The side lengths of a triangle are in the ratio 2:3:4. The longest side has a length of 24cm. What is the perimeter of the triangle? Give answer in centimetres.

Answer:

2x^3-5x^2-28x+15 -------------------------expanded

2x^3-5x^2-28x+15--------------------simplified

Step-by-step explanation:

\left(2x-1\right)\left(x+3\right)\left(x-5\right)

=2x^2x+2x^2\left(-5\right)+5xx+5x\left(-5\right)-3x-3\left(-5\right)

simplyfied

\left(2x^2+5x-3\right)\left(x-5\right)

=2x^2x+2x^2\left(-5\right)+5xx+5x\left(-5\right)-3x-3\left(-5\right)

=2x^3-5x^2-28x+15

Answer:

Step-by-step explanation:

the two other notations can be any from the following,

6/7, 6to7, 6<7

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

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:

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

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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a) Comparing the two functions, Scooter's business has a lower slope of 15 compared to Karissa's business with a slope of 35. b) Both Scooter and Karissa earned a total of $500 in that month. Therefore, they had an equal month in terms of total money earned.

a) In the context of Scooter's business, the slope of the function f(x) = 200 + 15x represents the rate at which the total money earned increases with each additional sale. Each unit increase in x (number of sales) results in an increase of $15 in total money earned. The y-intercept of 200 represents the initial amount of money Scooter already had before making any sales.

In the context of Karissa's business, the slope of the function f(x) = 80 + 35x represents the rate at which the total money earned increases with each additional repair job. Each unit increase in x (number of repairs) results in an increase of $35 in total money earned. The y-intercept of 80 represents the initial amount of money Karissa already had before doing any repair jobs.

Comparing the two functions, Scooter's business has a lower slope of 15 compared to Karissa's business with a slope of 35. This means that Karissa's business earns more money per repair job compared to Scooter's business per t-shirt sale.

b) To determine who had the better month, we need to calculate the total money earned by each individual based on the given sales/repairs.

For Scooter, with x = 20 (number of shirts sold):

f(x) = 200 + 15x

f(20) = 200 + 15 * 20

f(20) = 200 + 300

f(20) = 500

For Karissa, with x = 12 (number of repairs):

f(x) = 80 + 35x

f(12) = 80 + 35 * 12

f(12) = 80 + 420

f(12) = 500

Both Scooter and Karissa earned a total of $500 in that month. Therefore, they had an equal month in terms of total money earned.

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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 length of EB is approximately 14.6 units when rounded to the nearest tenth.

To find the length of EB, we can use the property of similar triangles in this diagram. By looking at triangle DFE and triangle CFB, we can see that they are similar triangles.

Using the similarity ratio, we can set up the proportion:

DF / CF = DE / EB

Plugging in the given values, we have:

13.1 / 10.9 = 17.5 / EB

To find EB, we can cross-multiply and solve for EB:

13.1 * EB = 10.9 * 17.5

EB = (10.9 * 17.5) / 13.1

EB ≈ 14.6

Therefore, the length of EB is approximately 14.6 units when rounded to the nearest tenth.

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Answer:

x = 28

Step-by-step explanation:

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

40x = 1120 ( divide both sides by 40 )

x = 28

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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The number of pounds in X kilograms is 2.2X pounds.

Given that,

Number of pounds in 1 kilogram = 2.2 pounds

1 kilogram = 2.2 pounds

X kilograms =?

After cross-multiplication,

Number of pounds = 2.2X

Therefore, there will be 2.2X pounds in 1 kilogram.

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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

Answer:

x = 53

Step-by-step explanation:

BCDE is a trapezoid

Therefore diagonals are equal

BD = EC

8x-27 = 2x+33

8x-2x = 33+27

6x = 60

x = 10

DB = 8x-27

= 8*10 - 27

= 80 - 27

= 53

The completed frequency table is:

Interval   |   Tally   |   Frequency

0-2       |   |||        |   6

3-5       |   |||        |   6

6-8       |   ||||||    |   8

9-11      |   |          |   1

12-14    |   ||        |   2

To complete the frequency table, we need to count the frequency of each interval. We can start by organizing the data in ascending order:

0, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 8, 9, 10, 10, 13, 13

Now, let's count the frequency for each interval:

0-2: 6 (0, 1, 1, 2, 2, 2)

3-5: 6 (3, 3, 4, 4, 5, 5)

6-8: 8 (6, 6, 6, 6, 6, 6, 7, 8)

9-11: 1 (9)

12-14: 2 (10, 10, 13, 13)

Now we can complete the frequency table:

Interval   |   Tally   |   Frequency

0-2       |   |||        |   6

3-5       |   |||        |   6

6-8       |   ||||||    |   8

9-11      |   |          |   1

12-14    |   ||        |   2

The tally marks represent the frequency count in each interval, where each vertical line represents one count. The total frequency is obtained by counting the number of tally marks in each interval.

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Answer:

1.) inverse proportion : 1kx = y

2) direct proportion. y = kx

The angle m∠JIX is 90 degrees.

IX is perpendicular to IJ. Therefore, angle m∠JIX is 90 degrees.

IG bisect CIJ. Hence,

m∠CIG ≅ m∠GIJ

Therefore,

m∠CIX = 150 degrees

Hence, let's find m∠JIX.

Therefore, m∠JIX is 90 degrees because IX is perpendicular to IJ. Perpendicular lines forms a right angle.

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The measure of angle m∠JIX is  estimated to be 90⁰.

You should understand that an angle is a figure formed by two straight lines or rays that meet at a common endpoint, called the vertex.

IX is perpendicular to IJ. Therefore, angle m∠JIX is 90⁰.

Frim the given parameters,

IG⊥CIJ.

But;  m∠CIG ≅ m∠GIJ

⇒ m∠CIX = 150⁰

Hence, let's find m∠JIX.

Therefore, m∠JIX is 90⁰ because IX is perpendicular to IJ. Perpendicular lines forms a right angle.

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The expression [tex]\displaystyle\sf \cos^{2}(x)+\sin^{2}(x)[/tex] represents the fundamental trigonometric identity known as the Pythagorean identity. It states that the square of the cosine of an angle plus the square of the sine of the same angle is always equal to 1.

Therefore, [tex]\displaystyle\sf \cos^{2}(x)+\sin^{2}(x)=1[/tex].

This identity holds true for any value of [tex]\displaystyle\sf x[/tex].

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

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:

  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:

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

Part 1: The base fee charged by the ride-share service is $4.

Part 2: The rate of increase in cost in dollars per mile is $2.7.

Part 3: The slope and y-intercept of the equation are $2.7 and $4 respectively.

Part 4: The slope-intercept equation of the line in the graph is y = 2.7x + .

Part 5: The cost of a 20-mile ride is $58.​

In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;

y = mx + b

Where:

Part 1.

The base fee is the initial fee or y-intercept, and based on the graph, it is equal to $4.

Part 2.

Rate of increase = (y₂ - y₁)/(x₂ - x₁)

Rate of increase = (12 - 4)/(3 - 0)

Rate of increase = $8/3 or $2.7.

Part 3.

The slope and y-intercept of the equation in this graph are $2.7 and $4 respectively.

Part 4.

Next, we would write the slope-intercept equation of the line as follows;

y = mx + b

y = 2.7x + 4

Part 5.

When x = 20 miles, the cost is given by;

y = 2.7x + 4

y = 2.7(20) + 4

y = $58.

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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

x = 301/56

Algebra is the branch of mathematics that deals with numbers and values which are represented with letters and symbols.

Sometimes, we do not want to mention a particular number, we can represent the number by a letter or a suitable symbol. This approach is algebraic.

For example, d + d = 2d

This is an example of an algebraic expression

Given the algebraic expression,

[tex]3\frac{1}{8} \div (x - 4 \frac{7}{28} ) = \frac{17}{18} + 1 \frac{5}{6} [/tex]

Step 1:

Simplify the expression by converting the mixed numbers into improper fractions:

25/8 ÷ ( x - 119/28) = 17/18 + 11/6

Step 2: Simplify the right side of the equation

25/8 ÷( x - 119/28) = 17/18 + 33/18

25/8 ÷( x - 119/28) = 50/18

Step 3: Multiply both sides of the equation by (x - 119/28)

25/8 = 50/18 × (x - 119/28)

Step 4: Multiply both sides of the equation by (18/50)

25/8 × (18/50) = (x - 119/28)

Step 5: Simplify the left side of the equation

9/8 = (x - 119/28)

Step 6: Add 119/28 to both sides of the equation

9/8 + 119/28 = x

Step 7: Calculate the value of x:

x = 9/8 + 119/28

= 63/56 + 238/56

= 301/56

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Answer:

The first mechanic worked 5 hours

The second mechanic worked 15 hours

======================================================

Work Shown:

x = number of hours the 1st mechanic worked

y = number of hours the 2nd mechanic worked

75x = total amount charged by the 1st mechanic

95y = total amount charged by the 2nd mechanic

75x+95y = total charged both mechanics combined

75x+95y = 1800 is the first equation to set up

The other equation is x+y = 20 since the total combined time spent was 20 hours.

Solve for y to get y = 20-x

Then plug it into the first equation to solve for x.

75x+95y = 1800

75x+95( y ) = 1800

75x+95(20-x) = 1800   ....... replace y with 20-x

75x+1900-95x = 1800

-20x+1900 = 1800

-20x = 1800-1900

-20x = -100

x = -100/(-20)

x = 5

Now we can determine the value of y

y = 20-x

y = 20-5

y = 15

The first mechanic worked 5 hours.

The second mechanic worked 15 hours.

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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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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