Solve the following equation. Then place the correct number in the box provided. 3x + 1 = 2x + 12

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

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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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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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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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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Answer:x=-1

Step-by-step explanation:

y+2=(x+3)

0+2=x+3

to find out x intercept make y 0

2=x+3

2-3=x

-1=x

Answer:

[tex]\displaystyle -\frac{3}{e^2}[/tex]

Step-by-step explanation:

Find f'(x)

[tex]\displaystyle f(x)=\frac{3x}{e^x}\\\\f(x)=3xe^{-x}\\\\f'(x)=3e^{-x}-3xe^{-x}[/tex]

Find f'(2) to get the slope

[tex]f'(2)=3e^{-2}-3(2)e^{-2}\\f'(2)=3e^{-2}-6e^{-2}\\f'(2)=-3e^{-2}\\f'(2)=-\frac{3}{e^2}[/tex]

Answer:

[tex]-\dfrac{3}{e^{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]f(x)=\dfrac{3x}{e^{x}}[/tex]

Differentiate the given function using the quotient rule.

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $f(x)=\dfrac{g(x)}{h(x)}$ then:\\\\\\$f'(x)=\dfrac{h(x)g'(x)-g(x)h'(x)}{(h(x))^2}$\\\end{minipage}}[/tex]

[tex]\textsf{Let\;$u=x^2}[/tex][tex]\textsf{Let\;$g(x)=3x$}\implies g'(x)=3[/tex]

[tex]\textsf{Let\;$h(x)=e^{x}$}\implies h'(x)=e^{x}[/tex]

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

[tex]\begin{aligned}f'(x)&=\dfrac{e^x \cdot 3 - 3x \cdot e^x}{(e^{x})^2}\\\\&=\dfrac{e^x(3 - 3x)}{(e^{x})^2}\\\\&=\dfrac{3 - 3x}{e^{x}}\end{aligned}[/tex]

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

[tex]\begin{aligned}f'(2)&=\dfrac{3 - 3(2)}{e^{2}}\\\\&=\dfrac{3 -6}{e^{2}}\\\\&=-\dfrac{3}{e^{2}}\\\\\end{aligned}[/tex]

Therefore, the slope of the line tangent to the graph of f(x) = 3x/eˣ at the point where x = 2 is:

[tex]\boxed{-\dfrac{3}{e^{2}}}[/tex]

[tex]\hrulefill[/tex]

Differentiation rules used:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Differentiating $ax$}\\\\If $y=ax$, then $\dfrac{\text{d}y}{\text{d}x}=a$\\\end{minipage}}[/tex]

[tex]\boxed{\begin{minipage}{4 cm}\underline{Differentiating $e^{x}$}\\\\If $y=e^{x}$, then $\dfrac{\text{d}y}{\text{d}x}=e^x$\\\end{minipage}}[/tex]

Answer:

x = 9

Step-by-step explanation:

Sum of angles in a polygon with n sides : (n - 2)*180

If the n sides are equal then each angle will be : sum/n

n = 6

sum = (6 - 2)*180

= 4*180

sum  = 720

each angle = 720/6 = 120

⇒ 11x + 21 = 120

⇒ 11x = 120 - 21

⇒ 11x = 99

⇒ x = 99/11

⇒ x = 9

The identity for cos in terms of cos²x is cos 2x = 2cos²x - 1. Solve the identity for cos²x to obtain the power-reducing identity, cos²x =  (1 + cos(2x)) / 2

We need to know that there are six different trigonometric identities.

They are listed as;

From the information given, we have that;

The identity for cos 2x in terms of cos^2x is given by the double-angle identity:

cos 2x = 2 × cos²x - 1

Also, following the power reducing identity, we have that;

cos²x = (1 + cos(2x)) / 2.

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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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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.

Answer:

x = 28

Step-by-step explanation:

40x + 42 = 1162 ( subtract 42 from both sides )

40x = 1120 ( divide both sides by 40 )

x = 28

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:

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:

To calculate the monthly take-home (net) pay, we need to subtract the tax rate from 100% and then multiply it by the gross wage. Let's assume the gross wage is X.

Calculation:

Net pay = (100% - tax rate) * gross wage

Net pay = (100% - 22.4%) * X

Net pay = 77.6% * X

From the given information, the take-home pay is 77.6% of the gross wage. Now, we can substitute the given values to find the monthly take-home pay.

Monthly take-home pay:

Given that the monthly income is the same as the monthly take-home pay, we can substitute the given answer choices to find the correct option:

a) 2,000: This is not the correct answer as it does not match the calculated monthly take-home pay.

b) 2,080: This is the correct answer as it matches the calculated monthly take-home pay of 77.6% of the gross wage.

c) 2,400: This is not the correct answer as it does not match the calculated monthly take-home pay.

d) 1,920: This is not the correct answer as it does not match the calculated monthly take-home pay.

Therefore, the correct answer is b) 2,080, which represents the monthly take-home (net) pay

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]

Answer:

10 root 2

Step-by-step explanation:

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

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

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

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

Answer:

50

Step-by-step explanation:

In trapezium STUV, SV║TU

Since ST = UV, this is an isosceles trapezium and by property,

∠SVU = ∠VST

⇒ ∠SVU = 50

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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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:

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

To seat all 58 children in the hall for the concert, 10 rows are needed.

1. Determine the number of children per row: Since there are 6 seats in a row, each row can accommodate 6 children.

2. Divide the total number of children by the number of children per row: Divide 58 (the total number of children) by 6 (the number of children per row).

  58 ÷ 6 = 9 with a remainder of 4

  This means that 9 full rows can be formed, and there will be 4 children left after the last full row.

3. Check if there are any remaining children: Since there are 4 children remaining, they cannot form a complete row by themselves.

4. Calculate the number of rows needed: Add 1 to the number of full rows to account for the remaining children.

  9 full rows + 1 row for the remaining children = 10 rows in total.

Therefore, to seat all 58 children in the hall for the concert, 10 rows are needed.

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A study based on a population sample that includes 2000 respondents would be the most reliable out of the given options.

In statistical analysis, the reliability of a study depends on the representativeness and size of the sample.

A larger sample size generally provides more reliable results as it reduces the sampling error and increases the precision of the estimates.

Given that the population contains 20,000 people, we need to consider which number of respondents would yield the most reliable study.

Option A: 200 respondents

This represents only 1% of the population.

While it is better than having just 2 respondents, it may not be sufficient to accurately capture the characteristics of the entire population.

Option B: 20 respondents

This represents only 0.1% of the population.

With such a small sample size, the study would likely suffer from a high sampling error and may not provide reliable results.

Option C: 2000 respondents

This represents 10% of the population.

While it is a larger sample size compared to the previous options, it still only captures a fraction of the population.

The study may provide reasonably reliable results, but there is room for potential sampling error.

Option D: 2 respondents

This represents an extremely small sample size, accounting for only 0.01% of the population.

With such a small sample, the study would be highly susceptible to sampling bias and would likely yield unreliable results.

Based on the options provided, option C with 2000 respondents would be the most reliable study.

Although it does not include the entire population, a sample size of 2000 respondents provides a larger representation of the population and reduces the potential for sampling error.

However, it's important to note that the reliability of a study depends not only on sample size but also on the sampling method, data collection techniques, and other factors that ensure representativeness.

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The solutions using few expressions are (b) x = 45 + n * 180 and x = -45 + n * 180

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

tan x = 1

Take the arc tan of both sides

So, we have

x = tan⁻¹(1)

Evaluate the arc tan

x = 45, 225, 405.....

When described in terms of n, we have

x = ±45 + n * 180

When expanded, we have

x = 45 + n * 180 and x = -45 + n * 180

Hence, the solutions are (b) x = 45 + n * 180 and x = -45 + n * 180

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Answer:

x = 7

Step-by-step explanation:

Each angle is a square is 90°

Also the diagonals bisect each angle

⇒ ∠ACB = 90/2

⇒ 11x - 32 = 45

⇒ 11x = 45 + 32

⇒ 11x = 77

⇒ x = 77/11

⇒ x = 7