Answer: Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle?
$\textbf{(A)}\ \frac{1}{6}\qquad \textbf{(B)}\ \frac{1}{4}\qquad \textbf{(C)}\ \frac{2-\sqrt{2}}{2}\qquad \textbf{(D)}\ \frac{1}{3}\qquad \textbf{(E)}\ \frac{1}{2}\qquad$
Solution
Let the center of the two circles be $O$. Now pick an arbitrary point $A$ on the boundary of the circle with radius $2$. We want to find the range of possible places for the second point, $A'$, such that $AA'$ passes through the circle of radius $1$. To do this, first draw the tangents from $A$ to the circle of radius $1$. Let the intersection points of the tangents (when extended) with circle of radius $2$ be $B$ and $C$. Let $H$ be the foot of the altitude from $O$ to $\overline{BC}$. Then we have the following diagram.
[asy] scale(200); pair A,O,B,C,H; A = (0,1); O = (0,0); B = (-.866,-.5); C = (.866,-.5); H = (0, -.5); draw(A--C--cycle); draw(A--O--cycle); draw(O--C--cycle); draw(O--H,dashed+linewidth(.7)); draw(A--B--cycle); draw(B--C--cycle); draw(O--B--cycle); dot("$A$",A,N); dot("$O$",O,NW); dot("$B$",B,W); dot("$C$",C,E); dot("$H$",H,S); label("$2$",O--(-.7,-.385),N); label("$1$",O--H,E); draw(circle(O,.5)); draw(circle(O,1)); [/asy]
We want to find $\angle BOC$, as the range of desired points $A'$ is the set of points on minor arc $\overarc{BC}$. This is because $B$ and $C$ are part of the tangents, which "set the boundaries" for $A'$. Since $OH = 1$ and $OB = 2$ as shown in the diagram, $\triangle OHB$ is a $30-60-90$ triangle with $\angle BOH = 60^\circ$. Thus, $\angle BOC = 120^\circ$, and the probability $A'$ lies on the minor arc $\overarc{BC}$ is thus $\dfrac{120}{360} = \boxed{\textbf{(D)}\: \dfrac13}$.
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just help me please. it might be easy but I kinda forgot what goes into where. Please help me out tho!!
The matching of the mathematical terms to the equations, and expressions is as follows;
Area of a circle → π•r²
Percent equation → P = %dec × w
Percent proportion → [tex] \displaystyle{ \frac{p}{w} = \frac{\%}{100} }[/tex]
Circumference → π × d
Pi → π
Base equation → y = k•x
Constant of proportionality → [tex] \frac{y}{x} [/tex]
% increase → [tex] \frac{Amount \: increase}{Original \: amount} [/tex]
%decrease →[tex] \frac{Amount \: decrease}{Original \: amount} [/tex]
Interest Earned → P×r×t
What is an expression in mathematics?An expression in mathematics consists of mathematical symbols and variables joined together operators in the proper manner
Area of a circleThe area of a circle is π × r²
Where;
r = The radius of the circle
Percent equationA percent equation equation is one of the sides of the equation consists of a percentage
Percent proportion;A percent proportion express a proportion as a percentage
Circumference;The circumference of a circle is the distance round the circle, which is given by the product of Pi and the radius of the circle
Pi.The symbol Pi is π
Base EquationThe base equation is one written in simple form
Constant of proportionalityThe constant of is the constant ratio between variables that are proportional
%increaseThis is the ratio of the increase to the original amount expressed as a percentage
%decreaseThe ratio of the decrease to the original amount expressed as a percentage
Interest EarnedThis is the amount added to an investment sum or amount and it is given by the product of the principal, P, rate, r and time, t
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help, i’ll give 20 points and thanks, and 5 stars
Answer: 8.75
Step-by-step explanation:
(0, 20) and (40, 370) any two coordinate points can be chosen
(370 - 20) / (40 - 0)
= 35/4 or 8.75
according to flightstats.com, american airlines flights from dallas to chicago are on time % of the time. suppose flights are randomly selected, and the number of on-time flights is recorded. (a) explain why this is a binomial experiment. (b) determine the values of n and p. (c) find and interpret the probability that exactly flights are on time. (d) find and interpret the probability that fewer than flights are on time. (e) find and interpret the probability that at least flights are on time. (f) find and interpret the probability that between and flights, inclusive, are on time.
According to the flight on certain routes, the experiment is a binomial experiment and this will be determined by observing that Flights on a certain route are on-time 80% of the time and 10 flights are randomly selected and the number of on-time flights is recorded.
(a) This is a binomial experiment since there are only two possible outcomes for each data that is a flight is either on time (p = 80% = 0.8) or late (q = 1 - p = 1 - 0.8 = 0.2).
(b) Using the formula:
P(a out of n) = (nCa)(pᵃ)(qⁿ⁻ᵃ), where n = 10 flights, a = the number of flights that arrive on time:
P(7/10) = (10C7)(0.8)⁷ (0.2)³= 0.2013
Therefore, 0.2013 chance is there that exactly 7 of 10 flights will arrive on time.
(c) Fewer than 7 flights are on time means that we must add up the probabilities for P(0/10) up to P(6/10).
P(0/10) + P(1/10) + ... + P(6/10) = 0.1209
This means that 0.1209 chance that less than 7 flights will be on time.
(d) The probability that at least 7 flights are on time is the exact opposite of part (c), where less than 7 flights are on time. So instead of calculating each formula from scratch, we can simply subtract the answer in part (c) from 1.
1 - 0.1209 = 0.8791.
So there is a 0.8791 chance that at least 7 flights arrive on time.
(e) For this, we must add up P(5/10) + P(6/10) + P(7/10), which gives us
0.0264 + 0.0881 + 0.2013 = 0.3158, so the probability that between 5 to 7 flights arrive on time is 0.3158.
According to the flight on certain routes, the experiment is a binomial experiment
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DISCLAIMER: The question provided is incomplete as some data are missing. The above answer is solved as per a relevant question
According to an airline, flights on a certain route are on time 80 % of the time. suppose 10 flights are randomly selected and the number of on-time flights is recorded. (a) explain why this is a binomial experiment. (b) find and interpret the probability that exactly 7 flights are on time. (c) find and interpret the probability that fewer than 7 flights are on time. (d) find and interpret the probability that at least 7 flights are on time. (e) find and interpret the probability that between 5 and 7 flights, inclusive, are on time.
Find the slope of the line passing through each of the following pairs of points and draw the graph of the line. (-4, 3), (2,3)
Answer: The slope is 0.
Step-by-step explanation:
We will use the slope formula to find the slope of the line going through these two points.
[tex]\displaystyle \frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]
[tex]\displaystyle \frac{3-3}{2--4}=\frac{0}{2+4} =0[/tex]
The slope of this line is 0.
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Suppose you want to convert kilometers to miles. Which unit should be in the
numerator of the conversion factor? Which unit should be in the denominator?
Explain your reasoning.
Answer:
see below
Step-by-step explanation:
km * miles/km
see how the 'km' then 'cancels out' and you are left with 'miles' the underlined shows the units of the conversion factor
You borrowed $1,690 for 5 1/2 years a at 5.7% compounded semi annually. What total will you pay back?
Answer: You would pay $2,302 back.
Step-by-step explanation:
Compound interest occurs when the interest is reinvested rather than paying it out. When it happens interest in the next period is then earned on the principal sum plus previously accumulated interest.
Question 1 (2 points)
(02.04)
For the function f(x) = -2(x + 3)2 - 1, identify the vertex, domain, and range.
Answer:
Vertex: (-3,-1)
Domain : (negative inf. , inf.)
Range: (negative inf. , 1]
a shipment of 22-inch pipes contains a pipe that is exactly 21.78 inches. What is the percent error, rounded to the nearest percent.
Please explain what you did step by the step. Thank you!
The percent error of the shipment pipes is 1.01%
Percent error:
Percent error is defined as a difference between estimated value and the actual value in comparison to the actual value and is expressed as a percentage.
It can be calculated using the formula,
PE = (|Estimated value-Actual value|/ Actual value) × 100
Given,
A shipment of 22-inch pipes contains a pipe that is exactly 21.78 inches.
Here we need to find the percent error of the shipment.
According to the percent error formula,
Here we have the values of the shipment are
Estimated value = 22
Actual value = 21.78
So, now we have to apply the values on the formula, then we get,
PE = (|22 - 21.78/21.78|) x 100
When we simplify it, then we get,
PE = 0.0101 x 100
PE = 1.01%
Therefore, the percent error in the shipment is 1.01%.
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Caroline’s cell phone plan costs $32 per month. Create a table that shows the sum of the amounts that Caroline will pay for her cell phone plan over the next 4 months. Can y'all please help me
Answer:
Month:
1 = 32$
2 = 64$
3 = 96$
4 = 128$
Step-by-step explanation:
The simplest form of 7x²-5x(x+2)
Answer:
factored form: [tex]2x(x-5)[/tex], standard form: [tex]2x^2-10x[/tex]
Step-by-step explanation:
so we're given the expression: [tex]7x^2 - 5x(x+2)[/tex]
In this case we want to use the distributive property, which essentially states that: [tex]a(b+c)=ab+ac[/tex]
So we can use this to rewrite:
[tex]-5x(x+2)=(-5x)(x)+(-5x)(2)[/tex]
which simplifies to
[tex]-5x^2-10x[/tex]
this just simplifies the last part of our expression, so using this instead of what we originally had gives us
[tex]7x^2-5x^2-10x[/tex]
Since the 7x^2 and the -5x^2 have the same variable, we can subtract their coefficients.
Another way to think of it is
[tex]7x^2-5x^2\implies x^2(7-5)[/tex]
so in a way we're "factoring" out the x^2, which means we can simply subtract the coefficients for this reason.
This of course gives us 2, by we're multiplying by x^2 so we get
[tex]2x^2-10x[/tex]
Now if you want you can factor out a 2x, to get
[tex]2x(x-5)[/tex]
Please please please help me it's due at 10:00 P.M today
Answer: 20x
Step-by-step explanation:
Answer:
20x
Step-by-step explanation:
just trust me because you have to mulitply
consider the function rpresented by 9x + 3y = 12 with x as the independant variable. How can this function be writte using function notation? f(y) = - 1/3y + 4/3, f(x) = -3x + 4, f(x) = - 1/3 x + 4/3 f(y), = - 3y + 4
The slope-intercept form of a function f(x) = mx + c therefore,f(x) = -3x + 4 is the correct notation of the function 9x + 3y = 12 so option (B) is correct.
What is a function?The function is a relationship between variables, and the nature of the relationship defines the function for example y = sinx and y = x +6 like that.
The machine will only accept specified inputs, described as the function's domain, and will potentially produce one output for each input.
As per the given function,
9x + 3y = 12
Subtract 9x both sides,
9x - 9x + 3y = 12 - 9x
3y = 12 - 9x
Divide by 3 into both sides,
y = 12/3 - 9x/3
y = -3x + 4
Since,y = f(x)
f(x) = -3x + 4 .Here -3 is the slope while 4 is the y-intercept.
Hence "The slope-intercept form of a function f(x) = mx + c therefore,f(x) = -3x + 4 is the correct notation of the function 9x + 3y = 12".
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find the slope 35 points
When we want to find the slope of a line,
We find two points and connect them to form a right triangle.
We then divide the increase in the [tex]y[/tex]-axis by the increase in the [tex]x[/tex]-axis to find the slope.
Since the line is tilted to the right, the slope is positive.
The answer must be 3.solutions to 2-variable equations 4x-1=3y+5
what 5+5+5+7= what is it
Add 5 x 3 + 7
5 x 3 + 7
15 + 7
22
Answer:
Alternatively, you may use addition to solve
Step-by-step explanation:
5+5 = 10 and 5+7 = 12, and we know that 10+10 = 20
Just add the number 2 from the number 12 to 20, and you have the same answer
A total of 77 tornadoes have been documented in tuscaloosa county in the period beginning on january 1, 1950 and ending on december 31, 2020. What is the recurrence interval for tornadoes in tuscaloosa county?.
The Recurrence interval for tornadoes in tuscaloosa county is 0.909
What is recurrence interval ?
The time period in which a new occurrence of something that happened or appeared before : a repeated occurrence or The time frame in which a previously occurring event or appearance first occurs: a recurring occurrence
Given that,
Total number of tornadoes is 77
Total time for Recurrence is 70 years
Using these data,
Recurrence = Total time for recurrence / Total number of tornadoes
Substituting we get,
Recurrence interval = 70/77
= (70/77)
= (0.909)
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he graph below shows the total cost for lunch, c, when Will and his friends buy a large salad to share and several slices of pizza, p.
A graph titled Lunch Cost. The x-axis shows the number of slices purchased, numbered 1 to 8, and the y-axis shows the cost in dollars, numbered 5 to 40. Blue diamonds appear at points (0, 7.5), (1, 10), (2, 12.5), (3, 15), (4, 17.5), (5, 20), (6, 22.5), (7, 25), (8, 27.5).
For each additional slice of pizza that is purchased, by how much does the total cost of lunch increase?
$1.50
$2.00
$2.50
$3.00
For each additional slice of pizza that is purchased, the amount by which the total cost of lunch increase is: C. $2.50.
What is a graph?A graph refers a type of chart that's commonly used to graphically represent data on both the horizontal and vertical lines of a cartesian coordinate, which are the x-axis and y-axis.
In Mathematics, the graph of any proportional relationship is characterized by a straight line with the points passing through the origin (0, 0) because as the values on the x-axis increases or decreases, the values on the y-axis increases or decreases simultaneously.
Next, we would calculate the difference between the total cost when an additional slice of pizza is purchased:
Difference = 10 - 7.5 = 12.5 - 10 = 15 - 12.5 = $2.50
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Solve the system of equations by elimination
From the equations we get 'x' as '-2' and 'y' as '-7'
What is an equation?
An equation is a formula in mathematics that expresses the equivalence of two expressions by linking them with the equals symbol =. The word equation and its cognates in various languages may have somewhat different definitions; for example, in French, an équation is defined as including one or more variables, but in English, an equation is any well-formed formula consisting of two expressions linked by an equals sign. An equation consists of two expressions joined by an equals sign ("="). The expressions on each side of the equals sign are referred to as the equation's "left-hand side" and "right-hand side." The right-hand side of an equation is frequently considered to be zero.
4x - 2y = 6....................equation (1)
-2x + 4y = -24...............equation (2)
Multiply equation (1) by 2 and add with equation (2)
8x - 4y = 12
-2x + 4y = -24
----------------------
6x = -12
or, x = -12/6 = -2
from equation (1),
4x - 2y = 6
or, 4(-2) - 2y = 6
or, -2y = 6 + 8
or, y = 14/-2 = -7
Hence, from the equations we get 'x' as '-2' and 'y' as '-7'
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Anya earns $150 less than three times Rick's monthly salary. If Rick's salary is represented by x which expression represents Anya's salary
A x-3- $150
B 3(x+$150)
C 3(x-$150)
D 3x-$150
E 3x+$150
Answer:
d
Step-by-step explanation:
3 multiplied by ricks salary minus 150
exapnd and simplify (3x+4)(2x+3)
Use the number line to add the fraction.
Drag and drop the answer into the box to match the sum
Answer:
the answer to the question is 1/8
please help asap
this is an assignment due today please help real quick
Answer:
A (3; 4)
B (5; 2)
C (4; 0)
D (-6; -2)
E (-2; -5)
I hope this helps!
k(x) = 2x²-x-1 find the roots
Answer:
x = -1/2 , 1
Step-by-step explanation:
Matthew wants his run today to take no more than 90 minutes. He already has run 30 minutes at a pace of 10 minutes per mile. Part A) Write an inequality that Matt could solve to find out how many more miles he can run if he keeps his current pace. Show or explain the reasoning you used to determine your answer.
The inequality that Matt could solve to find out how many more miles he can run if he keeps his current pace is less than 6 miles.
How to express the inequality?It should be noted an inequality is used to show the expressions that are not equal.
In this case, Matthew wants his run today to take no more than 90 minutes and he already has run 30 minutes at a pace of 10 minutes per mile.
Therefore, the appropriate inequality will be;
Let miles = m
m <= 90/10 - 30/10
m <= 9 - 3
m <= 6
He has about 6 miles more.
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A baseball Dimond is a square with sides of 90 feet. What is the shortest distance, to the nearest tenth of a foot, between first base and third base?
Using the pythagorean theorem we get that:
[tex]\begin{gathered} d^2=(90ft)^2+(90ft)^2=2(90ft)^2 \\ d=\sqrt[]{2}(90ft)=127.3ft \end{gathered}[/tex]Note: the shortest distance between two points in a plane is a stright line, that is why we calculated d.
What is the answer to
k + 4n+ 8=
Answer:
answer is 2 since k =1 and 4n will be 2 ×2 so 8 will be equivalent to 4 which will be 2
Sorry guys one last question
Complete the square to transform the expression x2 + 4x + 2 into the form a(x − h)2 + k. (1 point) (x + 2)2 − 2 (x + 2)2 + 2 (x + 4)2 − 2 (x + 4)2 + 2
By completing squares we will rewrite the equation as:
(x + 2)^2 - 2
How to complete squares?
Here we have the quadratic equation:
x^2 + 4x + 2
And we want to complete squares, remember that the square trinomial is:
(a + b)^2 = a^2 + 2ab + b^2
We can rewrite the quadratic as:
x^2 + 2*2*x + 2
Now, adding and subtracting 2 we get:
x^2 + 2*2*x + 2 + 2 - 2
x^2 + 2*2*x + 4 - 2
(x^2 + 2*2*x + 2^2) - 2
(x + 2)^2 - 2
So the first option is the correct one.
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Jaclyn says 7 5 has the same answer as 3/7 + 2/7 is she correct explain why or why not
3/7 + 2/ 7 = (3+2)/7 = 5/7
Then Jaclyn is WRONG
Because 5/7 is the inverse of 7/5
A solar panel covers the hood of a truck. The area is 0.8 the width of the panel is 1.6 meters, what is the length of the panel
The length of the panel is 0.5 meter.
What is the area of rectangle?To find the area of a rectangle, we multiply the length of the rectangle by the width of the rectangle.
Given that,
Area of the panel = 0.8 [tex]m^{2}[/tex]
width of the panel = 1.6 meter
Here, solar panel is in the form of rectangle.
Area of rectangle = length× width
0.8 = 1.6 × length
length = [tex]\frac{0.8}{1.6}[/tex]
length = 0.5 meter
Hence, The length of the panel is 0.5 meter.
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