Expand and simplify (2x - 1)(x + 3)(x - 5)

Answer:

[tex]\dfrac{5e^2}{2}[/tex]

Step-by-step explanation:

Differentiation is an algebraic process that finds the slope of a curve. At a point, the slope of a curve is the same as the slope of the tangent line to the curve at that point. Therefore, to find the slope of the line tangent to the given function, differentiate the given function.

Given function:

[tex]y=x^2\ln(2x)[/tex]

Differentiate the given function using the product rule.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Product Rule for Differentiation}\\\\If $y=uv$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}$\\\end{minipage}}[/tex]

[tex]\textsf{Let\;$u=x^2}[/tex][tex]\textsf{Let\;$u=x^2$}\implies \dfrac{\text{d}u}{\text{d}x}=2x[/tex]

[tex]\textsf{Let\;$v=\ln(2x)$}\implies \dfrac{\text{d}v}{\text{d}x}=\dfrac{2}{2x}=\dfrac{1}{x}[/tex]

Input the values into the product rule to differentiate the function:

[tex]\begin{aligned}\dfrac{\text{d}y}{\text{d}x}&=u\dfrac{\text{d}v}{\text{d}x}+v\dfrac{\text{d}u}{\text{d}x}\\\\&=x^2 \cdot \dfrac{1}{x}+\ln(2x) \cdot 2x\\\\&=x+2x\ln(2x)\end{aligned}[/tex]

To find the slope of the tangent line at x = e²/2, substitute x = e²/2 into the differentiated function:

[tex]\begin{aligned}x=\dfrac{e^2}{2}\implies \dfrac{\text{d}y}{\text{d}x}&=\dfrac{e^2}{2}+2\left(\dfrac{e^2}{2}\right)\ln\left(2 \cdot \dfrac{e^2}{2}\right)\\\\&=\dfrac{e^2}{2}+e^2\ln\left(e^2\right)\\\\&=\dfrac{e^2}{2}+2e^2\\\\&=\dfrac{5e^2}{2}\end{aligned}[/tex]

Therefore, the slope of the line tangent to the graph of y = x²ln(2x) at the point where x = e²/2 is:

[tex]\boxed{\dfrac{5e^2}{2}}[/tex]

Answer:

  x = 31.2

Step-by-step explanation:

You want the solution to the proportion 12/x = 5/13.

You can eliminate the fractions by multiplying this equation by the least common denominator. That value is 13x, the product of these denominators. Multiplying by 13x, we have ...

  [tex]\dfrac{12}{x}\times13x=\dfrac{5}{13}\times13x\\\\\\12\cdot13=5\cdot x\qquad\text{simplified}[/tex]

Now, the value of x is found by dividing both sides by its coefficient.

  [tex]\dfrac{12\cdot13}{5}=\dfrac{5x}{5}\\\\\\\dfrac{156}{5}=x\\\\\boxed{31.2=x}[/tex]

__

Additional comment

The first step we did, multiplying by 13x, is also sometimes called "cross multiplication." The result of that step is that each numerator is multiplied by the opposite denominator.

Multiplying both sides of the equation by the same value (13x) is supported by the multiplication property of equality. The term "cross multiplication" is descriptive of the result, but is not a recognized property of equality. It is a good idea to keep the math operations you do grounded in the properties of equality.

You will notice that the steps we did were "multiply by 13x" and "divide by 5". These can be done at once by "multiply by 13x/5". Of course, that operation is done to both sides of the equation.

Any proportion can be written 4 ways:

  [tex]\dfrac{12}{x}=\dfrac{5}{13}\qquad\dfrac{x}{12}=\dfrac{13}{5}\qquad\dfrac{x}{13}=\dfrac{12}{5}\qquad\dfrac{13}{x}=\dfrac{5}{12}[/tex]

These can be thought of as "upside down" and "sideways." We like the versions with the variable on top of a fraction, because the solution to that is simply multiplication by the variable's denominator.

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Answer:

B. [tex]2y = 24[/tex]

Step-by-step explanation:

Given the addition of equations:

[tex]\text{ }\ \, \left(\dfrac{}{}5x \ \ - y \ \: = -1\dfrac{}{}\right) \\ \underline{+ \left(\,-5x + 3y = 25\ \dfrac{}{}\right)} \\[/tex]

We can add each column of like terms ([tex]x[/tex]s, [tex]y[/tex]s, and constants).

Showing these in equation form:

[tex]0 + 2y = 24[/tex]

A. [tex]\boxed{2y = 24}[/tex]

If the random variable x is normally distributed with a mean equal to 0.45 and a standard deviation equal to 0.40, then P(x ≥ .75) is 0.9227.

A z-score describes the position of a raw score in terms of its distance from the mean when measured in standard deviation units. The z-score is positive if the value lies above the mean and negative if it lies below the mean.

Given the problem above, we need to find what the z-score is when P(x ≥ .75).

The formula for calculating a z-score is given by:

[tex]Z=\dfrac{\text{x}-\mu}{\sigma}[/tex]

Where:

Now,

[tex]Z=\dfrac{\text{x}-\mu}{\sigma}[/tex]

[tex]Z=P(\text{x} \geq 0.75) = \huge \text(\dfrac{P(Z \geq (0.75 - 0.45)}{0.40}\huge \text)[/tex]

[tex]Z= P\huge \text (\dfrac{Z \geq0.30}{0.40}\huge \text)[/tex]

[tex]Z=P(Z \geq 0.75) = 1 - P(Z < 0.75)[/tex]

[tex]Z=1 - 0.077337[/tex]

[tex]Z\thickapprox 0.9227[/tex]

Therefore, the z-score of P(x ≥ .75) is 0.9227.

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answer: 190%

step-by-step explanation:

hihi your problem is to find the percent increase. we can use the following formula!

percent Increase = ((new value - old value) / old value) * 100

given that bruce's initial typing speed was 10 words per minute (old value) and his final typing speed after the course was 29 words per minute (new value), we can substitute these values into the formula:

percent increase = ((29 - 10) / 10) * 100

simplifying the numerator:

percent increase = (19 / 10) * 100

dividing 19 by 10:

percent increase = 1.9 * 100

calculating the product:

percent Increase = 190

therefore, the percent increase in Bruce's typing speed after taking the course is 190%. this means that his typing speed improved by 190% compared to his initial speed.

hopefully this helped !!

Explanation:

Segment AW bisects angle CAD.

This leads to the smaller pieces (angles CAW and DAW) to be equal to one another. Both are 20 degrees each. That totals to 20+20 = 40 degrees.

Therefore, angle CAD = 40 degrees.

The supplement of this is angle DAX

(angle CAD) + (angle DAX) = 180

angle DAX = 180 - (angle CAD)

angle DAX = 180 - 40

angle DAX = 140 degrees

Two factors that need to be controlled to keep this test fair are the temperature and the sample volume.

The control variables are those variables that should be kept constant in order to avoid interfering with the free flow of the experiment.

For the provided test, it is important that the temperature of the environment where the test sample is kept is controlled throughout the duration of the experiment so that some results are not affected by this. Also, the sample volume should be controlled for fair outcomes.

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The profit function is P(x) = -500.523x^2 + 600x - 200.

To find the profit function, P(x), we need to subtract the cost function, C(a), from the revenue function, R(x).

Given:

Cost function: C(a) = 500x^2 + 300

Revenue function: R(x) = -0.523x^2 + 600x - 200 + 300

Profit function, P(x), is obtained by subtracting the cost function from the revenue function:

P(x) = R(x) - C(a)

P(x) = (-0.523x^2 + 600x - 200 + 300) - (500x^2 + 300)

Simplifying the expression:

P(x) = -0.523x^2 + 600x - 200 + 300 - 500x^2 - 300

P(x) = -500x^2 - 0.523x^2 + 600x + 300 - 200 - 300

P(x) = -500x^2 - 0.523x^2 + 600x - 200

Combining like terms:

P(x) = (-500 - 0.523)x^2 + 600x - 200

Simplifying further:

P(x) = -500.523x^2 + 600x - 200

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Answer:

Step-by-step explanation:

In the ration 2:3:4 the largest value is 4

24/4 is 6, so now we know the scale factor is 6

2 x 6 = 12

3 x 6 = 18

Now, we add all the values 24 + 12 +18 = 54

perimeter = 54 cm

Hope this helps!

Answer:

First, we need to calculate the volume of the box in cubic inches:

2 feet = 24 inches

3 feet = 36 inches

18 inches = 18 inches

Volume = 24 x 36 x 18 = 15,552 cubic inches

Next, we need to calculate the number of sugar cubes that can fit in the box:

Each sugar cube has a volume of 1.5 x 1.5 x 1.5 = 3.375 cubic inches

Number of sugar cubes that can fit in the box = 15,552 / 3.375 = 4,608

Since a bag of 5,000 sugar cubes costs $25, the value of 4,608 sugar cubes would be:

Value = (4,608 / 5,000) x $25 = $23.04

Therefore, the total value of sugar that the breeder is packing into the box is $23.04.

Step-by-step explanation:

Answer:

D. 4x = 14

Step-by-step explanation:

Given the subtraction of equations:

 (7x + 2y = 17)

- (3x + 2y = 3)

we can find the difference by subtracting like terms:

Putting these into an equation:

4x + 0 = 14

D. 4x = 14

Answer:

x = 9

Step-by-step explanation:

For this parallelogram to be a rhombus, all sides must be equal in size so:

6x-5 = 4x+13 (subtract 4x from both sides)

2x-5 = 13 (add 5 to both sides)

2x = 18 (divide by 2 for both sides)

x = 9

So x = 9

The average rate of change of the school's enrollment during this time period is -49 students per year. This means that on average, the enrollment at the college has been decreasing by 49 students per year.

To determine the average rate of change of the school's enrollment during the given time period, we can use the formula:

Average rate of change = (Change in enrollment) / (Change in time)

The change in enrollment is calculated by subtracting the initial enrollment from the final enrollment, while the change in time is calculated by subtracting the initial year from the final year.

Given that in 2004 there were 975 students enrolled and in 2009 there were 730 students enrolled, we can calculate the change in enrollment:

Change in enrollment = 730 - 975 = -245 students

The change in time can be calculated as:

Change in time = 2009 - 2004 = 5 years

Now we can calculate the average rate of change:

Average rate of change = (-245 students) / (5 years) = -49 students per year

Therefore, the average rate of change of the school's enrollment during this time period is -49 students per year. This means that on average, the enrollment at the college has been decreasing by 49 students per year.

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1. Total area of the wall: The expression is 2X square feet, obtained by multiplying the length of each section (X/5) by the wall's height (10 ft).

2. Area of each section: Divide the total area by 5 to find the area of each section, resulting in the expression 2X/5 square feet.

3. Expression for each section's area: The area of each section is 2X/5 square feet, derived from dividing the total area equally among the 5 sections.

To solve the problem, we'll break it down into three parts as stated in the questions:

1. Expression for the total area of the wall:

Since the wall is divided into 5 equal sections and the total length is X, we can determine the length of each section by dividing the total length by 5. So, the length of each section is X/5. The height of the wall is given as 10 ft. To find the total area of the wall, we multiply the length and height. Therefore, the expression for the total area of the wall is:

Total Area = (X/5) * 10 = 10X/5 = 2X square feet.

2. To find the area of each section, we divide the total area by the number of sections (which is 5 in this case). This is because the wall is divided equally into 5 sections, so each section has the same area. So, we need to perform the following calculation:

Area of Each Section = Total Area / Number of Sections = 2X / 5 square feet.

3. Expression for the area of each section of the wall:

As mentioned earlier, the area of each section is given by the expression 2X/5 square feet. This expression represents the equal division of the total area among the 5 sections of the wall.

In summary, the total area of the wall is expressed as 2X square feet, the area of each section is 2X/5 square feet, and these calculations are based on the given length of the wall and its equal division into 5 sections.

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The rate of the first athlete is greater than the rate of the second athlete.

To determine if the rate of the first athlete is greater than, less than, or equal to the rate of the second athlete, we need to compare their average speeds.

The average speed of an athlete is calculated by dividing the distance traveled by the time taken.

For the first athlete who runs 3 miles in 24 minutes, the average speed can be calculated as:

Speed1 = Distance1 / Time1 = 3 miles / 24 minutes = 1/8 miles per minute.

For the second athlete who runs 4 miles in 33 minutes, the average speed can be calculated as:

Speed2 = Distance2 / Time2 = 4 miles / 33 minutes.

To make a comparison, we need to convert both rates to a common unit. Let's convert both rates to miles per minute:

Speed2 =[tex](4 miles / 33 minutes) * (24 minutes / 24 minutes)[/tex] = (96 miles / 792 minutes) ≈ 0.1212 miles per minute.

Comparing the two rates, we see that Speed1 (1/8 miles per minute) is greater than Speed2 (0.1212 miles per minute). The rate of the first athlete is greater than the rate of the second athlete.

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Answer:

10 root 2

Step-by-step explanation:

root 200 = root 2 × root 100 = 10 root 2

The length of Sarah's rectangular backyard is 5 feet, that is equivalent to 60 inches.

1. We are given the perimeter formula for a rectangle: P = 2L + 2W, where P represents the perimeter, L represents the length, and W represents the width.

2. The perimeter of Sarah's backyard is given as 18 feet.

3. Since 1 foot is equal to 12 inches, we need to convert the given perimeter from feet to inches: 18 feet * 12 inches/foot = 216 inches.

4. Now we can substitute the values into the formula: 216 inches = 2L + 2(4 feet * 12 inches/foot).

5. Simplifying the equation, we have: 216 inches = 2L + 2(48 inches).

6. Distributing the multiplication, we get: 216 inches = 2L + 96 inches.

7. To isolate the term with L, we subtract 96 inches from both sides: 216 inches - 96 inches = 2L.

8. Simplifying further, we have: 120 inches = 2L.

9. To find the length, we divide both sides by 2: L = 120 inches / 2.

10. The length of the rectangular yard is therefore: L = 60 inches.

So, the length of Sarah's rectangular backyard is 5 feet, which is equivalent to 60 inches.

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Answer:

[tex](-\infty,2)[/tex]

Step-by-step explanation:

Since [tex]-3x+6\nless 0[/tex], then [tex]x\ngtr 2[/tex], therefore, the domain of the function is [tex](-\infty,2)[/tex].

The correct answer is that the data set {3, 7, 4} is represented in the given histogram.(option-a)

The given histogram represents the number of children in each age group who need to travel to school. Since the histogram has only three bars, we can conclude that there are only three age groups.

The first bar represents children aged 5-10, of which there are 3. The second bar represents children aged 11-13, of which there are 7. The third bar represents children aged 14-18, of which there are 4.

Therefore, the data set that is represented in the histogram is:

{3, 7, 4}

None of the other data sets given match the values in the histogram. The first data set has duplicate values and is not sorted by age group. The second data set includes ages that are not represented in the histogram. The third data set has values for ages 6, 11, 12, 13, 14, 15, 17, and 18, but the histogram does not have bars for all those ages. (option-a)

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Answer:

x = 18

Step-by-step explanation:

There are 2 ways to do this :

1) Method 1 :

sum of exterior angles of a triangle = 360

⇒ (9x - 31) + (4x + 33) + (7x - 2) = 360

⇒ 20x = 360

⇒ x = 360/20

⇒ x = 18

2) Method 2:

Since the three lines are straight lines, the interior angles of the triangle are given by:

∠CDE = 180 - (9x - 31) = 211 - 9x

∠DEC = 180 - (4x + 33) = 147 - 4x

∠ECD = 180 - (7x - 2) = 182 - 7x

The sum of angles in a triangle = 180

∠CDE + ∠DEC + ∠ECD = 180

⇒ 211 - 9x + 147 - 4x + 182 - 7x = 180

⇒ 540 - 20x = 180

⇒ 20x = 540 - 180

⇒ 20x = 360

⇒ x = 360/20

⇒ x = 18

The annualized return for the entire period is the closest to 10.5%.

To calculate the annualized return for the entire period, we need to consider the returns for each year and the return in the first quarter of Year 4. Since the returns are given for each period, we can use the geometric mean to calculate the annualized return.

The formula for calculating the geometric mean return is:

Geometric Mean Return = [(1 + R1) * (1 + R2) * (1 + R3) * (1 + R4)]^(1/n) - 1

Where R1, R2, R3, and R4 are the returns for each respective period, and n is the number of periods.

Given the returns:

Year 1 return: 5% or 0.05

Year 2 return: 12% or 0.12

Year 3 return: -6% or -0.06

First quarter of Year 4 return: 2% or 0.02

Using the formula, we can calculate the annualized return:

Annualized Return = [(1 + 0.05) * (1 + 0.12) * (1 - 0.06) * (1 + 0.02)]^(1/3) - 1

Annualized Return = (1.05 * 1.12 * 0.94 * 1.02)^(1/3) - 1

Annualized Return = 1.121485^(1/3) - 1

Annualized Return ≈ 0.105 or 10.5%

Therefore, the annualized return for the entire period is approximately 10.5%.

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Answer:

the object will fell 906.5 ft by the end of 7 seconds

Step-by-step explanation:

distance ∝ time²

74 ∝ 4

x ∝ 49

thus, 74/4 = x/49

74 * 49 = 4x

x = 3,626/4

x = 906.5 ft

Work Shown:

1 & 3/5 = 1 + (3/5) = (5/5) + (3/5) = 8/5

The mixed number 1 & 3/5 is the same as the improper fraction 8/5.

We'll divide this over 8 to determine the weight of each lot.

(8/5) ÷ 8

(8/5) ÷ (8/1)

(8/5) * (1/8)

(8*1)/(5*8)

1/5

Each lot of diamonds is 1/5 of a pound.

The square root of 76.8 is 8.76.

Square root of 76.8 is 8.746.

Given,

[tex]\sqrt{76.8}[/tex]

Now,

Simplifying the square root :

If square root of x is to be calculated then,

[tex]\sqrt{x}[/tex] = [tex]x^{1/2}[/tex]

Similarly,

[tex]\sqrt{76.8}[/tex] = [tex]76.8^{1/2}[/tex]

= 8.746

Thus the square root of 76.8 is 8.746.

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My recommendation to Carlos and Clarita would be to prepare for 3 cats and 3 dogs. This combination offers the highest profit potential based on the given prices and costs.

In order to maximize their profit, Carlos and Clarita should choose the combination of cats and dogs that yields the highest total profit.

In order to make a reasonable recommendation, we can create a profit table based on different values of x and y:

x y Total Profit

0 0 0

1 0 6

0 1 15

1 1 21

2 1 27

1 2 30

2 2 36

3 2 42

2 3 45

3 3 51

From the table, we can see that the highest total profit is achieved when they accommodate 3 cats and 3 dogs. This combination would result in a daily profit of $51.

Therefore, my recommendation to Carlos and Clarita would be to prepare for 3 cats and 3 dogs.

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Answer:

x = 6

Step-by-step explanation:

The diagonals of a Rhombus bisect the angles

⇒ 10x - 23 = 3x + 19

⇒ 10x - 3x = 19 + 23

⇒ 7x = 42

⇒ x = 6

Answer:

x = 23

Step-by-step explanation:

The diagonals of a rectangle are equal in size:

2x+10 = 56 (subtract 10 from both sides)

2x = 46 (divide by 2 for both sides)

x = 23

So x = 23