The trigonometric identity has been proven as 1 in the space below
How to prove the identityGiven the identity:
tan(1/2x) (2cot(x) + tan(1/2x)) = 1
To prove this identity, let's start by expressing cot(x) in terms of tan(x):
cot(x) = 1 / tan(x)
Now, replace cot(x) in the given identity:
tan(1/2x) (2(1/tan(x)) + tan(1/2x)) = 1
Now, let's use the double angle formula for tan(x):
tan(x) = 2 * tan(1/2x) / (1 - tan^2(1/2x))
Replace tan(x) in the equation:
tan(1/2x) (2(1/(2 * tan(1/2x) / (1 - tan^2(1/2x))) + tan(1/2x)) = 1
Simplify the equation:
tan(1/2x) ((1 - tan^2(1/2x)) + tan(1/2x) * tan(1/2x)) = 1
Now, notice that the expression in the parentheses is equal to 1:
(1 - tan^2(1/2x)) + tan^2(1/2x) = 1
So, we can simplify the equation further:
tan(1/2x) * 1 = 1
This directly leads us to the original identity:
tan(1/2x) = 1
Thus, the trigonometric identity is proven.
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Enter the correct answer in the box.
The function f(x) = 7x + 1 is transformed to function g through a horizontal compression by a factor of What is the equation of function g?
Substitute a numerical value for k into the function equation.
Answer:
The equation of function g after horizontal compression by a factor of 3 is:
g(x) = 7(3x) + 1 = 21x + 1
To substitute a numerical value for k into the function equation, we need to know the value of k. It is not given in the question, so we cannot provide a numerical answer.
Step-by-step explanation:
When a function is horizontally compressed by a factor of k, we divide the input (x) by k to obtain the new input for the transformed function.
The equation for the horizontally compressed function is:
g(x) = f(x/k)
Substituting f(x) = 7x + 1, we get:
g(x) = 7(x/k) + 1
Simplifying, we get:
g(x) = (7/k)x + 1
Thus, the equation for the transformed function g is obtained by replacing k with the given compression factor.
For example, if the compression factor is 3, then:
g(x) = (7/3)x + 1
And to substitute a numerical value for k, we need to know the value of k. If k = 2, for example, then:
g(x) = (7/2)x + 1
The diameter of a circle is 22 m. Find its area to the nearest whole number.
Answer:
380
Step-by-step explanation:
The formula for the area of a circle is A = pi r^2
radius is half the diameter so you would do 11 (half of diameter) to the second power, which is 121. Now you would do 121 times pi, which would be 380.132711084. When rounded to the nearest whole, it is 380.
SORRY IF THIS DIDN'T MAKE SENSE!
Calculate the length of HG. *
10
9
8
6
5
4
3
N
M
H
3
I
5
square root 15
square root 50
square root 8
square root 51
G
9 10
Using the Pythagorean theorem, we can find the length of HG:
HG² = HI² + IG²
Therefore, the length of HG is 5sqrt(3) units.
What is pythagorean theorem?The Pythagorean Theorem is a mathematical formula that states in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
What is Length?Length is a measurement of the extent of something along its longest dimension, from one end to the other. It is often expressed in units such as meters, feet, or inches.
According to the given information:
To calculate the length of HG, we can use the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of its two legs.
First, we can identify that triangle HGN is a right triangle, with HG being the hypotenuse. Using the Pythagorean theorem, we have:
(HG)² = (NG)² + (NH)²
We can also identify that triangle INH is also a right triangle, with IN being the hypotenuse. Using the Pythagorean theorem, we have:
(IN)² = (NH)² + (NI)²
Substituting (NH)^2 from the second equation into the first equation, we get:
(HG)² = (NG)² + (IN)² - (NI)²
Plugging in the values given in the diagram, we get:
(HG)² = 5² + 9² - 3²
(HG)² = 61
HG = √61
Therefore, the length of HG is the square root of 61.
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Utilizando los siguientes datos: 9, 9, 14, 7, 12, 8, 11, 19, 15, 11, se pueden calcular las siguientes medidas estadísticas:
Media: 11
Mediana: 11
Rango: 12
Moda: 11 (aparece dos veces, junto con el número 9)
Rango intercuartil (IQR): 6
Mínimo y máximo: 7 y 19, respectivamente.
using the same data as above, make the following data displays.
Box plot:
Line plot:
Answer: Box plot: | o
19 |
18 |
17 |
16 |
15 | o
14 | o
13 |
12 | o o
11 | o o o
10 |
9 | o o
8 | o
7 | o
|___________
Line plot: 19|
18|
17|
16|
15| o
14| o
13|
12| o o
11|o o o
10|
9|o o
8| o
7|o
|___________
1 2 3 4 5 6 7 8 9 10 (data index)
Step-by-step explanation:
find the values of variables, then find the lengths of the sides of each quadrilateral
I need help with this question!!!
Answer:
A. m∠X = 50°, AC = 3 cm
Step-by-step explanation:
If ABC is congruent to XYZ then the corresponding angle and sides are congruent so:
∠A = ∠X
∠B = ∠Y
∠C = ∠Z
and
AB = XY
BC = YZ
AC = XZ
Because we know m∠A is 50° and ∠A and ∠X are congruent then m∠X is also 50°
And because we know XZ is equal to 3 cm and XZ and AC are congruent then AC is also 3 cm
What is the image of the point (-1, 2) after a rotation of 270° counterclockwise about the origin?
The image of the point (-1, 2) after a rotation of 270° counterclockwise about the origin is (2, 1).
What is the image of the point?To rotate a point counterclockwise about the origin, we can use the following formulas:
(x', y') = (y,-x)
where (x, y) are the coordinates of the original point, (x', y') are the coordinates of the rotated point
In this case, we want to rotate the point (-1, 2) by 270° counterclockwise about the origin.
So, we have
(x', y') = (2,1)
Therefore, the image of the point (-1, 2) after a rotation of 270° counterclockwise about the origin is (2, 1).
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Factor Special Polynomials:
Factor each of the following expressions, please
Answer:
2) [tex](3x+2)^{2}[/tex]
3) [tex](2x+5)(2x-5)[/tex]
4) [tex](x^{2} +8)(x^{2} -8)[/tex]
Step-by-step explanation:
A method I was taught was the snowflake method whenever there were numbers in front of [tex]x^{2}[/tex] (ex [tex]4x^{2} and 9x^{2}[/tex]) picture shown below (you can search up some videos if you would like)
The snowflake method is really efficient and helps factor polynomials faster
For 3 and 4, you basically have to find the square root of the number without the [tex]x^{2}[/tex] and make sure (for example 25) one parenthesis had -5 and the other had +5. Number 4 had [tex]x^{4}[/tex], [tex]x^{2}*x^{2}[/tex] is [tex]x^{4}[/tex] so you would include that in the equation. To double check, use foil.
Mark to split a 50 pound bag of dog food between a seven dogs how many pounds of food should each dog get
Answer: 7 [tex]\frac{1}{7}[/tex] lbs
Step-by-step explanation:
We will divide the 50 lbs bag by 7, to represent te 7 dogs.
50 lbs food / 7 dogs = 7 [tex]\frac{1}{7}[/tex] lbs
CHED V11.1 A Select Class
babilities
What is the probability that either event will occur?
15
A
12
B
18
21
P(A or B)=P(A) + P(B) - P(A and B)
P(A or B) = [?]
Enter as a decimal rounded to the nearest hundredth.
Enter
The probability that either event will occur is 0.68
What is the probability that either event will occur?From the question, we have the following parameters that can be used in our computation:
The Venn diagram
On the Venn diagram, we have
Total = 15 + 12 + 18 + 12 - 12 + 21
Evaluate
Total = 66
This means that
P(A) = (15 + 12)/66
P(A) = 27/66
Also, we have
P(B) = (18 + 12)/66
P(B) = 30/66
Using the formula of the OR rule of probability:
P(A or B)=P(A) + P(B) - P(A and B)
Substitute the known values in the above equation, so, we have the following representation
P(A or B) = 27/66 + 30/66 - 12/66
Evaluate
P(A or B) = 45/66
This gives
P(A or B) = 0.68
Hence, the solution is 0.68
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please i need your help for today please i would really appreciate it help
The area of circle is represented by quadratic expression A = π · x², for a radius x = 5 units, the area of circle equals 25π square units.
How much is the area of a circle?
In this question we need to derive the area formula for a circle as a function of radius and compute the value of the figure for a given radius. According to geometry, the area of a circle (A), in square units, is described by following quadratic equation:
A = π · x²
Where x is the radius of the circle, in units.
If we know that x = 5, then the area of the circle is now computed:
A = π · 5²
A = 25π
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Adriel just got hired for a new job and will make $65,000 in his first year. Adriel was told that he can expect to get raises of $5,000 every year going forward. How much money in salary would Adriel make in his 29th year working at this job?
Using simple mathematical operations we can conclude that Adriel will earn $2,10,000 after 29 years working at the job.
What are mathematical operations?The order of operations is a rule that outlines the proper steps to take when analyzing a mathematical equation.
The steps that we can memorize using PEMDAS include parentheses, exponents, multiplication and division (from left to right), addition and subtraction (from left to right), and so on.
So, we know that:
Adriel will earn $65,000 this year.
She will get a raise of $5,000 each year.
Then, her salary after 29 years would be:
= (5000 * 29) + 65,000
= 1,45,000 + 65,000
= $2,10,000
Therefore, using simple mathematical operations we can conclude that Adriel will earn $2,10,000 after 29 years working at the job.
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Question 10
Find a polynomial of degree 5 with integer coefficients that has zeros -2, √3i, √2i, and y-intercept of 36.
a polynomial of degree 5 with integer coefficients that has zeros -2, √3i, √2i, and y-intercept of 36 is
[tex]f(x) = x^{5} + 5x^{4} + 13x^{3} + 21x^{2} - 6x + 84[/tex]
How to find the 5 degree polynomial?Let us begin by writing the polynomial factors using the supplied zeros:
Because -2 is a zero, the polynomial has a component of (x + 2).
Because [tex]\sqrt{ 3i }[/tex] is a zero, its conjugate -[tex]\sqrt{ 3i }[/tex] is likewise a zero, implying that the polynomial has a component of (x - [tex]\sqrt{3i}[/tex])(x + [tex]\sqrt{3i}[/tex]) = (x2 + 3).
Similarly, because [tex]\sqrt{2i}[/tex] is a zero, its conjugate -[tex]\sqrt{ 2i }[/tex] is also a zero, implying that the polynomial has a component of (x - [tex]\sqrt{ 2i }[/tex])(x + [tex]\sqrt{ 2i }[/tex]) = (x2 + 2).
Finally, we know that the y-intercept is 36, which means that the polynomial's constant term is 36 when x=0.
When we add these components together, we get:
[tex]f(x) = (x + 2)(x^2 + 3)(x^2 + 2) + 36[/tex]
Extending on this, we get:
[tex]f(x) = x^{5} + 5x^{4} + 13x^{3} + 21x^{2} - 6x + 84[/tex]
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A cafe sells small, medium and large mugs of coffee. A small mug has a capacity of 100 ml. A medium mug has double the capacity of a small mug. A large mug has double the capacity of a medium mug. Work out the capacity, in litres, of a large mug.
Answer:
Answer:
12 ounces
Step-by-step explanation:
Let x = Small Mug
& y = Large Mug
Then we write the given statement in an equation form as below:
2x + 1y = 20 ounces of coffee ---------------> EQUATION 1
3y - 1x = 32 ounces of coffee ----------------> EQUATION 2
Let us solve the above equation for y;
Multiply Equation 2 by '2', we get;
-2x + 6y = 64 ----------------------> Equation 3
Add Equation 1 and Equation 3, we get;
(2x + y) + (-2x + 6y) = 64 + 20
7y = 84
Divide the above equation by 7, we get;
y = 84 / 7
or, y = 12
Hence, One Large mug can hold 12 ounces of coffee.
Step-by-step explanation:
SHARE
Express the ratio below in its simplest form.
4.5: 3 1/2
Answer:
9 : 7
Step-by-step explanation:
4.5 : 3.5 are decimals so we can multiply them both by 2 to get whole numbers. 4.5 doubled is 9 and 3.5 doubles is 7. Therefore the ratio becomes 9:7
I need help with this question
(4x+3)5=2x/6
Consider the following equation. 5x+3y=8 Step 1 of 2 : Determine the missing coordinate in the ordered pair (3,?) so that it will satisfy the given equation.
Answer: (3, -2.333) or (3, -7/3)
Step-by-step explanation:
We're given an x value (x=3) when looking at the coordinate, so we're trying to find what y is equal to given the equation. We can plug in x=3 into 5x+3y=8 and isolate for y.
[tex]5(3)+3y=8\\15 + 3y=8\\3y=-7\\y=-\frac{7}{3}[/tex]
Answer:
y= -2 1/3
Step-by-step explanation:
Since you have x, you put x into the equation and work through the problem.
5(3)+3y=8
15+3y=8, then subtract 15 from both sides
3y=(-7), then divide both sides by 3
y=(-7)/3 or (-2 1/3)
Which phrase can be represented by the expression 25.75x + 10?
The phrase represents that the expression is the cost of a product that is priced at $25.75 per unit with an additional fixed cost of $10.
Which phrase represents the expressionThe expression 25.75x + 10 represents a phrase that involves a numerical value that is dependent on a variable x.
We can think of this expression as a linear function, where x is the independent variable and 25.75x is the dependent variable that is multiplied by a coefficient of 25.75.
The constant term of 10 represents a fixed value that is added to the result of the multiplication.
One possible phrase that can be represented by this expression is the cost of a product that is priced at $25.75 per unit with an additional fixed cost of $10.
The variable x in this case would represent the number of units of the product being purchased.
Multiplying the price per unit by the number of units purchased (25.75x) gives us the variable component of the cost, and adding the fixed cost of $10 to this gives us the total cost of the purchase.
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i need help please also need the answer.
The correct answer as a trinomial in descending degree order is -2x^2 + 6x + 4
How to solve
Here's the step-by-step solution with the figures:
(5x^2 + 10x - 5) - (7x^2 + 4x - 9)
= 5x^2 + 10x - 5 - 7x^2 - 4x + 9 (distribute the negative sign)
= (5x^2 - 7x^2) + (10x - 4x) + ( -5 + 9) (collect like terms)
= -2x^2 + 6x + 4 (simplify and arrange in descending degree order)
Therefore, the simplified expression in descending degree order is -2x^2 + 6x + 4.
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In a microscope, the image of a 0.25-millimeter paramecium appears to be 12 millimeters long. What is the scale factor of the dilation?
The scale factor of the dilation is 48.
The scale factor is a mathematical concept that describes the ratio of the lengths, areas, or volumes of two similar geometric figures. Similar figures have the same shape but may have different sizes.
When one figure is a dilation (a type of transformation that changes the size of a figure without changing its shape) of another figure, the scale factor is the ratio of the corresponding side lengths or dimensions of the two figures.
The scale factor of the dilation is the ratio of the length of the image to the length of the original object. In this case, the length of the image is 12 millimeters and the length of the original object is 0.25 millimeters. Therefore, the scale factor is:
scale factor = length of image/length of the original object
scale factor = 12 mm / 0.25 mm
scale factor = 48
So the scale factor of the dilation is 48.
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Chanasia buys a plastic bucket. The bucket weighs 460 g. The label on the bucket says made with 20% recycled plastic. How many grams of recycled plastic are used to make Chanasia’s bucket?
show work please
Using percentages we know that the bucket which is of 460 grams is made up of 92 grams of recyclable plastic.
What is the percentage?A% is a mathematical figure or ratio that denotes a percentage of 100.
Ratios, fractions, and decimals are some of the numerous ways to represent a dimensionless connection between two numbers.
The symbol "%" is generally written after the integer to denote percentages.
A figure or ratio that is stated as a fraction of 100 is referred to as a percentage in mathematics.
So, in the given situation we will use percentages to get what % of recyclable plastic is used as follows:
= 460/100 * 20
= 4.60 * 20
= 92 grams
Therefore, using percentages we know that the bucket which is of 460 grams is made up of 92 grams of recyclable plastic.
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4. Which of the following angles is vertical to angle /DEB?
A
C
109°
E
D
B
Answer:
A
Step-by-step explanation:
-2
-3
1
Attempt: 1 of
Initial
height
5
A ball is thrown from an initial height of 6 feet with an initial upward velocity of 44 ft/s. The ball's height h (in feet) after t seconds is given by the following.
h=6+44t-16r²
Find all values of t for which the ball's height is 26 feet.
Round your answer(s) to the nearest hundredth.
(If there is more than one answer, use the "or" button.)
ground
IH
seconds
DD
X
S
The values of t for which the height of the ball is of 26 feet are given as follows:
t = 0.57 s and t = 2.18 s.
How to model the situation?The quadratic function that models the height of the ball after t seconds is given as follows:
h(t) = -16t² + 44t + 6.
The height of the ball is of 26 feet when:
h(t) = 26
Hence:
-16t² + 44t + 6 = 26
16t² - 44t + 20 = 0.
The coefficients are given as follows:
a = 16, b = -44, c = 20.
Using a quadratic function calculator, the solutions to the quadratic equation are given as follows:
t = 0.57 s and t = 2.18 s.
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NO LINKS!! URGENT HELP PLEASE!!!
Express the statement as an inequality part 8a^2
Answer:
(g) D, (h) A--------------------
Part (g)'At most' means less than or equal, therefore the quotient of p and q is:
p/q ≤ 6The matching choice is D.
Part (h)'At least' means more than or equal, therefore the reciprocal of w is:
1/w ≥ 9The matching choice is A.
Answer:
g) The correct option is: p/q ≤ 6
The phrase "the quotient of p and q" means p divided by q, which is expressed as p/q. The statement "the quotient of p and q is at most 6" means that the value of p/q cannot exceed 6, but it can be less than or equal to 6. Therefore, we use the less than or equal to a symbol (≤) to represent this statement.
here is an explanation of each option:
p/q = 6: This statement indicates that the quotient of p and q is exactly 6. In other words, p/q can only be equal to 6 and not less than or greater than 6.p/q ≥ 6: This statement indicates that the quotient of p and q can be equal to 6 or greater than 6. In other words, p/q can be any value that is greater than or equal to 6.p/q > 6: This statement indicates that the quotient of p and q can be greater than 6, but not equal to 6. In other words, p/q can be any value that is greater than 6.p/q ≤ 6: This statement indicates that the quotient of p and q can be equal to 6 or less than 6. In other words, p/q can be any value that is less than or equal to 6.p/q < 6: This statement indicates that the quotient of p and q can be less than 6, but not equal to 6. In other words, p/q can be any value that is less than 6.Here's how to express each statement as an inequality in terms of 8a^2:
p/q ≤ 6:
Multiplying both sides by q, we get:p ≤ 6q
Multiplying both sides by 8a^2, we get:8a^2p ≤ 48a^2q
Therefore, the inequality for p/q ≤ 6 in terms of 8a^2 is:8a^2p ≤ 48a^2q
h) The correct option is: 1/w ≤ 9
The reciprocal of a number w is 1/w. The phrase "the reciprocal of w is at least 9" means that the value of 1/w cannot be less than 9, but it can be greater than or equal to 9. Therefore, we use the less than or equal to symbol (≤) to represent this statement.
Here's an explanation of each option:
1/w ≥ 9: This statement indicates that the reciprocal of w is greater than or equal to 9. In other words, 1/w can be any value that is greater than or equal to 9.1/w < 9: This statement indicates that the reciprocal of w is less than 9. In other words, 1/w can be any value that is less than 9.1/w ≤ 9: This statement indicates that the reciprocal of w cannot be less than 9, but it can be greater than or equal to 9. In other words, 1/w can be any value that is greater than or equal to 9.1/w > 9: This statement indicates that the reciprocal of w is greater than 9. In other words, 1/w can be any value that is greater than 9.1/w = 9: This statement indicates that the reciprocal of w is exactly 9. In other words, 1/w can only be equal to 9 and not less than or greater than 9.Here's how to express each statement as an inequality in terms of 8a^2:
1/w ≤ 9:
Multiplying both sides by w, we get:1 ≤ 9w
Subtracting 1 from both sides, we get:0 ≤ 9w - 1
Multiplying both sides by 8a^2, we get:0 ≤ 72a^2w - 8a^2
Therefore, the inequality for 1/w ≤ 9 in terms of 8a^2 is:0 ≤ 72a^2w - 8a^2
Note that the inequality for 1/w ≤ 9 involves a quadratic term with respect to "w," whereas the inequality for p/q ≤ 6 only involves linear terms with respect to "p" and "q."
10 workers can create 20 products in 40days. all workers are equally productive. A company has only 5 workers.
how many days will it take to create all 20 products?
The number of days it will take 5 workers to create 20 products, found using the work rate for 1 worker is 80 days
What is the work rate of 1 worker?Work rate is the rate at which a specified work is completed.
The number of days it will take 10 workers to create 20 products = 40 days
Therefore;
The number of days it will take 1 worker to create 20 products = 40 × 10 days = 400 days
Where 1 worker can create 20 products in 400 days, then;
The work rate for one worker indicates;
The number of days it will take 5 workers to create 20 products can be found by dividing the number of days it will take 1 worker to create the 20 products by 5
5 workers will take = 400 days ÷ 5 = 80 days
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HELP!!!!!!!!!!!!!!!!!
What did I do wrong on 6,7, and 8??
Once there is an answer… Please no one else answer!
The transformation in words for picture 6 is a rotation of 90° counterclockwise or 90° counterclockwise rotation.
The algebraic rule for the above transformation is (x, y) → (-y, x).
The transformation in words for picture 7 is a reflection across the line x = -3.
The algebraic rule for the above transformation is [tex]Ref_{x=-3} \triangle ABC[/tex].
The transformation in words for picture 8 is a reflection across the x-axis.
The algebraic rule for the above transformation is (x, y) → (x, -y).
What is a rotation?In Mathematics and Geometry, the rotation of a point 90° about the center (origin) in a counterclockwise (anticlockwise) direction would produce a point that has these coordinates (-y, x).
By applying a rotation of 90° counterclockwise to the vertices of quadrilateral RSTUVW, the coordinates of W of the image is as follows:
(x, y) → (-y, x)
Ordered pair W = (-1, 3) → Ordered pair R' = (-(3), -1) = (-3, -1).
What is a reflection?In Mathematics and Geometry, a reflection over the x-axis is modeled by this transformation rule (x, y) → (x, -y). Therefore, a reflection over the x-axis would maintain the same x-coordinate while the sign of the x-coordinate changes from positive to negative or negative to positive.
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____ are characteristics of samples, whereas ______ are characteristic of populations.
AConcepts; statistics
BParameters; statistics
CStatistics; parameters
DParameters; estimations
Find the sum or difference.
1. -80+77 =
2. 77 + 160 =
3. -64+ (-33) =
4. 104-(-92) =
5. -105-(-122) =
6. 185-(-154) =
7. -53-(-59) =
8. -6+ (-35) =
9. 15-(-26)-(-39) =
10. -93 +191+ (-179)
Find the product or quotient.
11. 60+ 12 =
12. -194+ (-2)=
13. 88 (-2) =
14. -12 10 =
15. -10 (-11) =
16. 90+ (-6)=
17. 3 (-59) =
18. -7 (-2) =
19. 100 (0) =
20.100/0=
After answering the presented question, we may conclude that the solution of the expressions are as follows.
what is expression ?In mathematics, you can multiply, divide, add, or subtract. An expression is constructed as follows: Number, expression, and mathematical operator A mathematical expression (such as addition, subtraction, multiplication, or division) is made up of numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression or algebraic expression is any mathematical statement that has variables, integers, and an arithmetic operation between them. For example, the phrase 4m + 5 has the terms 4m and 5, as well as the provided expression's variable m, all separated by the arithmetic sign +.
the solution of the expressions are as follows -
-3237-97196173396-4152-8172-196-176-12011084-177140undefined (division by zero is undefined)To know more about expressions visit :-
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What is the area of the regular pentagon?
Answer:300
Step-by-step explanation:
We can find the answer the simple way.
There are 10 triangles in a regular pentagon
We used pythagorean theorm to find side b, 12
12*5*0.5 is 30
30*10 is 300
Help me on number 4. 14 points
Answer:[tex]18\frac{3}{8}[/tex] yards
Step-by-step explanation:
Formula:
Base x Height ---> [tex]3\frac{1}{2}[/tex] x [tex]5\frac{1}{4}[/tex] = [tex]18\frac{3}{8}[/tex]