Answer:
Step-by-step explanation:
To find the arc length of the polar curve r = 8cosθ between 0 and π/4, we can use the formula for arc length:
L = ∫a^b √(r^2 + (dr/dθ)^2) dθ
Substituting in the values for r and dθ, we get:
L = ∫0^(π/4) √(8^2cos^2θ + (-8sinθ)^2) dθ
= ∫0^(π/4) √(64cos^2θ + 64sin^2θ) dθ
= ∫0^(π/4) √(64) dθ
= ∫0^(π/4) 8 dθ
= 8(π/4 - 0)
= 8π/4
= 2π
Therefore, the arc length of the polar curve r = 8cosθ between 0 and π/4 is 2π.
Liam works at a zoo. He was looking at some data showing the masses of their
5
55 African elephants. The mean mass of the elephants was
3
,
800
kg
3,800kg3, comma, 800, start text, k, g, end text, and the median mass was
3
,
600
kg
3,600kg3, comma, 600, start text, k, g, end text. The smallest elephant, named Lola, weighed
2
,
700
kg
2,700kg2, comma, 700, start text, k, g, end text.
Lola then got very sick and lost weight until her mass reached
1
,
800
kg
1,800kg1, comma, 800, start text, k, g, end text.
How will Lola's mass decreasing affect the mean and median?
Median won't change, however Mean will fall by 180.
What do mean and median?By summing all the numbers in a data collection and dividing by the total number of values in the set, one may get the mean (average) of the data set. When a data collection is sorted from least to largest, the median represents the midpoint value.
African elephants total 555 in number.
Elephants had a 3800 kg average mass.
Elephants had a 3600 kg median mass.
Lola, the elephant that was the smallest, weighed 2700 kg.
Later, Lola became really ill and lost a lot of weight, reaching a mass of 1800 kg.
What effect will Lola's weight loss have on the mean and median as well as Currently, the median is 3600 and the lowest is 2700.
now 2700 becomes 1800
so Median remains the same as 3600
No change in Median
The mean mass of the elephants was 3800 kg
=> total weight = 5 x 3800 = 19000 kg
2700 kg becomes 1800 kg
Hence new total mass = 19000 - 2700 + 1800 = 181000 kg
New Mean = 18100/5 = 3620 kg
Mean reduced by 3800 - 3620 = 180 kg
No change in Median weight
Mean reduced by 180 kg
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In the diagram below of triangle PQR, S is the midpoint of PR and T is the midpoint of Q R. If ST=-2x+32, and PQ=7x-2, what is the measure of PQ ?
The measure of PQ in the given triangle is 24.
What is triangle?
Having three edges and three vertices, a triangle is a three-sided polygon. The fact that a triangle's internal angles add up to 180 degrees is its most crucial characteristic. This characteristic is known as the triangle's angle sum property.PQ = ST
7x-2 = -2x+32
7x + 2x = 32 + 2
9x = 34
x = 34/9
The measure of PQ:
PQ = 7x-2
= 7(34/9) - 2
= (238 - 18) / 9
= 24.4
PQ ≅ 24
Hence, the measure of PQ in the given triangle is 24.
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Carter invested $70,000 in an account paying an interest rate of 1.125% compounded continuously. Savannah invested $70,000 in an account paying an interest rate of 1.375% compounded quarterly. To the nearest dollar, how much money would Carter have in his account when Savannah's money has doubled in value?
Step 1: Figure out when Savannah will double her account value:
We need to solve [tex]140000 = 70000\left(1+\frac{0.01375}{4}\right)^{4t}[/tex].
Divide by 70000:
[tex]2 = \left(1+\frac{0.01375}{4}\right)^{4t}[/tex]
Take the natural log of both sides and bring the exponent down:
[tex]\ln(2) = 4t\cdot\ln\left(1+\frac{0.01375}{4}\right)[/tex]
Divide by that mess multiplied by t:
[tex]\dfrac{\ln(2)}{4\ln\left(1+\frac{0.01375}{4}\right)} = t[/tex]
Throw that into a calculator and you get about 50.497297884.
Depending on how picky your teacher is, we'd need to round this to the next time the interest is compounded, since it's only compounded each quarter, so t = 50.5. (The fact is that Savannah's account will never exactly double in value.)
Step 2:
Now, we need to evaluate Carter's continuously compounded investment for 50.5 years:
[tex]B = 70000e^{0.01125\cdot(50.5)}\approx123546.825994[/tex]
B ≈ 123547 to the nearest dollar.
As a comparison, here's the other calculation with the more precise t value.
[tex]B = 70000e^{0.01125\cdot(50.497297884)}\approx123543.070375[/tex]
B ≈ 123543 to the nearest dollar.
Again, I would say the 50.5 calculation is actually more correct, since Savannah's account only compounds the interest each quarter, but you'll have to decide what your teacher would say.
Answer:
$123,543 (nearest dollar)
Step-by-step explanation:
Find the length of time Savannah has to invest her money for it to double.
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+\frac{r}{n}\right)^{nt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]
Given:
A = $140,000P = $70,000r = 1.375% = 0.01375n = 4 (quarterly)Substitute the given values into the formula and solve for t:
[tex]\implies 140000=70000\left(1+\dfrac{0.01375}{4}\right)^{4t}[/tex]
[tex]\implies 140000=70000\left(1.0034375\right)^{4t}[/tex]
[tex]\implies2=\left(1.0034375\right)^{4t}[/tex]
[tex]\implies \ln2=\ln\left(1.0034375\right)^{4t}[/tex]
[tex]\implies \ln2=4t\ln\left(1.0034375\right)[/tex]
[tex]\implies t=\dfrac{\ln 2}{4 \ln\left(1.0034375\right)}[/tex]
[tex]\implies t=50.49729788[/tex]
Therefore, it would take approximately 50.5 years for Savannah's principal investment to double.
[tex]\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Formula}\\\\$ A=Pe^{rt}$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\ \phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}[/tex]
Given:
P = $70,000r = 1.125% = 0.01125t = 50.49729788...Substitute the given values into the formula along with the found value of t and solve for A:
[tex]\implies A=70000e^{0.01125 \times 50.4972...}[/tex]
[tex]\implies A=123543.0704...[/tex]
Therefore, the amount of money that Carter would have in his account when Savannah's money has doubled in value is $123,543 (nearest dollar).
The coordinates of the vertices of AABC are A(0, 0), B(8, 12), and C(16, 6). The points D, E, and F are the midpoints of the
sides BC, AC, and AB respectively. Select all the statements that are true.
The slope of the median CF is
3
5
The median BE is parallel to y-axis.
The length of the median AD is 15 units.
The length of the median AD is 21 units.
The centroid of the triangle is the point (12, 9)
The centroid of the triangle is the point (8, 6)
The correct statements for the triangle are given as follows:
The median BE is parallel to y-axis.The length of the median AD is 15 units.The centroid of the triangle is the point (8, 6).How to obtain the correct statements?The vertices of the triangle are given as follows:
A(0,0), B(8,12), C(16,6).
The midpoints of each side are given by the mean of the coordinates at the extrema of these sides, hence:
D: (12,9).E: (8,3).F: (4,6).The segment BE is a vertical segment, as the x-coordinate is fixed at x = 8, hence ti is parallel to the y-axis.
Given two points, the slope is given by the change in y divided by the change in x. Then the slope of CF is given as follows:
m = (6 - 6)/(16 - 4) = 0.
The length of the segment AD is given as follows:
Length of AD = square root(12² + 9²) = 15 units.
(formula for the distance between two points).
The centroid of a triangle is given by the mean of it's three coordinates, hence:
x = (0 + 8 + 16)/3 = 24/3 = 8.y = (0 + 12 + 6)/3 = 18/3 = 6.More can be learned about the centroid of a triangle at https://brainly.com/question/7644338
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A racetrack charges $85 for each seat in the lower section, $60 for each seat in the upper sections, and $35 for field tickets. There are three times the amount of seats in the upper section as compared to the lower section. The revenue from selling all 22,800 seats is $948,000. How many seats are in the upper section of the racetrack?
Based on a system of equations, the number of seats in the upper section of the racetrack is 3,600.
What is a system of equations?A system of equations or simultaneous equations is two or more equations concurrently or simultaneously solved.
There are four methods for solving simultaneous equations:
MatrixGraphicalEliminationSubstitution.The charge per lower-section seat = $85
The cost per upper-section seat = $60
The fee per field ticket = $35
Let the number of lower-section seats = x
Let the number of upper-section seats = 3x
Let the number of field tickets = y
Equation 1: x + 3x + y = 22,800 or 4x + y = 22,800
85x + 60(3x) + 35y = 948,000
Equation 2: 85x + 180x + 35y = 948,000
From equation 1, y = 22,800 - 4x ...Equation 3
Substitute Equation 3 in Equation 2 to eliminate y:
85x + 180x + 35(22,800 - 4x) = 948,000
85x + 180x + 798,000 - 140x = 948,000
125x = 948,000 - 798,000
125x = 150,000
x = 1,200
The number of seats for each racetrack section:
Lower section seats, x = 1,200
Upper section, 3x = 3,600 (1,200 x 3)
Field tickets, y = 22,800 - 4x
y = 22,800 - 4(1,200)
= 18,000
Check:
85x + 180x + 35y = 948,000
85(1,200) + 180(1,200) + 35(18,000) = 948,000
102,000 + 216,000 + 630,000 = 948,000
948,000 = 948,000
Total revenue:
Lower section seats = $102,000 (85 x 1,200)
Upper section seats = $216,000 (60 x 3 x 1,200)
Field tickets = $630,000 (35 x 18,000)
Total revenue = $948,000
Thus, based on simultaneous equations, we can conclude that there are 3,600 seats in the upper section.
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A container is a hemisphere of radius 30cm.
Sand fills the container at a rate of 4000 cm³ per minute.
Does it take less than a quarter of an hour to fill the container?
You must show your working.
Since the time is 0.24 hour. Yes, it takes less than a quarter of an hour to fill the container
How to determine if it takes less than a quarter of an hour to fill the container?Given: A container is a hemisphere of radius 30cm
Sand fills the container at a rate of 4000 cm³ per minute
A hemisphere is half a sphere. Thus:
The volume of a hemisphere = 1/2 × 4/3 πr³ = 2/3 πr³
The volume of a hemisphere = 2/3 × π × 30 = 56548.67 cm³
Rate = Volume / Time
Time = Volume/Rate
Time = 56548.67 /4000 = 14.14 minutes
In hours: Time = 14.14/60 = 0.24 hour
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Please help! :] 50 Points and awarding Brainliest. Consider the following right pyramid and right prism. Both figures have a base that is a regular pentagon.
Heidi tried to use Cavalieri's principle to show that the two figures have the same volume.
"The base areas are the same. Therefore, corresponding cross-sections have the same area. The figures have the same height. So the volumes must be the same."
What is the first mistake Heidi made?
Choose 1 answer:
(Choice A)
A
The base areas are not the same.
(Choice B)
B
It isn’t true that corresponding cross-sections have the same area.
(Choice C)
C
The heights are not the same.
Answer:
The answer to your question is,
B) It isn’t true that corresponding cross-sections have the same area.
Step-by-step explanation:
They have the same areas and the same heights. but the corresponding cross-section is not the same
{The top of the first figure is a point, but the second one is a flat}
I hope this helps :)
Answer:
B) It isn’t true that corresponding cross-sections have the same area.
Step-by-step explanation:
Cavalieri's principle
If two three-dimensional figures have the same height and the same cross-sectional area at every point along that height, they have the same volume.
From inspection of the given diagram:
The base area of both figures is the same.The height of both figures is the same.The cross-sectional area at every point along the height is not the same in both figures.
16) How many more line segments will be needed to be added to Step 3 to create Step 4 of the
sequence?
Step 1
Step 2
Step 3
Solve the equation A = L • W for W.
Answer:
[tex]W=\frac{A}{L}[/tex]
Step-by-step explanation:
Divide both sides of the equation by [tex]L[/tex].
You brake your car from a speed of 55 mph. The table shows data that represent your car's speed versus the amount of time elapsed from the moment that you began to brake.
Elapsed Time (seconds)
Speed (mph)
1
47
2
39
3
31
4
23
5
23
6
23
Evaluate the following graph based on the data in the table.
A graph has elapsed time (seconds) on the x-axis, and speed (miles per hour) on the y-axis. A line goes through points (1, 47), (2, 39), (3, 31), (4, 23), (5, 23), (6, 23).
For what interval of time is the speed increasing?
For what interval of time is the speed decreasing?
For what interval of time is the speed constant?
a.
None
0 to 4 seconds
4 to 6 seconds
c.
None
0 to 5 seconds
5 to 6 seconds
b.
0 to 5 seconds
None
5 to 6 seconds
d.
0 to 4 seconds
None
4 to 6 seconds
Answer:
Step-by-step explanation:
d. 0 to 4 seconds, None, 4 to 6 seconds.
Based on the data in the table, the speed of the car is increasing for the interval of time from 0 to 4 seconds. After 4 seconds, the speed remains constant at 23 mph for the remainder of the time period. Therefore, the speed is not decreasing at any point in the given time interval.
(100 POINTS AND BRAINLYEST GURANTEE PLEASE I NEED HELP)
Answer everything in the image listed below please.
To prove parallelograms is a parallelograms we have show that, diagonals cut each other in half, and one angle adds to both following angles .
What is parallelograms ?One particular type of quadrilateral that is created by parallel lines is a parallelogram. A parallelogram can have any angle between its adjacent sides as long as its opposite sides are parallel.
The sum of all the sides' lengths determines a rhombus' perimeter. A rhombus's perimeter is equal to four times the length of each side because all of its sides are equal. A diamond's perimeter is given in 4x units, for instance, if one of its sides is x units.
P=4a
The two opposing side pairs must be parallel, the two opposite side pairings must coincide, the two opposite angle pairs must coincide, the diagonals must cut each other in half, the first angle must add to both the following angles, and one set of opposing sides must be both parallel AND congruent. These are the six basic properties of parallelograms.
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12 Work out an expression for the nth term for each sequence.b,20,19.8,19.6,19.4
Answer:
18.2
Step-by-step explanation:
20 + (n-1)-.2
n is the number of the term.
-.2 is the common difference
20 + (10-1) -.2
20 + 9(-.2)
20 - 1.8
18.2
Graph the linear equation by plotting points. x +y = 1
On solving the provided question, we got to know that - the points (0,1) and (1,0) to be used to plot the graph .
What is a linear equation?An algebraic equation of the form y=mx+b (where m is the slope and b is the y-intercept) with one constant and only one first-order (linear) term is called a linear equation. The above is sometimes called a "linear equation in two variables" where x and y are variables.
To draw a straight line, just two points are needed.
Finding the points where the graph intersects is one approach.
[tex]the intercepts of the x and y axes[/tex]
[tex]Let x = 0 in the y-intercept equation.\\ Let y = 0 in the x-intercept equation.[/tex]
x =0→y=0+y=1⇒y=1←y-intercept
y =0→x+0=1⇒x=1←x-intercept
plot the points (0,1) and (1,0)
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Carlos sells tickets for a concert. He earns $200 plus
$2.50 per ticket he sells. So far, Carlos has earned
$300. How many more tickets does Carlos need to sell
to earn $350?
A. 20
B. 50
C. 125
D. 150
Answer:
A. 20
Step-by-step explanation:
300 ÷ 2.5= 120
So
50÷2.5 = 20
Please hurry please help me but correct answer
Answer:
increase of 100%
Step-by-step explanation:
50 cents to 1 dollar is an increase.
The percent increase can be calculated by dividing the increased amount by the original amount.
$1.00 - $0.50 = $0.50 increase
% increase = 100%
Solve the system of equations by using the addition method.
5x+-7y=-2
10x+5y=15
Answer:
y=1, x=1
Step-by-step explanation:
no need for step by step cause i know
The perimeter of a playing field for a certain sport is 302 ft. The field is a rectangle, and the length is 47 ft longer than the width. Find the dimensions. What is the width?
Answer:
To find the dimensions of a rectangle
Step-by-step explanation:
perimeter=302
let width=x
length=x+47
p=2(L+W)
302=2(x+x+47)
302=2(2x+47)
302=4x+94
subtracting 94 from the other side we have
302-94=4x
208=4x
x=52
length=99
width=52
find the circumference of the circle use 3.14 r=6
The circumference of the circle with the dimensions radius = 6 and π = 3.14 is 37.68
What is the circumference of a circle?The circumference of a circle refers to the line that bounds a circle or other two-dimensional figure.
Circumference of a circle = 2πr
Where,
π = 3.14Radius, r = 6Circumference of a circle = 2πr
= 2 × 3.14 × 6
= 37.68
Therefore, the circumference of the circle is given by 37.68
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Which of the following statements are true?
i. It is okay to use the median to estimate the mean since both are a measure of the center of a
distribution.
ii. It is okay to use the standard deviation to estimate the IQR since both are measures of variability.
iii. Point estimates based on a sample are sometimes far from a parameter's value.
Oi only
O ii only
O iii only
Oi and ii
Oi and iii
O i, ii, and iii
Check Answer
The correct answer is c) iii only as statement 1 and statement 2 are false.
What do mean and standard deviation tell us?
In statistics, the measurement of variability known as the standard deviation (SD) is frequently utilized. It demonstrates how different things are from the norm (mean). While a high SD shows that the data are dispersed throughout a wide range of values, a low SD suggests that the data points tend to be close to the mean.
What is relationship between mean and standard deviation?
The standard deviation is a measure that summarizes how far away from the mean each observation is. The positive would exactly balance the negative if the discrepancies themselves were totaled together, making their aggregate equal to zero. The squares of the discrepancies are then added as a result.
What is median?
The value that, when a dataset is arranged, falls exactly in the middle is the median. It is a measure of central tendency that distinguishes between the values' lowest and maximum 50%. Depending on whether you have an odd or an even number of data points, the processes for calculating the median change.
Statement 1: False, it only works if distributiion is syummetric, for non- symmetric distribution, median is not unbiased estimator population in mean.
Statement 2: False standard deviation is not equal to IQR, and is heavily influenced by sample mean and outliers.
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(sec²0 - 4/3) (2sin0 + 1)=0
csc³0 + 2csc²0 - 4/3csc²0 - 8/3=0
2sin²0 + 2sin0 + 1/2 =0
Answer:
2sin²0 + 2sin0 - 1/2 = 0
(2sin0 + 1)(sin²0 - 1/2)= 0
2sin0 + 1 = 0 or sin²0 - 1/2 = 0
2sin0 = -1 or sin0 = ±1/√2
Lauren was given a gift card for a coffee shop. Each morning, Lauren uses the card to buy one cup of coffee. The original amount of money on the gift card was $10 each cup of coffee costs $2. Make a table of values and then write an equation in terms of x, representing the amount money remaining on the card after buying x cups of coffee.
The amount of money left on the gift card, in term of the number of cups, a Lauren has bought is represented by the equation would be A = 10 - 2a.
What is an equation?
An equation is an expression that shows the relationship between two or more numbers and variables.
We are given that from the task content that the total amount of money Lauren has initially is $10.
Hence, each cup of coffee costs $2.
Since the amount of money Lauren has left each time,
That is related to the number of coffee cups bought by the equation;
A = 10 -2a.
where a ranges be between 0 and 5.
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Solve for x. 4 (x + 1 1/3) = 9 2/3 Enter your answer as a mixed number in simplest form in the box.
An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
The value of x is 6.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
4(x + 11/3) = 9(2/3)
Divide both sides by 4.
x + 11/3 = 29/3
Subtract 11/3 on both sides.
x = 29/3 - 11/3
x = (29 - 11) / 3
x = 18/3
x = 6
Thus,
The value of x is 6.
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5
ARK
Simplify each of the following expressions:
4+ (-9) + 3 =
Determine algebraically whether the given function is even, odd, or neither.
f(x) = -5x² + |2x|
well, as you already know, to check if a function is EVEN, we simply check what's f( -x ), if f( -x ) = f( x ), then voila!!. and if f( -x ) = -f( x ), then oddly enough, is an ODD function.
well, let's plug in " -x " to see what we get anyway
[tex]f(x)=-5x^2 + |2x|\implies \boxed{f(x)=-5x^2 + 2x} \\\\[-0.35em] ~\dotfill\\\\ f(-x)=-5(-x)^2 + |2(-x)|\implies f(-x)=-5(-x)(-x) + |-2x| \\\\\\ f(-x)=-5(+x^2) +2x\implies \boxed{f(-x)=-5x^2 + 2x}~\hfill \stackrel{\textit{\LARGE Even}}{f(-x)=f(x)}[/tex]
Find the intercept of the line 4x−3y=17
Answer:
see attachment
Step-by-step explanation:
What is mZQPR?
I need help please help me
Answer: 68
Step-by-step explanation:
Angles in a triangle add to [tex]180^{\circ}[/tex], meaning that [tex]m\angle TRS +m\angle TSR=136^{\circ}[/tex].
Furthermore, since [tex]\triangle TSR[/tex] is isosceles, [tex]m\angle TRS=m\angle TSR[/tex].
Therefore, [tex]m\angle TSR=68^{\circ}[/tex].
Mrs. Silva owns a cake shop and she is working on two wedding cakes this week. The first cake consists of 3 small tiers and 5 medium tiers, which will serve a total of 291 guests. The second one includes 1 small tier and 5 medium tiers, which is enough servings for 247 guests. How many guests does each size of tier serve.. ?
Answer:
Step-by-step explanation:
3 small tiers + 5 medium tiers= 291 guests
1 small tier+5 medium tiers=247 guests
The 2 cakes have the same medium tiers(5 medium tiers). If we subtract 291-247 we will have the number of guest that 2 small tiers serve.
291-247= 44 guests serve per 2 small tiers
44:2=22 guests serve per one small tier.
So if we subtrack the number of guest that one small tier one the cake that has only one small tier we will obtain the number of guests that the 5 medium tiers serve.
247-22=225 guests serve per 5 medium tiers
225:5= 45 guests per 1 mediem tier.
Answer:45 guests serve in a medium tier22 guests serve in a small tier.Sorry, because english isn't my mother tongue.
ABCD is a rhombus and side AE Is congruent to EC prove triangle AEB is congruent to CEB
ABCD is a rhombus and side AE Is congruent to EC. Then triangle AEB is congruent to CEB.
What is a rhombus?A rhombus is one type of parallelogram and has 4 sides and 4 vertices.
Every side is equal in length and opposite angles are equal.
Given that the side AE is congruent to EC of rhombus.
That means, AE ≅ EC.
And the diagonals of rhombus bisects each other.
Both the triangle AEB and triangle CEB share the same side which is BE.
That means BE ≅ BE, by the reflexive property.
And the sides AB and BC are the sides of rhombus.
So, AE ≅ BC.
Therefore, if ABCD is a rhombus and side AE Is congruent to EC. Then triangle AEB is congruent to CEB.
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PLEASE HELP ASAP HURRY PLEASE!! I'LL GIVE BRAINLIEST TO FIRST PERSON TO ANSWER!!!
4y = 164, solve for y.
The value of y that satisfies the equation is 41.
What is the arithmetic operation?
Arithmetic operations involve the study of numbers, the operation of numbers that are useful in all the other branches of mathematics. It basically comprises operations such as Addition, Subtraction, Multiplication, and Division.
To solve this equation for y, you can divide both sides by 4:
4y/4 = 164/4
This simplifies to:
y = 41
Hence, the value of y that satisfies the equation is 41.
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f(x)=x+7 solve for f(3)
Answer:
f(3) = 10
Step-by-step explanation:
f (x) = x +7
plug in 3
f (3) = 3 + 7
add
f(3) = 10