200.96 is the circle's circumference .
What is a circle, exactly?
A circle is a closed, two-dimensional object in which the distances between every point in its plane and its centre are all equal. The creation of the reflection symmetry line is influenced by each line that crosses the circle.
Additionally, any angle surrounding the centre exhibits rotational symmetry. A figure with a circular form that lacks both sides and edges is referred to as a circle. A circle can be referred to in geometry as a closed form, a two-dimensional shape, or a curved shape.
C = 2πr
C = 2 *π * 32
C = 2 x 3.14 x 32
C = 200.96
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To purchase $14,500 worth of restaurant equipment for her business, Debra made a down payment of $1300 and took out a business loan for the rest. After 2 years of paying monthly payments of $585.04, she finally paid off the loan.
(a) What was the total amount Debra ended up paying for the equipment (including the down payment and monthly payments)?
(b) How much interest did Debra pay on the loan?
The total amount Debra ended up paying for the equipment was $28,541.60 and the amount of interest Debra paid on the loan was $14,041.60
(a) To find the total amount Debra ended up paying for the equipment (including the down payment and monthly payments), we need to add the down payment to the total amount of the loan, and then add the total amount of the monthly payments made over the two years.
Total amount of the loan = $14,500 - $1,300 (down payment) = $13,200
Total amount paid = Down payment + Total amount of the loan + Total amount of monthly payments
Total amount paid = $1,300 + $13,200 + ($585.04 x 24) [since there are 24 monthly payments in 2 years]
Total amount paid = $1,300 + $13,200 + $14,041.60
Total amount paid = $28,541.60
Therefore, the total amount Debra ended up paying for the equipment (including the down payment and monthly payments) was $28,541.60.
(b) To find the amount of interest paid on the loan, we need to subtract the total amount borrowed from the total amount paid, and then subtract the down payment. This will give us the total amount of interest paid over the two years.
Total interest paid = Total amount paid - Total amount borrowed - Down payment
Total interest paid = $28,541.60 - $13,200 - $1,300
Total interest paid = $14,041.60
Therefore, the amount of interest Debra paid on the loan was $14,041.60.
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4. Your teacher decides to determine groups in your class by pulling names
out of a hat and not replacing them. What type of event is this?
O independent event
dependent event
Answer:
dependent
Step-by-step explanation:
its dependent because the names getting pulled depends on the names pulled first
which diagram represents the expression |-5|
A diagram that represents the expression |-5| include the following: A. graph A.
What is a number line?In Mathematics and Geometry, a number line simply refers to a type of graph with a graduated straight line which comprises both positive and negative numbers that are placed at equal intervals along its length.
This ultimately implies that, a number line primarily increases in numerical value towards the right from zero (0) and decreases in numerical value towards the left from zero (0).
Since the absolute value of -5 or |-5|, an inequality that matches this graph is represented by the first diagram.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Convert the rectangular coordinates (-√√2, -√2) into polar form.
Express the angle using radians in terms of 7 over the interval
0 ≤0 < 27, with a positive value of r.
The polar form of the rectangular coordinates (-√√2, -√2) is (2√(1 + √2), 15π/28)
Converting into polar formTo convert the rectangular coordinates (-√√2, -√2) into polar form, we first need to find the value of r (the radius) and θ (the angle).
r = √((-√√2)^2 + (-√2)^2) = √(2 + 2√2) = 2√(1 + √2)
To find the value of θ, we can use the following formula:
θ = atan(y/x)
where atan is the inverse tangent function, and (x, y) are the rectangular coordinates.
θ = atan(-√2/(-√√2)) = atan(√2) = π/4 radians
However, we need to express the angle in terms of 7 over the interval 0 ≤ θ < 2π/7, with a positive value of r.
To do this, we can add a multiple of 2π/7 to the value of θ until we get an angle in the desired interval.
θ = π/4 + 2π/7 = (7π + 8π)/28 = 15π/28 radians
So the polar form of the rectangular coordinates (-√√2, -√2) is:
(2√(1 + √2), 15π/28)
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Addition Property of Equality:
If AB = CD, then AB + EF =
The Addition Property of Equality allows us to add EF to both sides of the equation without changing its truth, giving us AB + EF = CD + EF.
We have,
The Addition Property of Equality states that if we add the same quantity to both sides of an equation, the equation remains true.
In the given equation,
AB = CD, we can add the same quantity, EF, to both sides of the equation to obtain:
AB + EF = CD + EF
This is because if two quantities are equal, and we add the same amount to both of them, the result is still equal.
Therefore,
The Addition Property of Equality allows us to add EF to both sides of the equation without changing its truth, giving us AB + EF = CD + EF.
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Mr. Greene created a data set that represents the daily high temperatures, in degrees Fahrenheit, in his city over a week. The values in the table describe the data set. Minimum 73 Mean 77.86 Mean Absolute Deviation 2.73 Median 79 Interquartile Range 6 Maximum 82 Mr. Greene wants to use a single number that summarizes the temperatures for each day. Which value should he use? A. interquartile range B. mean absolute deviation C. mean D. maximum
The correct option is C. mean, that is Mr. Greene should use the mean temperature, which is 77.86°F.
What is a mean value?In mathematics, the mean value is a measure of central tendency of a set of numbers. It is also known as the arithmetic mean and is found by adding all the numbers in a set and dividing the sum by the total number of values. The mean is commonly used to represent the typical or average value of a set of data. It is a useful tool in statistics and other branches of mathematics for analyzing data and making predictions. The mean value can be affected by outliers or extreme values in a set of data, which can skew the results. However, in many cases, the mean provides a reliable and useful measure of central tendency.
Mr. Greene should use the mean temperature, which is 77.86°F, to summarize the temperatures for each day. The mean is a good measure of central tendency when the data set is approximately symmetric and has no significant outliers, which appears to be the case here. Additionally, using the mean will provide a single number that is representative of the entire data set. The interquartile range and mean absolute deviation are measures of spread and do not provide a summary of the temperatures for each day, and the maximum temperature is just one value and may not be representative of the entire data set.
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Trigonometry help me please!
As per the question of trigonometry the another equation that describes the given graph is
y = -cos5x. And after verifying the equation given in the question as well as the new equation we conclude that both equations are equal.
In trigonometry what are the domain and range of the standard functions?The sine, cosine, tangent, cotangent, secant, and cosecant functions are the fundamental geometrical operations in trigonometry. These functions' domain and range are as follows:Sine Function (sin x): All real numbers are its domainRange: [-1, 1]The domain of the cosine function (cos x): all real numbersRange: [-1, 1]Domain: All real numbers, excluding x = (n + 1/2), where n is an integer. Tangent Function (tan x):the entire real number rangeDomain: All real numbers except for x = n, where n is an integer. Cotangent Function (cot x)the entire real number rangeDomain: All real numbers, excluding x = (n + 1/2), where n is an integer. Secant Function (sec x)Range: (-∞, -1] U [1, ∞)Domain: All real numbers except x = n, where n is an integer. What distinguishing characteristics can be found in the sine and cosine function graphs?The sine and cosine function graphs' main characteristics are:
The sine and cosine functions are both periodic, which means that their graphs repeat at regular intervals. The maximum distance between the function value and the horizontal axis is determined by the sine and cosine functions' amplitudes. Zeros: The positions where the sine and cosine functions intersect the horizontal axis are known as the zeros.The functions' maximum and minimum values occur at the period's endpoints. For charting and analyzing the behavior of the sine and cosine functions, it is crucial to comprehend these fundamental characteristics.a. We can use trigonometric identities to simplify the given equation in order to describe the graph using a different equation.
[tex]cos((a+b)/2) = 2 cos(sin(a-b)) sin((a-b)/2)[/tex][tex]Sin((a+b)/2) = cos(a) - cos(b) sin((a-b)/2)[/tex]Using this identity, the given equation can be made simpler:[tex][(2 cos(5x) sin(-4x))/(-2 sin(5x) sin(-4x)] gives y.\\y = -cos(5x)[/tex]So, the equation [tex]y = -cos(5x)[/tex] can be used to describe the graph.b. We can simplify the following equation and compare it with [tex]y = -cos(5x)[/tex]to see if the two equations are equivalent.
We obtain this by applying the same trigonometric identity as before.
Y is equal to [tex][((sinx - sin9x)/(cosx - cos9x)][/tex]Y is equal to [tex][(2 cos5x sin(-4x))/(-2 sin5x sin(-4x)][/tex] (using the identities [tex]cos(a) - cos(b) and sin(a) - sin(b))[/tex][tex]y = -cos5x[/tex]As we can see, the following equation's shortened form is equal.To know more about trigonometric functions visit:
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3. Gabe has $100. He plans to buy a jersey for $55. He will spend the rest of his money on T-shirts, which cost $15 each. Which inequality could Gabe use to find s, the number of T-shirts he can buy?
A 55 + 15s ≤ 100
B 15 + 55s ≤ 100
C 55 + 15s ≤ 100
D 15 + 55s ≤ 100
Question 1(Multiple Choice Worth 1 points)
(06.05 MC)
PLEASE HELP ASAP THIS IS DUE TODAY!
If f(x) = 3x + 1 and g(x) = x − 3, find the quantity f divided by g of 8.
125
25
20
5
The value of the function when the quantity f divided by g of 8 is given by option D. 5.
The function are equal to,
f(x) = 3x + 1
g(x) = x - 3
The expression 'f divided by g of 8' means it is required to evaluate the function f(x) divided by g(x) at x=8.
First, find the individual values of function f(8) and g(8) is equal to,
f(x) = 3x + 1
Substitute the value of x = 8 we get,
⇒ f(8) = 3(8) + 1
⇒ f(8) = 25
Now, for the function g(x),
g(x) = x - 3
Substitute the value of x = 8 we get,
⇒ g(8) = 8 - 3
⇒ g(8) = 5
Then, we can find the value of f(x) divided by g(x) at x=8 which is equal to,
f(x) / g(x)
= ( 3x + 1 )/ ( x - 3 )
= f(8) / g(8)
= 25 / 5
= 5
Therefore, the quantity function f divided by g of 8 is 5, which is option (D).
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Use the Internet to briefly define each of the following terms and explain how they are examples of annuities.
a) RRSP
A registered investment account, or RRSP, enables you to put money aside for retirement while delaying taxes on investment gains.
What is annuities?An annuity is a financial product that provides regular payments over a specified period of time in exchange for a lump sum of money. It's often used as a retirement income source and comes in different types such as fixed, variable, immediate, and deferred annuities. It's important to understand the terms, fees, and charges before purchasing an annuity.
What is the full form of RRSP?Registered Retirement Savings Plan is the full form of RRSP.
In Canada, both workers and independent contractors can save and invest during their retirement through Registered Retirement Savings Plans (RRSP).
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Can anybody help me
The dot plot shows the height, in feet, of a group of trees being studied. How many trees are being studied? Enter the answer in the box.
Answer:
13
Step-by-step explanation:
By counting the amount of dots there is on the graph you would end up with 13 trees.
Need help on this question please! I don’t understand :,(
The area of a parallelogram with base of 10 inches and height of 5 inches is 50 in²
How to solve an equation?An equation is an expression that uses mathematical operations to show how numbers and variables are related to each other.
The area of a parallelogram is the product of its base and height. It is given as:
Area = base * height
For a parallelogram with base of 10 in and height of 5 in:
Area = base * height
Area = 10 in * 5 in = 50 in²
The area is 50 in²
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For a moving object, the force acting on the object varies directly with the object's acceleration. When a force of 27 N acts on a certain object, the acceleration of the object is 9 m/s². If the force is changed to 24 N, what will the object's acceleration be?
*Worth 100 points and will award brainliest for the first correct answer.*
Given:
[tex]\vec F_{1} =27 \ N[/tex]
[tex]\vec F_{2} =24 \ N[/tex]
[tex]\vec a_{1} = 9 \ m/s^2[/tex]
Find:
[tex]\vec a_{2} = ?? \ m/s^2[/tex]
We know that [tex]\vec F= m \vec a[/tex]. Use [tex]\vec F_{1}[/tex] and [tex]\vec a_{1}[/tex] to find mass, m.
[tex]\Longrightarrow \vec F_{1} = m \vec a_{1} \Longrightarrow 27= m(9) \Longrightarrow m= \frac{27}{9} \Longrightarrow m= \boxed{ 3 \ kg }[/tex]
We now know the mass of the moving object we can now find [tex]\vec a_{2}[/tex].
[tex]\Longrightarrow \vec F_{2} = m \vec a_{2} \Longrightarrow 24= (3) \vec a_{2} \Longrightarrow \vec a_{2}= \frac{24}{3} \Longrightarrow \vec a_{2}= \boxed{ 8 \ m/s^2 } \ \therefore \ Sol.[/tex]
A quadratic equation with opposite solutions can be found by multiplying ______
The equation will have ________.
A quadratic equation with opposite solutions can be found by multiplying a variable by its negative value. Specifically, if we start with a quadratic equation of the form:
ax^2 + bx + c = 0
We can find an equation with opposite solutions by multiplying either a or c by -1. For example, if we multiply c by -1, we get:
ax^2 + bx - c = 0
This equation will have opposite solutions if the discriminant, b^2 - 4ac, is negative. This is because the solutions of a quadratic equation are given by the formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
Thus, if the discriminant is negative, then the square root term will be imaginary, which means that the solutions will be complex conjugates of each other.
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A chemical company makes two brands of antifreeze. The first brand is 35% pure antifreeze, and the second brand is 60% pure antifreeze. In order to obtain 70 gallons of a mixture that contains 55% pure antifreeze, how many gallons of each brand of antifreeze must be used?
Therefore , the solution of the given problem of percentage comes out to be 14 gallons of the first brand and 56 gallons of the second brand must be utilised.
What is percentages?In statistics, a figure or metric than may be presented as a percentage or 100 is denoted by the abbreviation "a%". Another unusual spelling is "pct," "pct," and "pc." The percent symbol ("%") is the method that is most usually used for this. Additionally, there are no indications or predetermined ratios of any component to the whole. Numbers are effectively integers since they frequently add up to 100.
Here,
Let's write "x" for the first brand's (35% pure antifreeze) number of gallons and "y" for the second brand's (60% pure antifreeze) number of gallons.
Given:
70 gallons of mixture are required in total.
Desired antifreeze content in the combination is 55%
Based on the information provided, we can construct the following system of equations:
=> Equation 1: x + y = 70
=> Equation 2: 0.5*70 = 0.55x + 0.60y
=> x = 70 - y
=> 0.35(70 - y) + 0.60y = 0.55 * 70
=> 24.5 - 0.35y + 0.60y = 38.5
=> 0.25y = 14
=> y = 56
=> x = 70 - 56
=> x = 14
In order to get 70 gallons of a mixture that contains 55% pure antifreeze, 14 gallons of the first brand and 56 gallons of the second brand must be utilised.
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100 points! Algebra question, photo attached. Please show as much work as possible. Thank you!
Given:
[tex]\sqrt{z+5}+4\leq 13[/tex]
Move 4 to the right side:
[tex]\sqrt{z+5}\leq 9[/tex]
Simplify
[tex]z+5 \leq 81[/tex]
Move 5 to the right side:
[tex]z\leq 76[/tex]
Find singularity points
Find non-negative values for radicals: [tex]z\geq -5[/tex]
Combine the intervals
[tex]z\leq 76 \ \text{and} \ z\geq -5[/tex]
Merge overlapping intervals
[tex]-5\leq z\leq 76[/tex]
Answer:
[tex]\longrightarrow \boxed{\bold{-5\leq z\leq 76}}[/tex]Explain your plan for improving your math skills.
Answer:
To improve my math skills, I plan on doing the following:
1. Practice, practice, practice: I will practice math problems regularly, starting with simple problems and gradually increasing the difficulty level.
2. Use online resources: There are many online resources available that provide tutorials, practice problems, and videos to help with math skills. I will utilize these resources to supplement my learning.
3. Work with a tutor: I will consider working with a tutor if I need additional help with math concepts or if I'm struggling with specific problems.
4. Study regularly: I will set aside time each day to study math, even if it's just for a few minutes.
5. Ask questions: I will ask my teacher or tutor for help if I don't understand a concept or if I'm struggling with a problem.
6. Stay positive: I will maintain a positive attitude towards math and believe in my ability to improve my skills.
By following these steps, I'm confident that I can improve my math skills and become more confident in my abilities.
Answer:
"My plan for improving my math skills is to focus more on understanding and going over new concepts and practice problems. I also will try studying math concepts online and using a method of solving extra problems and applying math in the real world to better understand."
Step-by-step explanation:
Mia hired a moving company. The company charged $500 or its services, and Mia gives the movers a 16% tip.
The requreid, as per the given percentage condition Mia tips the movers $80.
To find the amount of the tip, we need to calculate 16% of the total cost of the services, which is $500.
16% of $500 = 0.16 x $500 = $80
Therefore, Mia tips the movers $80.
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what is the value of sec^-1(1.4)
The calculated value of sec^-1(1.4) is 44.41 degrees
What is the value of sec^-1(1.4)The inverse secant function, denoted as sec^-1(x), returns the angle whose secant is x.
In other words, if secθ = x, then sec^-1(x) = θ.
To find the value of sec^-1(1.4), we need to find the angle whose secant is 1.4.
However, we know that the range of the secant function is [-∞, -1] ∪ [1, ∞], which means that sec^-1(x) is only defined for values of x that fall within this range.
Uisng a graphint tool, we have
sec^-1(1.4) = 44.41
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Match the orange graph above to the corresponding equation.
The equation for the orange graph is the one in the last option:
f(x) = -3x³
Which is the equation for the orange graph?We can see that the orange graph seems to be:
A cubic function.It has a negative leading coefficient, because when x > 0 the graph goes downwards.The graph is narrower than the parent cubic function, then the leading coefficient is larger than 1.From the given options, the only one that matches all these options is the last one:
f(x) = -3x³
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Marta tiene un terreno de forma triangular cuya área es menor a 40 m¨2. La altura de su terreno triangular mide 2 metros más que su base.
Para esta situación, ¿Cuál de las siguientes afirmaciones es correcta? Escoge 1 respuesta:
a) La altura del terreno puede medir entre 2 metros y 10 metros.
b) El menor valor que puede tomar la base del terreno triangular es 2 metros.
c) La base del terreno triangular se encuentra entre 3 metros y 8 metros.
d) El mayor valor entero que puede tomar la altura del terreno es 8 metros.
The correct statement is option a) The height of the terrain can measure between 2 meters and 10 meters.
How to determine triangular terrain?Let the base of the triangular piece of land be x meters. Then, the height of the triangular plot is (x+2) meters.
The area of a triangle is given by the formula: A = (1/2)bh, where b is the base and h is the height.
Substituting the given values:
A = (1/2)x(x+2)
Simplifying:
A = (1/2)(x² + 2x)
Since the area is less than 40 m², write:
(1/2)(x² + 2x) < 40
Multiplying both sides by 2:
x² + 2x < 80
Rearranging and factorizing:
x² + 2x - 80 < 0
(x+10)(x-8) < 0
The roots of the quadratic equation are -10 and 8. Since the coefficient of x² is positive, the parabola opens upwards and is negative in between the roots. Therefore, the inequality is true when -10 < x < 8.
However, x cannot be negative, so the base of the triangular terrain is between 0 and 8 meters.
Substituting the values in the expression for the height, the height is between 2 and 10 meters.
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Is (3, 5) a solution to this system of equations?
y = 5
Y= -5/3x+10
yes
E
Answer:
no
Step-by-step explanation:
-5/3x+10=5
-5/3x=5-10=-5
3x=-5/-5=1
=>x=1/3
Solve the following triangles:
A = 60 degree, a = 4, b = 5
The solution of missing sides and angle is: A = 60°, B = 34.8°, C = 85.2° a = 4, b = 5, c = √21
What is the main purpose of trigonometry and how many kinds of trigonometry are there?Trigonometric functions are used to obtain unknown angles and distances from known or measured angles in geometric drawings.
Basics of Trigonometry deals with measuring angles and problems related to angles. There are six basic trigonometric ratios: sine, cosine, tangent, cosecant, secant, and cotangent. All important trigonometric concepts are based on these trigonometric relationships or functions.
Solving a triangle means determining all its angles and sides. In this case, we are given angle A and sides a and b, and we want to find the remaining angles and sides.
First, we can use the law of cosines c to find the third side:
c² = a²+ b² - 2ab cos(A)
c² = 4² + 5² – 2(4)(5) cos(60)
c² = 16 + 25 - 40 cos (60)
c² = 41-20
c² = 21
c = √21
Now that we have all three sides, we can use the law of cosines to find the remaining angles:
cos(B) = (a²+ c² – b²) / (2ac)
cos(B) = (4² +(√21)² – 5²) / (2(4) (√21))
cos(B) = (16 + 21 - 25) / (8√21)
cos(B) = 12 / (8√21)
B = cos⁻¹ (12 / (8√21))
cos(C) = (a²+ b² – c²) / (2ab)
cos(C) = (4² + 5² – (√21)²) / (2(4)(5))
cos(C) = (16 + 25 - 21) / 40
cos(C) = 20/40
C = cos⁻¹ (20/40)
Simplifying trigonometric functions:
B ≈ 34.8°
C = 85.2°
Therefore, the solution is:
A = 60°, B = 34.8°, C = 85.2°
a = 4, b = 5, c = √21
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Jim recorded the points a group of students scored during a gaming session in the dot plot below. Which measure of center would be best to use for this distribution? A. interquartile range B. mean C. mode D. median
Answer: median
Step-by-step explanation:
A is not a center of measure and nethier ia mode and since the data is skewd it is the median and if it is not skewd it is the mean
Multiply the product
(m+3)(m-3)
Answer:
The answer to (m+3)(m-3) is m^2-9.
Step-by-step explanation:
We will use distributive law to solve this question.
m(m-3)+3(m-3)
= m^2-3m+3m-9
= m^2-9 (since, 3m-3m=0)
As an alternative method, we can use the direct rule learned in algebra: (a+b)(a-b)=a^2-b^2.
so, here a=m and b=3.
(m+3)(m-3)
=m^2-3^2
=m^2-9
Use the figure shown to the right to find the value of x.
x=.
The distance from the center are equal,
x = 6
How to find xThe value of x is solved using the relationship existing between chords and the bisectors.
It will be noted that the line from the center
are perpendicular to the chord and bisects the chord into equal halvesIn the figure, since the chords are equal and the line from the center are perpendicular bisectors then the line from the center are equal.
we can therefore say that x = 6
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Three expressions that are equivalent to 3/5a+10?
Really need on 1 through 10
The locations of the angles in the right triangle, and the number pairs indicates that we get;
ΔMKJ ~ ΔMLK, ΔMKJ ~ ΔKLJ, ΔMLK ~ ΔKLJΔXZW ~ ΔXYZ, ΔXZW ~ ΔZYW, ΔXYZ ~ ΔZYWx = 4.8x = 14 14/29x = 18 6/13x = 11 121/32512·√36·√516·√38·√6What is a right triangle?A right triangle is a triangle that has one 90° interior angle.
1. The three similar triangles are; ΔMKJ, ΔMLK, and ΔKLJ
2. The three similar triangles are; ΔXZW, ΔXYZ, ΔZYW
3. The area of the triangle is; (1/2) × 8 × 6 = 24, which indicates;
(1/2) × 10 × x = 24
x = 24/((1/2) × 10) = 4.8
x = 4.8
4. The area of the triangle is; (1/2) × 21 × 20 = 210, which indicates;
(1/2) × 29 × x = 210
x = 210/((1/2) × 29) = 420/29 = 14 14/29
x = 14 14/29
5. The area of the triangle is; (1/2) × 20 × 48 = 480, which indicates;
(1/2) × 52 × x = 480
x = 480/((1/2) × 52) = 240/13 = 18 6/13
x = 18 6/13
6. The area of the triangle is; (1/2) × 13.2 × 22.4 = 147.84, which indicates;
(1/2) × 26 × x = 147.84
x = 147.84/((1/2) × 26) = 3696/325 = 11 121/325
x = 11 121/325
The geometric mean of a pair of numbers can be obtained as follows;
7. The geometric mean is; √(16 × 27) = √(432)
√(432) = 12·√3
8. The geometric mean is; √(5 × 36) = √(180)
√(180) = 6·√5
9. The geometric mean of the number is; √(24 × 32) = √(768)
√(768) = 16·√3
10. The geometric mean of the numbers is; √(8 × 48) = √(384)
√(384) = 8·√6
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Is the range a subset or a proper subset of the codomain?
A function's codomain is a subset of its range. Although it is not a true subset, it is conceivable for the range to be equal to the codomain, in which case it is also a subset of the codomain.
What is a subset?A set that includes all or a portion of another set's elements is known as a subset. For instance, because every even integer is also an integer, the set of even integers is a subset of the set of integers. A subset that contains some, but not all, of the members of another set is said to be a proper subset. Because it contains some, but not all, of the positive integers, the set of positive integers less than 10 is a valid subset of the set of positive integers.
A function's codomain is a subset of its range. Although it is not a true subset, it is conceivable for the range to be equal to the codomain, in which case it is also a subset of the codomain.
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