Find those values of x for which the given function is increasing and those values of x for which it is decreasing, y = 12x – x^3 • Increasing for x < 2, decreasing for x > 2 • Increasing for 4 4 • Increasing for X <-2.x > 2, decreasing for -2 2

The woman performed 100 joules of work to push the cart of groceries over a distance of 10 meters.

To find the work performed by the woman, we will use the work-energy theorem which states that work (W) is equal to the change in energy. In this case, the woman applies a 10 Newton force and uses 100 joules of energy. The formula for work is:

W = F × d × cos(θ)

Where W is work, F is force, d is the distance over which the force is applied, and θ is the angle between the force and the direction of motion. Since the woman used 100 joules of energy, we can rewrite the equation as:

100 J = 10 N × d × cos(θ)

We don't have information about the angle θ, but if we assume that she applied the force horizontally, which is in the same direction as the motion, the angle θ would be 0 degrees, and the cosine of 0 is 1. Therefore, the equation becomes:

100 J = 10 N × d

To find the distance (d), we can now solve for d:

d = 100 J / 10 N

d = 10 meters

So, 100 joules of work was performed by the women to push the cart of groceries over a distance of 10 meters.

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Since ln(1/6) is a finite value, the sequence does not diverge. It converges to ln(1/6) as n approaches infinity.

To determine if the sequence diverges, we need to take the limit of the expression as n approaches infinity.

Using the logarithmic identity ln(a/b) = ln(a) - ln(b), we can simplify the expression as follows:

[tex]ln(n^3 8) - ln(6n^3 13n) = ln(n^3) + ln(8) - ln(6n^3) - ln(13n)[/tex]

= [tex]ln(n^3) - ln(6n^3) + ln(8) - ln(13n)[/tex]

= [tex]ln(n^3/6n^3) + ln(8/13n)[/tex]

=[tex]ln(1/6) + ln(8/13n)[/tex]

As n approaches infinity, ln(8/13n) approaches 0, so the limit of the expression is:

lim n→∞ [ln(1/6) + ln(8/13n)]

= ln(1/6)

Since ln(1/6) is a finite value, the sequence does not diverge. It converges to ln(1/6) as n approaches infinity.

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If Mateo jogged 25 9/10 miles last week and jogged the same course all 7 days, then he jogged an average of (25 9/10) / 7 = 3 11/14 miles per day.

To convert this mixed number to an improper fraction, we can multiply the whole number by the denominator of the fraction and add the numerator, then place the result over the denominator:

3 * 14 + 11 = 53

53/14

So, Mateo jogged an average of 53/14 miles per day last week

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To use Newton's method to approximate a root of the equation

In (4x) = arctan(x -0.1),

we will need to find the first derivative of the function f(x) = In(4x) - arctan(x-0.1). f(x) = In(4x) - arctan(x-0.1) f'(x) = 4/(4x) - 1/(1+(x-0.1)^2) Using the initial approximation x1 = 0.1,

We can find the second approximation x2: x2 = x1 - f(x1)/f'(x1) x2 = 0.1 - [In(4*0.1) - arctan(0.1-0.1)] / [4/(4*0.1) - 1/(1+(0.1-0.1)^2)] x2 = 0.1076

We can repeat this process to find the third approximation x3: x3 = x2 - f(x2)/f'(x2) x3 = 0.1076 - [In(4*0.1076) - arctan(0.1076-0.1)] / [4/(4*0.1076) - 1/(1+(0.1076-0.1)^2)] x3 = 0.1078

Now we can find the fourth approximation x4: x4 = x3 - f(x3)/f'(x3) x4 = 0.1078 - [In(4*0.1078) - arctan(0.1078-0.1)] / [4/(4*0.1078) - 1/(1+(0.1078-0.1)^2)] x4 = 0.1078

Finally, we can find the fifth approximation x5: x5 = x4 - f(x4)/f'(x4) x5 = 0.1078 - [In(4*0.1078) - arctan(0.1078-0.1)] / [4/(4*0.1078) - 1/(1+(0.1078-0.1)^2)] x5 = 0.1078

Therefore, the fourth approximation x4 is 0.1078 and the fifth approximation x5 is also 0.1078.

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The price of one teddy bear is $7 and the price of one doll is $14.

Let's use matrices to solve this system of equations:

First, we need to define the variables:

x = price of one teddy bear

y = price of one doll

Then we can write the system of equations:

14x + 3y = 158

8x + 12y = 296

system of matix:

| 14   3 |   | x |   | 158 |

|  8  12 | * | y | = | 296 |

To solve for x and y, we can use matrix multiplication and inversion:

| x |   | 12  -3 |   | 158 |   |  99 |

| y | = | -8  14 | * | 296 | = | -14 |

So, x = $7 and y = $14. Therefore, the price of one teddy bear is $7 and the price of one doll is $14.

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The sum of the series under the interval (3, 6) will be negative 4.

Given that:

Series, ∑ (2k - 10)

A series is a sum of sequence terms. That is, it is a list of numbers with adding operations between them.

The sum of the series under the interval (3, 6) is calculated as,

∑₃⁶ (2k - 10) = (2 x 3 - 10) + (2 x 4 - 10) + (2 x 5 - 10) + (2 x 6 - 10)

∑₃⁶ (2k - 10) = - 4 - 2 + 0 + 2

∑₃⁶ (2k - 10) = -4

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If combine harvester able to cut a 2450 square meters field in 5 hours and other one can do the same job in 7 hours then the area that the two combines cut in 9 hours is equals to the 7,560 square meters.

We have two harvester which can cut a area into different time.

The time taken by first harvester cuts into 2450 square meters field =5 hours.

The time taken by second harvester cuts into 2450 square meters field = 7 hours.

We have to determine the area can the two combines cut in 9 hours.

Here, total work = 2450 square meters

Work ability or efficiency of first harvester = 2450/5 = 490

Work efficiency of second harvester

= 2450/7= 350

Efficiency of both harvester in combine

= 350 + 490 = 840

So, area they cut into 9 hours = 840 × 9

=7560 square meters

Hence, required area is 7,560 square meters.

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The probability of event A or B occurring i.e.,  P(A or B) is 0.67.

Given the table of outcomes and their probabilities, you need to find P(A or B), where A is the event "the outcome is greater than 1" and B is the event "the outcome is greater than or equal to 2".

1: Identify the outcomes that satisfy A or B.

A: Outcomes greater than 1: {2, 3, 4, 5}

B: Outcomes greater than or equal to 2: {2, 3, 4, 5}

A or B: Outcomes greater than 1 or greater than or equal to 2: {2, 3, 4, 5}

2: Calculate the probability of each outcome in the combined set A or B.

Outcome 2: Probability 0.19

Outcome 3: Probability 0.13

Outcome 4: Probability 0.31

Outcome 5: Probability 0.04

3: Add up the probabilities of each outcome in the combined set A or B.

P(A or B) = P(2) + P(3) + P(4) + P(5)

P(A or B) = 0.19 + 0.13 + 0.31 + 0.04

P(A or B) = 0.67

Therefore, the probability of event A or B occurring, where A is "the outcome is greater than 1" and B is "the outcome is greater than or equal to 2", is 0.67.

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Total outcomes when we spin the spinner and flip a coin = 12

In the figure

In spinner the labelled number are from 1 to 6

And for a coin there are two outcomes head and tail

Therefore,

total number of outcomes for spinner = 6

total number of outcomes for a coin   = 2

Then the number of outcomes when both are performed once

= 6x2

= 12

Hence total outcomes = 12

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The correct equation is,

⇒ (5 - 6i) (2 + 4i)

Let us assume that;

⇒ (a + bi) (c + di)

⇒ ac + adi + bci + bdi²

⇒ (ac − bd) + (ad + bc)i

Matching coefficients:

30 < ac − bd < 80

ad + bc = 0

Hence, We need to pick four integers between -9 and 9 such that these two equations are satisfied.  One possible combination is:

a = 8, b = -4, c = 6, d = 3

The number would be:

ac − bd = (8)(6) − (-4)(3) = 60

Puzzle 2

Using the result from Puzzle 1:

ac − bd = 34

ad + bc = 8

Like before, it makes sense to assume b is negative.  With some trial and error, one possible answer is:

a = 5, b = -6, c = 2, d = 4

Thus, The correct equation is,

⇒ (5 - 6i) (2 + 4i)

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if one side of triangle is approximately 10 feet, the height of the attic is approximately 8.66 feet.

To find the height h of an equilateral triangle, we need to know the length of one of its sides. However, this information is not given in the question.

We can use the Pythagorean theorem to find an approximate value of h. Let's assume that the length of one side of the equilateral triangle is 10 feet (this is just an arbitrary value for illustration purposes).

Then, the height h can be found using the formula:

h = √(10² - (10/2)²)

h = √(100 - 25)

h = √75

h ≈ 8.66 feet

So, if one side of the equilateral triangle is approximately 10 feet, the height of the attic is approximately 8.66 feet. However, if the length of the side is different, the height will also be different.

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To express the number 2 as a logarithm with a base of 4, you would write it as log₄(16). This is because 4² = 16.

In general, the logarithm function is the inverse of exponentiation. When we write logₐ(b) = c, it means that a raised to the power of c equals b.

In your example, you want to find the logarithm of 2 with a base of 4, which means you are looking for the exponent to which 4 must be raised to obtain 2.

So, log₄(2) represents the exponent c such that 4 raised to the power of c equals 2.

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The surface area is 72 square inches.

To find the surface area of a square pyramid, we need to calculate the area of the base and the four triangular faces.

Given that the base is 4 inches wide, the area of the square base is:
Base area = side² = 4² = 16 square inches.

The slant height is 7 inches. To find the area of one triangular face, we use:
Triangle area = (base * slant height) / 2

Each triangle has the same base length as the square base, which is 4 inches. Therefore, the area of one triangular face is:
Triangle area = (4 * 7) / 2 = 14 square inches.

Since there are four triangular faces, their total area is:
4 * Triangle area = 4 * 14 = 56 square inches.

Finally, add the base area and the total area of the triangular faces to get the surface area:
Surface area = Base area + Total triangular faces area = 16 + 56 = 72 square inches.

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The cat is 3/8 inches taller than the dragon.

To find the difference in height between the cat and the dragon, we need to subtract the height of the dragon from the height of the cat.

The cat is 7/8 inch tall, and the dragon is 1/2 inch tall.

To subtract fractions, we first need a common denominator. The least common denominator (LCD) of 2 and 8 is 8.

So, we'll convert the fractions to equivalent fractions with a denominator of 8.

1/2 = 4/8 (multiply both the numerator and the denominator by 4)

Now, we can subtract the fractions:

(7/8) - (4/8) = (7 - 4)/8 = 3/8

So, the cat is 3/8 inches taller than the dragon.

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Elijah and Riley are playing a board game. Elijah chooses the dragon for his game piece, and Riley chooses the cat for hers. The dragon is about 1 2 inch tall, and the cat is about 7 8 inch tall. How much taller is the cat than the dragon?

(a) Identifying the population, parameter, sample, and statistic for a study on the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts before and after a campaign.

(b) Stating the null and alternative hypotheses for a significance test on whether the percentage of male drivers not using seatbelts has decreased after the campaign.

(a) Population: All male drivers between the ages of 19 and 29 in the major urban area.

Parameter: The percentage of male drivers between the ages of 19 and 29 in the major urban area who do not regularly use seatbelts after the radio and television campaign and stricter enforcement by the local police.

Sample: 100 male drivers between the ages of 19 and 29 who were polled.

Statistic: The percentage of male drivers between the ages of 19 and 29 in the sample who did not wear their seatbelts, which is 24%.

(b) The null hypothesis is that the percentage of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area has not decreased after the radio and television campaign and stricter enforcement by the local police.

The alternative hypothesis is that the percentage of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area has decreased after the radio and television campaign and stricter enforcement by the local police.

Mathematically, the hypotheses can be stated as follows:

H0: p >= 0.28

Ha: p < 0.28

where p is the proportion of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area after the radio and television campaign and stricter enforcement by the local police.

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The question is -

In a major urban area, the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts was 28%. After a major radio and television campaign and stricter enforcement by the local police, researchers want to know if the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts has decreased.  They polled a random sample of 100 males between the ages of 19 and 29 and find the percentage who didn’t wear their seatbelts was 24%.

(a) Identify the population, parameter, sample, and statistic.

(b) State appropriate hypotheses for performing a significance test.

The mean and standard deviation of the sampling distribution of p are μp = 0.42 and σp = 0.0868, respectively.

Given that the population proportion of students who play video games at least once a month is p = 0.42 and the sample size is n = 30.

The mean of the sampling distribution of the sample proportion is given by:

μp = p = 0.42

The standard deviation of the sampling distribution of the sample proportion is given by:

σp = sqrt[p(1-p)/n] = sqrt[(0.42)(0.58)/30] ≈ 0.0868

Therefore, the mean and standard deviation of the sampling distribution of p are μp = 0.42 and σp = 0.0868, respectively.

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Therefore , the solution of the given problem of expressions comes out to be  7n.

Instead of using random estimates, it is preferable to use shifting numbers that may also prove increasing, reducing, variable or blocking. They could only help one another by trading tools, information, or solutions to issues. The justifications, components, or quantitative comments for tactics like further disagreement, production, and blending may be included in the assertion of truth equation.

Here,

By dividing the number of rows by the number of trees in each row, one can calculate the total number of trees in Mr. Lee's apple orchard. Here are two expressions that demonstrate various approaches to determining the overall number of trees:

There are 7n =  trees in all.

=> Total number of trees = (Number of rows) x (Number of trees in each row) = 7n

The total number of trees in the orchard is the outcome of both expressions.

While the second statement more directly depicts the multiplication, the first expression merely merges the two elements into a single term.

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Answer:

Step-by-step explanation:

(a)

x + y = -1

3x = 4 - 3y

from 1st equation we can write x = -1 - y and put this x in the 2nd equation

3(-1 -y) = 4 -3y

-3 -3y = 4 -3y

now here y is getting cancel so that means this two equation has no common solution.

(b)

3x - 4y +2 = 0

10 -10y = 10y -15x

from 1st equation we can write x = (4y - 2)/3 and put this x in the 2nd equation

10 = 10y +10y -15((4y - 2)/3)

2 = 2y +2y - 3((4y -2)/3)

2 = 4y - (4y - 2)

again here 4y and 4y getting cancel so both the equation has no common solution.

We can make 24 bouquets of flowers for a wedding, each with 10 roses and 7 lilies.

To determine the largest number of identical bouquets that can be made using 240 roses and 168 lilies, we need to find the greatest common factor (GCF) of these two numbers.

The prime factorization of 240 is 2^4 x 3 x 5, while the prime factorization of 168 is [tex]2^3 * 3 * 7[/tex]. To find the GCF, we can take the product of the common prime factors raised to the smallest exponent they appear in either number. Therefore, the GCF of 240 and 168 is [tex]2^3 * 3[/tex] = 24.

This means that we can make 24 identical bouquets using 240 roses and 168 lilies. To do so, we would use 10 roses and 7 lilies in each bouquet, since 10 and 7 are the largest numbers that divide both 240 and 168 without remainder, respectively. So, we can make 24 bouquets, each with 10 roses and 7 lilies.

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The location of all the relative and absolute extrema is (0, 0) (local minimum); (1, 1) (local maximum); (3, 3) (absolute maximum)

To find the relative and absolute extrema of the function f(x) = 2x - x^2 on the domain (0,3), we first take the derivative:

f'(x) = 2 - 2x

Setting this equal to zero, we find the critical point:

2 - 2x = 0
x = 1

To determine the nature of the critical point, we need to examine the second derivative:

f''(x) = -2

Since the second derivative is negative at x = 1, this critical point is a local maximum. To find the absolute extrema, we also need to examine the endpoints of the domain, x = 0 and x = 3:

f(0) = 0
f(3) = 3

So the function has an absolute maximum at x = 3 and an absolute minimum at x = 0. Therefore, the location of all the relative and absolute extrema, from smallest to largest x, is:

(0, 0) (local minimum)
(1, 1) (local maximum)
(3, 3) (absolute maximum)

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In seven years, the student population at the Brentwood Student Center will be approximately 4,174.

Using the given terms, the current student population at the Brentwood Student Center is 2,500 and the enrollment increases at a rate of 6% each year. To find the student population closest to seven years from now, we'll use the formula for exponential growth:

Future Population = Current Population × (1 + Growth Rate)^Number of Years

In this case, the future population will be:

Future Population = 2,500 × (1 + 0.06)^7

After calculating, we get:

Future Population ≈ 4,174

So, to the nearest whole number, the student population at the Brentwood Student Center will be approximately 4,174  in seven years.

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The relationship between the areas of the three squares is that square A plus square B equals the area of square C.

The Pythagorean theorem is a fundamental idea in geometry that states that for any right-angled triangle, the square of the length of the longest side (opposite the right angle) is equal to the sum of the square of the lengths of the two remaining sides. This equation can be expressed as:

[tex]a^2 + b^2 = c^2[/tex]

Thus, the relationship between the areas of the three squares is that square A plus square B equals the area of square C.

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The per capita birth rate of the given population of manned wolves 246 with number of birth as 83 is equals to 0.34  births per individual per year.

Number of births = 83

Initial population = 246

Time period = 1 year

Number of deaths = 42

The per capita birth rate is calculated as the number of births per individual in the population.

Typically expressed as a rate per unit time.

Per capita birth rate as follows,

Per capita birth rate

= (Number of births / Initial population) × (Time period / 1 year)

Substituting these values into the formula, we get,

Per capita birth rate

= (83 / 246) × (1 / 1)

= 0.3374

Rounding this to the hundredths place, we get,

Per capita birth rate = 0.34

Therefore, the per capita birth rate for this population of maned wolves is 0.34 births per individual per year.

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To find the length of the entire perimeter of the region inside r = 7 sin Ф but outside r = 2, we first need to determine the boundaries of the region.

The equation r = 7 sin Ф represents a curve that forms a flower-like shape, while the equation r = 2 represents a circle with radius 2.

To find the region inside r = 7 sin Ф but outside r = 2, we need to find the points where these two curves intersect.

Setting the two equations equal to each other, we get:

7 sin Ф = 2

Solving for sin Ф, we get:

sin Ф = 2/7

Using a calculator, we can find the two values of Ф that satisfy this equation to be approximately 0.304 and 2.837 radians.

Thus, the region inside r = 7 sin Ф but outside r = 2 is bounded by the angles 0.304 and 2.837 radians.

To find the length of the entire perimeter of this region, we need to integrate the length element around this curve:

L = ∫(from 0.304 to 2.837) √[r² + (dr/dФ)²] dФ

Using the equation r = 7 sin Ф, we can substitute and simplify the expression under the square root:

L = ∫(from 0.304 to 2.837) √[49sin²(Ф) + 49cos²(Ф)] dФ

L = ∫(from 0.304 to 2.837) 7 dФ

L = 7(2.837 - 0.304)

L = 16.1

Therefore, the length of the entire perimeter of the region inside r = 7 sin Ф but outside r = 2 is approximately 16.1 units.
To find the length of the entire perimeter of the region inside r = 7 sin Ф but outside r = 2, we must first identify the points of intersection between the two curves.

1. Set the equations equal to each other:
7 sin Ф = 2

2. Solve for Ф:
sin Ф = 2/7
Ф = arcsin(2/7)

Now, we must determine the length of the perimeter of each curve in the region of interest:

3. Length of the perimeter of r = 7 sin Ф (half of the curve, since it's within the specified region):
For a polar curve r = f(Ф), the arc length L is calculated using the formula L = ∫√(r² + (dr/dФ)²) dФ.

In this case, f(Ф) = 7 sin Ф, so dr/dФ = 7 cos Ф. Integrating over the range [0, arcsin(2/7)], we can find the half-length of this curve.

4. Length of the perimeter of r = 2 (portion of the circle outside the region):
Since we know the points of intersection from step 2, we can find the central angle of the circular segment using Ф. The central angle is 2 * arcsin(2/7), and the circumference of the circle is 2π * 2. The portion of the perimeter is given by the ratio of the central angle to 2π, multiplied by the circumference.

Finally, add the lengths obtained in steps 3 and 4 to get the total length of the entire perimeter of the region.

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The null hypothesis (H₀) and the alternative hypothesis (H₁) in symbolic form for this scenario are:

H₀: p = 0.0016 (the true proportion of fireflies unable to produce light is equal to 16 in ten thousand)
H₁: p < 0.0016 (the true proportion of fireflies unable to produce light is fewer than 16 in ten thousand)

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The answers to the questions on linear combination have been solved below

1. We can start by multiplying both sides by -2 to eliminate the x-term:

2x + 4y = 0

-2(2x + 4y) = -2(0)

-4x - 8y = 0

Now, we have:

-4x - 8y = 0

9x + 4y = 28

We can now use linear combination by adding these two equations to eliminate the y-term:

(-4x - 8y) + (9x + 4y) = 0 + 28

5x = 28

Dividing both sides by 5, we get:

x = 28/5

Now, we can substitute this value of x into either of the original equations to solve for y. Let's use the second equation:

2x + 4y = 0

2(28/5) + 4y = 0

56/5 + 4y = 0

4y = -56/5

y = -14/5

Therefore, the solution to the system of equations is:

x = 28/5

y = -14/5

We can check that these values satisfy both equations:

9x4y = 28

9(28/5)(-14/5) = 28

-352/25 = 28/25 (true)

2x + 4y = 0

2(28/5) + 4(-14/5) = 0

56/5 - 56/5 = 0 (true)

Therefore, the solution is verified.

2. The system of equations is:

5x + 3y = 41

3x - 6y = 9

We can simplify the second equation by dividing both sides by 3:

3x - 6y = 9

x - 2y = 3

Now we can use linear combination by multiplying the first equation by 2 to eliminate the y-term:

2(5x + 3y) = 2(41)

10x + 6y = 82

(x - 2y) + (10x + 6y) = 3 + 82

11x = 85

Dividing both sides by 11, we get:

x = 85/11

Now we can substitute this value of x into either of the original equations to solve for y. Let's use the first equation:

5x + 3y = 41

5(85/11) + 3y = 41

425/11 + 3y = 41

3y = 41 - 425/11

3y = (451 - 425)/11

y = 26/33

Therefore, the solution to the system of equations is:

x = 85/11

y = 26/33

We can check that these values satisfy both equations:

5x + 3y = 41

5(85/11) + 3(26/33) = 41

425/11 + 26/11 = 41

451/11 = 41 (true)

3x - 6y = 9

3(85/11) - 6(26/33) = 9

255/11 - 52/11 = 9

203/11 = 9 (true)

Therefore, the solution is verified.

3.

3(x - 2y) = 3(-8)

3x - 6y = -24

3x - 6y = -12

3x - 6y = -24

Subtracting the second equation from the first equation, we get:

0 = 12

This is a contradiction, since 0 cannot equal 12. Therefore, there is no solution that satisfies both equations.

This means that the system is inconsistent, and there are no solutions.

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After about 15.7 months, the apartment complex will have 800 occupied units.

Logistic growth is a type of growth in which the growth rate of a population decreases as the population size approaches its maximum value. In this case, the apartment complex has a maximum capacity of 1500 units.

Starting with 15 occupied units and growing at a rate of 10% per month, the number of occupied units can be modeled by a logistic function.

To find the number of months it takes to reach 800 occupied units, we need to solve for the time when the logistic function equals 800.

Let P(t) be the number of occupied units at time t (in months), then we have:

P(t) = 1500 / (1 + 1485[tex]e^{(-0.1t)}[/tex])

We want to find t such that P(t) = 800. Solving for t, we get:

t = -10 ln(1 - 4/37) ≈ 15.7 months

This means that after about 15.7 months, the apartment complex will have 800 occupied units.

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The package that requires more wrapping paper to cover is the cube. The total amount of wrapping paper Susan must use to cover both packages is 286 square inches.

Let's find the surface area of both packages to determine which requires more wrapping paper and the total amount needed.

1. Rectangular prism:
Surface area = 2lw + 2lh + 2wh
where l = length, w = width, h = height
Surface area = 2(4)(2) + 2(4)(10) + 2(2)(10)
Surface area = 16 + 80 + 40 = 136 square inches

2. Cube:
Surface area = 6s²
where s = side length
Surface area = 6(5)² = 6(25) = 150 square inches

The cube requires more wrapping paper to cover as its surface area is 150 square inches, compared to the rectangular prism's 136 square inches. The total amount of wrapping paper Susan must use for both packages is 136 + 150 = 286 square inches.

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The sample space of the number of outcomes is A = 12

Given data ,

To create a tree diagram and list the sample space, we need to consider the possible outcomes at each stage of the event.

First, we have three options for the spinner: Red, Blue, and Yellow.

Now, let's consider the possible outcomes when a letter is chosen from the word MATH

The total number of outcomes A = 12 outcomes

where A = { RM , BM , YM , RA , BA , YA , RT , BT , YT , RH , BH , YH }

       Red             Blue            Yellow

       / \             / \             / \

      M   A           M   A           M   A

     /     \         /     \         /     \

    T       H       T       H       T       H

Hence , the number of outcomes is A = 12 and the tree diagram is solved

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