the only critical point within the interval is x = 0.

There is a local minimum at x = 0, and no local maximum. Therefore, the answer is:

A. The local minimum is at x = 0 and there is no local maximum. (Type an integer or a simplified fraction.)

B. For the absolute maximum and minimum, we need to evaluate the function at the endpoints and the critical point within the interval.

f(-2) = (-2)²(-2 - 2) = 16
f(0) = (0)²(0 - 2) = 0
f(2) = (2)²(2 - 2) = 0

The absolute maximum occurs at x = -2 and the absolute minimum occurs at x = 0 and x = 2. The answer is:

B. The absolute maximum is at x = -2 and the absolute minimum is at x = 0, 2. (Use a comma to separate answers as needed. Type integers or simplified fractions.)

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