A circle of radius 1 inch is inscribed in an equilateral triangle. A smaller circle is inscribed at each vertex, tangent to the circle and two sides of the triangle. The process is continued with progressively smaller circles. What is the sum of the circumference of all the circles.
Wrong answers will be reported
Answers
\( \texttt{Refer to the solution above } \)~
\( \texttt{The required sum [ i.e sum of } \)\( \texttt{circumference of all circles } \)\( \texttt{possible ] will be : 3π sq.ft } \)
The sum of the circumferences of all the circles is 4π inches
What is a circle?A circle is a two-dimensional figure with a radius and circumference of 2pir.
The area of a circle is πr².
We have,
Let's start by finding the radius of the second circle.
Since the big circle is inscribed in an equilateral triangle, its diameter is equal to the altitude of the triangle, which is 2 times the radius of the equilateral triangle.
So the diameter of the big circle is 2 inches, and its radius is 1 inch.
The smaller circle touches two sides of the equilateral triangle and the big circle, so it forms a 30-60-90 triangle with the side of the equilateral triangle and the radius of the big circle.
Therefore, the radius of the second circle is half the radius of the big circle or 0.5 inches.
We can repeat this process to find the radius of the third circle, which is half the radius of the second circle, or 0.25 inches.
We can continue to find the radius of the fourth circle, which is half the radius of the third circle, or 0.125 inches, and so on.
The sum of the circumferences of all the circles is:
C = 2πr₁ + 2πr₂ + 2πr₃ + ...
where r₁ is the radius of the big circle, r₂ is the radius of the second circle, r₃ is the radius of the third circle, and so on.
Substituting the values we found, we get:
C = 2π(1) + 2π(0.5) + 2π(0.25) + 2π(0.125) + ...
This is a geometric series with first term a = 2π and common ratio r = 1/2.
The sum of an infinite geometric series with first term a and common ratio r (|r| < 1) is a/(1 - r).
Therefore, the sum of the circumferences of all the circles is:
C = 2π/(1 - 1/2) = 4π inches
Thus,
The sum of the circumferences of all the circles is 4π inches
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Related Questions
find the point on the curve r(t)=(2cost 2sint e^t)
Answers
To find the point on the curve r(t)=(2cost, 2sint, e^t) we need to evaluate the x, y, and z coordinates at a specific value of t. Let's choose t=0 for simplicity.
So, plugging in t=0 we have:
r(0) = (2cos(0), 2sin(0), e^0)
r(0) = (2, 0, 1)
Therefore, the point on the curve at t=0 is (2, 0, 1).
To find the point on the curve r(t) = (2cos(t), 2sin(t), e^t) for a specific value of t, you can plug in the value of t into the parameterized equation:
1. Replace t with the specific value you're interested in (e.g., t = a).
2. Compute 2cos(a) for the x-coordinate.
3. Compute 2sin(a) for the y-coordinate.
4. Compute e^a for the z-coordinate.
The point on the curve will be (2cos(a), 2sin(a), e^a).
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In ΔSTU, m∠S=(11x+3)∘,∠T=(2x+16)∘, and m∠U=(x+7)∘. Find .
Answers
Answer:
x = 11
Step-by-step explanation:
In ΔSTU ,
m∠S = (11x+3)° ; m∠T = (2x+16)° ; m∠U = (x+7)°
According to angle sum property of a triangle , sum of all the interior angles of the triangle is 180°.
So,
\(11x + 3 + 2x + 16 + x + 7 = 180\)
\( = > 14x + 26 = 180\)
\( = > 14x = 180 - 26 = 154\)
\( = > x = \frac{154}{14} = 11\)
What is c so that x + 2 is a factor of 3x³ + cx²-3x - 2 ?
Answers
Answer:
c = -3 for x + 2 to be a factor of 3x³ + cx² - 3x - 2.
Answer:
C = -6
Step by step explanation:
If x + 2 is a factor of 3x³ + cx² - 3x - 2, it means that when we divide 3x³ + cx² - 3x - 2 by x + 2, the remainder is 0. We can use polynomial long division to find the value of c that satisfies this condition.
The division looks like:
3x^2 - 6x + c
x + 2 | 3x^3 + cx^2 - 3x - 2
- (3x^3 + 6x^2)
---------------
-5x^2 - 3x
+(-5x^2 - 10x)
-------------
7x - 2
Since the remainder is not zero, we need to add a term to the dividend to get a zero remainder. We can add the term 5x + 10 to the dividend, which gives:
3x^2 - x + c - 1
x + 2 | 3x^3 + cx^2 - 3x - 2 + 5x + 10
- (3x^3 + 6x^2)
---------------
-5x^2 - 3x
+(-5x^2 - 10x)
-------------
7x - 2 + 10
--------
7x + 8
Now, the remainder is 7x + 8. For x + 2 to be a factor, the remainder must be 0. Therefore, we need to choose c such that 7x + 8 is divisible by x + 2. We can do this by setting x = -2 and solving for c:
7(-2) + 8 = -14 + 8 = -6
Therefore, we need to choose c = -6 so that x + 2 is a factor of 3x³ + cx² - 3x - 2
How would I do this?
Answers
Evaluate the expression below when x = -2. x^2 − 8 / x + 6
Answers
Answer:
14
Step-by-step explanation:
Substitute -2 for x in the two places where x occurs:
8
(-2)^2 - ------- + 6 = 4 + 4 + 6 = 14
-2
A probability is a numerical value that indicates the chance, or likelihood, of a specific event occurring.
Select one:
True
False
Answers
This statement is true
A probability is a numerical value that indicates the chance, or likelihood, of a specific event occurring. Probability is the measure of the likelihood of a random event happening, and it is expressed as a number between 0 and 1, with 0 implying that the event would never occur and 1 implying that it is certain to happen.
The probability of an event happening is referred to as its likelihood. Probability can be expressed as a fraction, percentage, or decimal, and it is a vital tool in statistics. It is frequently utilized in mathematics to estimate real-world situations that involve random variables such as coin flips, weather patterns, stock market trends, and even sporting events.
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please solve this fast
Write the following equation in standard form. Identify the related conic. x' + y2 - 8x - 6y - 39 = 0
Answers
The standard form of the equation is (x' - 8x) + (y - 3)^2 = 48
To write the equation in standard form and identify the related conic, let's rearrange the given equation:
x' + y^2 - 8x - 6y - 39 = 0
Rearranging the terms, we have:
x' - 8x + y^2 - 6y - 39 = 0
Now, let's complete the square for both the x and y terms:
(x' - 8x) + (y^2 - 6y) = 39
To complete the square for the x terms, we need to add half the coefficient of x (-8) squared, which is (-8/2)^2 = 16, inside the parentheses. Similarly, for the y terms, we need to add half the coefficient of y (-6) squared, which is (-6/2)^2 = 9, inside the parentheses:
(x' - 8x + 16) + (y^2 - 6y + 9) = 39 + 16 + 9
(x' - 8x + 16) + (y^2 - 6y + 9) = 64
Now, let's simplify further:
(x' - 8x + 16) + (y^2 - 6y + 9) = 8^2
(x' - 8x + 16) + (y - 3)^2 = 8^2
(x' - 8x + 16) + (y - 3)^2 = 64
Now, we can rewrite this equation in standard form by rearranging the terms:
(x' - 8x) + (y - 3)^2 = 64 - 16
(x' - 8x) + (y - 3)^2 = 48
The equation is now in standard form. From this form, we can identify the related conic. Since we have a squared term for y and a constant term for x (x' is just another variable name for x), this equation represents a parabola opening horizontally.
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let's say you have a bag with 14 cherries, 10 sweet and 4 sour. if you pick a cherry at random, what is the probability that it will be sweet? write your answer as a reduced fraction.
Answers
Therefore, the probability of picking a sweet cherry at random from the bag is 5/7.
How does probability operate using an example?It is based on the possibility that something might happen. One important aspect of probability theory is the justification for probability. For instance, the theoretical chance of getting a head while tossing a coin is 1 in 2.
The probability of picking a sweet cherry at random can be calculated by dividing the number of sweet cherries by the total number of cherries in the bag:
Probability of picking a sweet cherry = Number of sweet cherries / Total number of cherries
Probability of picking a sweet cherry = 10 / (10 + 4)
Probability of picking a sweet cherry = 10/14
Simplifying the fraction by dividing both numerator and denominator by their greatest common factor (which is 2), we get:
Probability of picking a sweet cherry = 5/7
Therefore, the probability of picking a sweet cherry at random from the bag is 5/7.
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a Find the Q2 and Q; of the following data. 5500, 6800, 4200, 3900, 7800, 6150, 4400, 7500 5 and 12 + 1 are in
Answers
Step-by-step explanation:
Quartile Formula:
Formula for Lower quartile (Q1) = N + 1 multiplied by (1) divided by (4)
Formula for Middle quartile (Q2) = N + 1 multiplied by (2) divided by (4)
Formula for Upper quartile (Q3) = N + 1 multiplied by (3) divided by (4)
Formula for Interquartile range = Q3 (upper quartile) – Q1 (lower quartile)
i don't know how to work it out but i gave you some information.
a six-sided die (with numbers 1 through 6) and an eight-sided die (with numbers 1 through 8) are rolled. what is the probability that exactly one 6 is rolled? express your answer as a common fraction.
Answers
When a six-sided die and an eight-sided are rolled then the probability that exactly one 6 is rolled is 1/4.
Given, a six-sided die (with numbers 1 through 6) and an eight-sided die (with numbers 1 through 8) are rolled.
we have to find the probability that exactly one 6 is rolled.
First we have to check the total number of outcomes on rolling both the dices.
Now, the total number of outcomes be, 48
As, we the favorable outcomes be, the outcome with exactly one 6 on the dices.
Favorable Outcomes = { (1,6) , (2,6) , (3,6) , (4,6) , (5,6) , (6,1) , (6,2) , (6,3) , (6,4) , (6,5) , (6,7) , (6,8)}
The total number of favorable outcomes be, 12
So, the probability that exactly one 6 is rolled is
P(E) = Number of favorable outcomes / Total number of outcomes
P(E) = 12/48
P(E) = 1/4
Hence, the probability that exactly one 6 is rolled is 1/4.
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Determine whether the statement is true or false. Prove the statement directly from the definitions if it is true, and give a counterexample if it is false.
For all integers a, b, and c, if a is a factor of c and b is a factor of c then ab is a factor of c.
Answers
The statement "For all integers a, b, and c, if a is a factor of c and b is a factor of c then ab is a factor of c." is true.
To prove this statement directly from the definitions, we need to show that if a and b are factors of c, then ab is also a factor of c. By definition, a factor of a number divides that number evenly without any remainder.
Let's assume that a and b are factors of c, which means that c can be expressed as c = ma and c = nb, where m and n are integers. Multiplying these equations, we get:
c x c = (ma) x (nb)
Expanding the right-hand side of this equation, we get:
\(c^2\) = mnb x ab
Since m, n, and ab are all integers, we can conclude that ab is also a factor of \(c^2\). Now, we know that c is a factor of\(c^2\) , since any number is a factor of itself. Therefore, if we divide both sides of the equation \(c^2\) = mnb x ab by c, we get:
c = (mn) x ab
This shows that ab is also a factor of c, since c can be expressed as the product of (mn) and ab.
Therefore, we have shown that if a and b are factors of c, then ab is also a factor of c, which means the statement is true.
Counterexample: If we take a = 3, b = 5, and c = 7, we can see that a and b are both factors of c (since 7 is a prime number and its only factors are 1 and 7), but ab = 15 is not a factor of c. This provides a counterexample to the statement and shows that it is false in general.
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What dose rotations mean in math
Answers
Answer:
A rotation is a transformation that turns a figure about a fixed point called the center of rotation. • An object and its rotation are the same shape and size, but the figures may be turned in different directions. • Rotations may be clockwise or counterclockwise.
Step-by-step explanation:
Answer:it means the movitem of a figure to somewhere else
Step-by-step explanation:
Express as integer: a deposit of $100
Answers
Answer:
Well if you deposit something you won't have it so negative. If you earn something then it is positive. So it is negative
Step-by-step explanation:
:)
Lionel walks 3/5 mile in 1/3 of a day. Cristiano walks 5/8 mile in 2/6 of a day.
Who walked farther, Cristiano or Lionel? How much did each boy walk in 1 day?
Answers
Answer:
Cristiano walked farther In 1 day: Lionel walked 1.8 mile Cristiano walked 1.875 mileStep-by-step explanation:
2/6 = 1/3 so we just need to know what is more 3/5 or 5/8
3/5 = (3•8)/(5•8) = 24/40
5/8 = (5•5)/(8•5) = 25/40
25 > 24 so 5/8 > 3/5
so Cristiano walked farther
They walk for 1/3 of a day so in a whole day they walk three times the distance
3/5 • 3 = 9/5 = 1 4/5 = 1.8 mile
5/8 • 3 = 15/8 = 1 7/8 = 1.875 mile
In a density curve, proportion of data is represented by__
A. area above the curve. B. area under the curve.
C. the shape of the curve.
D. intervals of values on the vertical axis.
E. intervals of values on the horizontal axis.
F. the length of the curve.
Answers
Option B. Area under the curve. In a density curve, the proportion of data is represented by the area under the curve.
The area under the curve represents the proportion of data points that fall within a certain range of values, and the height of the curve at any given point represents the relative frequency or density of data points at that value. The total area under the curve is equal to 1, indicating that the sum of the proportions of data points in all possible intervals is equal to 100%.
It's important to note that the total area under the curve is equal to 1, indicating that the sum of the proportions of data points in all possible intervals is equal to 100%. This means that the density curve provides a complete picture of the distribution of the data, and the shape of the curve can be used to make inferences about the distribution, such as whether it is symmetrical, skewed, bimodal, etc.
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Aaron answered 7 out of 12 questions correctly on his last English quiz. Find his grade
to the nearest whole percent.
Answers
which would give you 0.58
so that’s 58%
Answer:
Step-by-step explanation:
58%
7/12 into a percentage
2. Todos los días hay muchos vuelos de Guadalajara a Monterrey. Los datos de abajo muestran los minutos de retraso (o anticipación) de un vuelo a Monterrey, de una muestra de 5 vuelos. Para explicarlo, un número positivo indica que el vuelo está retrasado, un valor de 0 señala que llegó a tiempo, y un número negativo, que llegó con anticipación. Así que el primer vuelo llegó 4 minutos tarde, y el último vuelo con 10 minutos de anticipación. 4 12 -9 6 -10 a) Determine la cantidad media de vuelos que llegaron tarde (o antes). b) Determine la cantidad mediana de vuelos que llegaron tarde (o antes).
Answers
Hay 5 vuelos en total, por lo que el valor en el medio es el tercer valor. La cantidad mediana de vuelos que llegaron tarde (o antes) es de 4 minutos, ya que es el tercer valor ordenado.
a) Determine la cantidad media de vuelos que llegaron tarde (o antes): Para determinar la cantidad media de vuelos que llegaron tarde (o antes), es necesario sumar todos los tiempos de los vuelos y luego dividir esa suma por la cantidad total de vuelos. Si el resultado es un número positivo, entonces la cantidad media de vuelos llegaron tarde; si es negativo, entonces la cantidad media de vuelos llegaron temprano.
4 + 12 - 9 + 6 - 10 = 3 minutos
La cantidad media de vuelos que llegaron tarde es de 3 minutos.b) Determine la cantidad mediana de vuelos que llegaron tarde (o antes):
La mediana es el valor que está en el medio de un conjunto de datos ordenados. Para encontrar la mediana de los tiempos de los vuelos, primero se deben ordenar los datos de menor a mayor.-10, -9, 4, 6, 12
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Which of the following equations shows how substitution can be used to solve the following system of equations?
Answers
By substituting the first equation into the second, we get:
2x + 5*(2x +3) = 3
The correct option is the second one.
Which equation shows how substitution?Here we have the system of equations:
y = 2x + 3
2x + 5y = 3
You can see that the variable "y" is already isolated in the first equation, so we can just substitute it in the second equation:
2x + 5*(2x +3) = 3
This is what we can see in the second option, so we conclude that the second option is the correct one.
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If an artile bought for US $ 1 is sold for sterling £1, what is the gain percent? (Given, \$ 1 = Rs. 105.52, £1= Rs. 155.63)
Answers
Answer:
47.49%
Step-by-step explanation:
profit / selling price ×100% = gain percent
profit =selling price - cost price
therefore profit = 155.63 - 105.52 = 50.11
50.11/105.52 ×100% = gain percent
47.49% = gain percent
Can someone pls help me :))
Answers
Answer:
Angle three and six has the same angle. This is because when the triangle is place on a parralel line, the angle 3 and 6 which is place at ziz zag as the same value. Besides, angle1 and 2 has the same value with angle 7 with using the same format. Thank you and please rate me as brainliest.
Chess top uses the periodic inventory system. For the current month, the beginning inventory consisted of 200 units that cost p65 each. During the month, the company made two purchases: 300 units at p68 each and 150 units at p70 each. Chess top also sold 500 units during the month. Using the average cost method, what is the amount of ending inventory?.
Answers
The ending inventory amount is p68,000: (200 x 65) + (300 x 68) + (150 x 70) - (500 x 68) = 68,000.
1. Calculate the total cost of the beginning inventory: 200 units x p65 = p13,000
2. Calculate the total cost of the first purchase: 300 units x p68 = p20,400
3. Calculate the total cost of the second purchase: 150 units x p70 = p10,500
4. Calculate the total cost of the units sold: 500 units x p68 = p34,000
5. Calculate the total cost of the ending inventory amount : (13,000 + 20,400 + 10,500) - 34,000 = p68,000
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I NEED AN ANSWER ASAP PLEASE
Answers
Step-by-step explanation:
the correct answer is option a 6a-7
When is the function positive?
Answers
When f(x) > 0.
Write the linear equation in slope-intercept form ( y = mx + b )
Answers
Answer:
B. Y= 2x - 4
Step-by-step explanation:
I think, so sorry if wrong
I needes y on its own so I devided the equation by 3
From the diagram below, given the side lengths marked, and if we know that < C is congruent to < E, we can say that ___.
Select one:
a.
the two triangles are similar by SAS
b.
the two triangles are not similar
c.
the two triangles are congruent
d.
the two triangles are similar by AA
Answers
Answer:
d) the two triangles are similar by AA
Step-by-step explanation:
Someone please help with this question
Tony and his cousin are playing a game where they pick up colored sticks. Tony currently has 12 points and likes to pick up the yellow sticks, earning 10 points every turn. His cousin just lost all his points on the previous turn, and has a strategy to catch up by getting all the blue ones, earning 14 points per turn. In a certain number of turns, the score will be tied. How many points will they each have? How long will that take?
Tony and his cousin will each have __ points in __ turns.
Answers
Answer:
Tony and his cousin will each have 42 points in 3 turns.
Step-by-step explanation:
10 x 3 = 30
30 + 12 = 42
14 x 3 = 42
Sarah is building a birdhouse the nails she uses are 1 inch long the wood board is 1 foot long how many times smaller is the nails compared to the wood
Answers
answer for this 3+4×5+7
Answers
Answer:
Answer is 84
Step-by-step explanation:
3+4×5+7
7×12
84
I hope it's helpful!
marked a increments of 5 s and the yertical axil in marked in increments st 1mil. (a) o th 10.00÷ min (8) 6 in 20−00= (c) 10.0000000.00 mes: (d)20.00 to 35.00 s miss (ie) 0 to 40.00 s
Answers
The given graph is a rectangular hyperbola graph because the product of the variables, that is x and y, is constant. The equation of a rectangular hyperbola is y=k/x. k is the constant value. The variables x and y are inversely proportional to each other.
Thus, as x increases, y decreases, and vice versa.GraphA rectangular hyperbola graph with labeled axesThe horizontal axis is labeled in increments of 5s. The vertical axis is labeled in increments of 1mil. a) On the graph, 10.00 ÷ min is 0.1mil. Thus, 10.00 ÷ min corresponds to a point on the graph where the vertical axis is at 0.1mil.b) At 6 in 20-00, the horizontal axis is 6, which corresponds to 30s.
The vertical axis is 20-00 or 2000mil, which is equivalent to 2mil. The coordinates of the point are (30s, 2mil).c) At 10.0000000.00 mes, the horizontal axis is at 100s. The vertical axis is 0, which corresponds to the x-axis. The coordinates of the point are (100s, 0).
d) From 20.00 to 35.00s, the vertical axis is at 4mil. From 20.00 to 35.00s, the horizontal axis is at 3 increments of 5s, which is 15s. The coordinates of the starting point are (20.00s, 4mil). The coordinates of the ending point are (35.00s, 4mil). The point on the graph is represented by a horizontal line segment at y=4mil from x=20.00s to x=35.00s. Similarly, from 0 to 40.00s, the coordinates of the starting point are (0, 10mil).
The coordinates of the ending point are (40.00s, 0). The point on the graph is represented by a curve from (0, 10mil) to (40.00s, 0).
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I NEED HELP!!
Pls help
Answers
Answer:
Number 4
Step-by-step explanation:
A theoretical probability chance is a probably where all the other chances are equal. In this case, you can roll 6 different types of numbers on the dice: 1, 2, 3, 4, 5, and 6. And the number of times the dice was rolled was 60 times. So, we can divide 60/6 because there were 60 rolls and 6 different possibilities. 60/6=10, so 10 is the theoretical probability. As you can see, the graph shows that the number 4 was rolled 10 times, which means that number 4 is the experimental probability that matched the theoretical probability.