Answer:
10 square inches per second.
Step-by-step explanation:
The radius of the circle is given by the equation:
r(t) = (1/π in/s)*t
Where time in seconds.
Remember that the area of a circle of radius R is written as:
A = π*R^2
Then the area of our circle will be:
A(t) = π*( (1/π in/s)*t)^2 = π*(1/π in/s)^2*(t)^2
Now we want to find the rate of change (the first derivation of the area) when the radius is equal to 5 inches.
Then the first thing we need to do is find the value of t such that the radius is equal to 5 inches.
r(t) = 5 in = (1/ in/s)*t
5in*(π s/in) = t
5*π s = t
So the radius will be equal to 5 inches after 5*π seconds, let's remember that.
Now let's find the first derivate of A(t)
dA(t)/dt = A'(t) = 2*(π*(1/π in/s)^2*t = (2*π*t)*(1/π in/s)^2
Now we need to evaluate this in the time such that the radius is equal to 5 inches, we will get:
A'(5*π s) = (2*π*5*π s)*((1/π in/s)^2
= (10*π^2 s)*(1/π^2 in^2/s^2) = 10 in^2/s
The rate of change is 10 square inches per second.
Damian works after school. Each day he earns a set amount, plus an hourly wage. The function f(x)=12 x+10 models the amount he earns each day for working x hours. How much does Damian earn on Friday if he works for 2.75 hours?
$=________
The amount earned by Damian on Friday, if The function f(x) = 12 x + 10, and he works for 2.75 hours, is $43.
What is a function?Each element of X receives exactly one element of Y when a function from one set to the other is used. The sets X and Y are collectively referred to as the function's domain and co-domain, respectively.
Given:
The function f(x) = 12 x + 10, where x is working hours,
Calculate the amount by putting the value of x = 2.75 in the function as shown below,
f(x) = 12 × 2.75 + 10
f(x) = 33 + 10
f(x) = $43
Thus, the amount earned is $43.
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division answer in feet and inches 8 divided by 34ft 8in
Answer:
since 12inches=1foot
x inches=34feet
34feet converted to inches =408inches
total inches=408+8
T=416inches
dividing 416 by 8 we have
52 inches left
converting back to feet we get
4ft 4in
If x = 29°, find the measures of angles 1, 2, and 3.
Answer:
75.5
Step-by-step explanation:
Question
Solve the equation.
−2w=−9
w=
w=9/2 =4,5
Step-by-step:
:)
(20points) Let A be a symmetric positive define matrix with Cholesky decomposition A = LLT = RTR. Prove that the lower triangular matrix L(or that the upper triangular matrix R) in the factorization is unique.
In the factorization, the bottom triangular matrix L is the only one.
What is Cholesky decomposition?The exponent of a number indicates how many times it has been multiplied by itself.
Exponent-related issues can be solved using either exponent laws or exponent characteristics. When a number is repeated numerous times by itself, writing the product without the use of exponents becomes very difficult. Major exponentiation rules are also thought of as having these characteristics.
Given that A is a positively defined symmetric matrix with a Cholesky decomposition,
A = L[tex]L^{T}[/tex] = [tex]R^{T} R[/tex]
Let A be our positive define symmetric matrix,
suppose A has Cholesky decomposition,
A = L₁L₂ = L₁L₂ , for L₁, L₂ lower triangular matrix with positive diagonal entries,
thus (A x ,x) = (Lx, Lx) = (Lx, Lx)
pick x = eₓ the last coordinate vector,
then (Aₓ, x) = Axx = || L₁x||² = || L₂x||²
Given that L1, and L2 are lower triangular, which necessitates that their lower right entry be the same,
The Kth entry is currently the last row.
given by (L₁, eₓ, eₓ) = 1/√Axx (L₁eₓ, L₂eₓ) = 1/ Aₓ = (Aeₓ, eₓ)
so the last row of L₁, L₂ be the same,
we reduce our A to new (n - 1)x(n - 1) submatrix,
which will likewise be true, and then repeat the process to discover each row's unique value.
Thus, the inductive process shows that L1 = L2.
As a result, the factorization's bottom triangular matrix L is exclusive.
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Mariah has a total of $15,000 invested in two accounts. The total amount of interest she earns from the accounts in the first year is $1540. If one account pays 8% per year and the other pays 12% per year, how much did she invest in each account?
The amount invested in the account that earns 8% interest is $6500
The amount invested in the account that earns 12% interest is $8500
How much did she invest in each account?a + b = $15,000 equation 1
0.08a + 0.12b = $1540 equation 2
Where:
a = amount invested in the account that earns 8% interest
b = amount invested in the account that earns 12% interest
The elimination method would be used to determine the required values:
Multiply equation 1 by 0.08
0.08a + 0.08b = 1200 equation 3
Subtract equation 3 from equation 2
0.04b = 340
Divide both sides of the equation by 0.04
b = 340 / 0.04
b = $8500
a + 8500 = 15,000
a = $15,000 - $8,500
a = $6500
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The question is in the picture
Answer:
-6/6 or -1
Step-by-step explanation:
To find the equation of the line (or the slope, assuming that's what equation means, ) we need to write the formula Y1-Y2/X1-X2
So, we need to fill in the numbers, which gives us the fraction -6-0/-2-(-8)
When we solve for the numerator (-6-0), we get -6.
When we solve for the denominator (-2 - {-8}), we get 6.
So, we have -6/6, which, when simplified, is -1.
(If the problem is not asking for slope when it says equation, then sorry!)
-7 x 3^0.25x = -10
Which of the following is the solution of the equation?
Rewrite the expression as a single logarithm
Please help will give brainlist only if correct!
The correct option is (d) i.e. log [ √x × (x-1)^3 / ∛ (x+1)^2 ] is as the single logarithm.
What is logarithm?
Logarithm is a mathematical operation which is used to determine the exponential power of a number. It is the inverse operation of exponentiation. Logarithm is commonly used in calculus, algebra, and other areas of mathematics. It can also be used to solve problems involving exponents, roots, and powers. Logarithm is denoted by the symbol log, and is read as "log to the base". The base is the number which the logarithm is taken to. For example, log2 10 is the logarithm to the base 2 of 10. This is equivalent to saying "2 to the power of what equals 10?" The answer is that 2 raised to the power of 3 (2^3) would equal 10.
Given, log x / 2 + 3 [ log(x-1) - 2/9 log(x+1) ]
= log √x + 3 log(x-1) - 2/3 log (x+1) {∵ log x^a = a log x }
= log √x + log (x-1)^3 - log (x+1)^2/3
= log √x + log [(x-1)^3 / (x+1)^2/3] {∵ log a - log b = log a/b }
= log [√x × (x-1)^3 / (x+1)^2/3 ]
= log [ √x × (x-1)^3 / ∛ (x+1)^2 ]
Hence, (d) is the correct option.
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Use the model to help you find 1/(8/12)
1/(8/12)=___
Answer:
3/2 or 1.5 or 1 1/2 in mix number
Is (6, 8) a solution to this system of equations?
y =1/6x+5
y=5/6x+3
Yes
No
Answer:No
Step-by-step explanation:
When we substitute x=6 into the first equation we get
y=1/6 *6+5=6 so we get (6, 6)
When we substitute x=6 into the second equation we get
y=5/6*6+3=8 so we get (6, 8) . it works for the second equation but not the first. It must be true for both equations
Ravi sells real estate. Based on previous data, he knows that 5% of home tours result in a sale. Assume that the results of these tours are independent from each other. Which of the following choices are binomial random variables? Choose all answers that apply: A. Take a random sample of 30 tours and let L = the number of tours that result in a sale. B. Take a random sample of 3 tours and let K = the number of tours that result in a sale. C. Take a random sample of 3 tours and let M = the amount of sales (in dollars) generated by the tours
The following choices are binomial random variables: A. Take a random sample of 30 tours and let L = the number of tours that result in a sale. B. Take a random sample of 3 tours and let K = the number of tours that result in a sale.
Choice A is a binomial random variable because it represents the number of tours that result in a sale in a random sample of 30 tours. The trials are independent, as the results of one tour do not affect the results of other tours, and the probability of success (a sale) is constant at 5%.
Choice B is also a binomial random variable because it represents the number of tours that result in a sale in a random sample of 3 tours. The trials are independent and the probability of success is constant at 5%.
Choice C is not a binomial random variable because it represents the number of sales generated by a random sample of 3 tours, which is a continuous variable. Binomial random variables are always discrete, as they represent the number of successes in a sequence of trials.
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Does meditation cure insomnia? Researchers randomly divided 400 people into two equal-sized groups. One group meditated daily for 30 minutes, the other group attended a 2-hour information session on insomnia.
At the beginning of the study, the average difference between the number of minutes slept between the two groups was about 0. After the study, the average difference was about 32 minutes, and the meditation group had a higher average number of minutes slept.
To test whether an average difference of 32 minutes could be attributed to chance, a statistics student decided to conduct a randomization test. She wrote the number of minutes slept by each subject in the study on an index card. She shuffled the cards together very well, and then dealt them into two equal-sized groups, representing those who meditated and those who attended the information session.
Which of the following best describes the outcome of the randomization test.
O The average difference between the two values on the two stacks of cards is expected to be about 0 minutes.
O If meditation is effective, the average difference between the values on the two stacks of cards is expected to be more than 32 minutes.
O The average difference between the two values on the two stacks of cards is expected to be about 32 minutes.
1. The average difference between the two values on the two stacks of cards is expected to be about 0 minutes best describes the outcome of the randomization test.
This randomization test was conducted to test whether the average difference of 32 minutes between the two groups could be attributed to chance. The test essentially involved shuffling the index cards containing the minutes slept by each subject and then dealing them into two equal-sized groups.
This process allows for a fair comparison between the two groups and ensures that any differences in the average minutes slept are due to chance. Since the cards are shuffled randomly, the average difference between the two stacks of cards is expected to be about 0 minutes.
If meditation is effective, then the difference between the values on the two stacks of cards would be expected to be more than 32 minutes.
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You are packing books into a box. The box can hold at most 10 books. The function y=5.2x represent the weight y (in pounds) of x books
Is 52 the range?
Is 45 the range?
Is 15 the domain?
Is the domain discrete or continuous?
Answer:
52 is range
Step-by-step explanation:
Range is the 'y' values the function can have
from 0 to 10 books would give a range of 0 - 52
Domain is the x-values 0, 1,2,3,4,56,7,8,9,10
discrete, because you cannot have anything but the whole numbers listed for the values of 'x'
Answer:
The domain is the input. It is the number of books. You can have 0-10 books. These would be the domains.
The range is the outputs. The outputs can be from 0 to 52 pounds.
The domain is discrete. The domain is the number of books. We cannot have half or a book or 3/4 of a book. So this will be discrete. The number of books can only be whole numbers up to 10.
Step-by-step explanation:
In the article, Attitudes About Marijuana and Political Views (Psychological Reports, 1973), researchers reported on the use of cannabis by liberals and conservatives during the 1970's.
To test the claim (at 1% significance) that the proportion of voters who smoked cannabis frequently was lower among conservatives, the hypotheses were
In the hypothesis test about cannabis use by conservatives and liberals, the test statistic was z = -4.27, with a corresponding p-value of about 0.00001.
Which conclusion is most appropriate in the context of this situation?
The data do not support the claim that a lower proportion of conservatives smoke cannabis when compared to liberals.
The data support the claim that the proportion of conservatives who smoke cannabis is no different that the proportion for liberals.
The data support the claim that a lower proportion of conservatives smoke cannabis when compared to liberals.
The z-test statistic is used to test the claim that the proportions of the two populations differ. A normal test statistic, the z-test statistic can also be used to measure the proportion of a single population.
Let's say that:
p1: Liberal voters' proportion of the population who smoked cannabis, p. 2:
Conservative voters' proportion of the population who smoked cannabis The null hypothesis is H 0:
The alternative hypothesis is H 0: p 1 = p 2.
p 1 p 2 P-value = 0.00001 The P-value is less than the significance level of 1%. The null hypothesis is rejected at the 1% significance level.
We can conclude that conservative voters had a lower percentage of voters who regularly smoked cannabis.
What is theory trying?Two distinct hypotheses are tested by all analysts using a random population sample: the alternative hypothesis and the null hypothesis. Typically, the null hypothesis is a hypothesis that all population parameters are equal; For instance, a null hypothesis might assert that the population mean return is zero.
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Cooper has 35 video games in his collection, and Maria has 15 in hers. Cooper decides to add 10 video games to his collection each month. Maria decides to add 8 video games to her collection each month. Part A Drag the values to the positions in the table to show how many video games each of them will have at the ends of months 1, 2, 3, and 4. 314533436529 Month Cooper Maria Start 35 15 1 23 2 55 3 39 Part B Which of the following is a true statement about the relationship between the number of video games collected by Cooper and Maria after each month? A. Cooper will always have 20 more video games than Maria. B. Maria will always have 20 more video games than Cooper. C. Cooper will always have 28 more video games than Maria. D. There is no constant relationship between the number of video games collected by Cooper and Maria.
From the values that we have here, the statement that can be said to be true here would be that: There is no constant relationship between the number of video games collected by Cooper and Maria. Last option
What is a constant relationship?A relationship with a fixed ratio between two quantities is referred to as proportionate. The graph will therefore be linear, or straight.
When two variables are directly or indirectly proportional to one another, their relationship can be expressed using the formulas y = kx or y = k/x, where k specifies the degree of correspondence between the two variables. The proportionality constant, k, is often used.
The values between these two does not show any constant relationship hence the last option
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Answer: part a
cooper- 35 45 55 65
maria 15 23 31 39
part B
there is no constant relationship between the number of video games collected by coopeer and maria
Step-by-step explanation:
Solve for x -13x<65 simplify
The value of x in the inequality, -13x < 65, is simplified as, x > 5.
How to Solve an Inequality?Any inequality given can be solved by find the value of the variable in the inequality, which will make it a true statement. To do this, isolate the given variable to make it stand alone on one side.
Given the inequality, -13x < 65, solve as shown below:
-13x < 65 [given]
Divide both sides by -13:
-13x/-13 < 65/-13
x > -65/13 {the sign changes from < to > because we divided both sides by a negative quantity]
x > 5
The value of x is simplified as x > 5.
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Solve the equation b/16 = -4 for b
-64
-4
4
64
Answer: -64
Step-by-step explanation:
To solve this equation, you need to isolate the b term on one side of the equation. You can do this by dividing both sides of the equation by 16. This will give you:
b/16 = (-4)
You can then divide both sides of the equation by -4 to get:
b/16 = -4
b/16 * -4 = -4 * -4
b = -64
Therefore, the solution to the equation is b = -64.
The graph of the linear function passes through points (2, 44) and (5, 80). What is the equation of the function? PLEASE HELP!! I WILL MARK BRAINLIEST!!!
Answer:
y=12x+20
Step-by-step explanation:
Your question and a graph of picture are different. So I just considered your question.
m= (80-44)/(5-2) = 36/3=12
(y- y_1)=m(x-x_1)
(y-44)= 12(x-2)
y-44=12x-24
y= 12x-24+44
y=12x+20
What is the area of the triangle with vertices at A(2, 2), B(4, 5) and C(7, 5)? Express your
answer as a decimal to the nearest tenth.
Answer:
Therefore, the area of the triangle with vertices at A(2, 2), B(4, 5), and C(7, 5) is approximately 4.5, to the nearest tenth.
Step-by-step explanation:
To find the area of a triangle with vertices at A(2, 2), B(4, 5), and C(7, 5), you can use the Shoelace Theorem.
The Shoelace Theorem states that the area of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) is given by the following formula:
A = 1/2 * |(x1y2 + x2y3 + ... + xn-1yn + xny1) - (y1x2 + y2x3 + ... + yn-1xn + ynx1)|
To apply the Shoelace Theorem to a triangle with vertices at A(2, 2), B(4, 5), and C(7, 5), you can plug in the coordinates of these vertices into the formula:
A = 1/2 * |(25 + 45 + 72) - (24 + 57 + 52)|
= 1/2 * |(10 + 20 + 14) - (8 + 35 + 10)|
= 1/2 * |44 - 53|
= 1/2 * |-9|
= 1/2 * 9
= 4.5
A composite figure is represented in the image.
What is the total area of the figure?
A: 192 m2
B: 216 m2
C: 288 m2
D: 336 m2
The total area of the figure is equal to 192 square meters. The correct option is A.
What is an area?The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the rectangle in a two-dimensional plane is called the area of the rectangle.
Calculate the area of the rectangle by the formula written below,
Area of the rectangle = L x W
Area of the rectangle = 8 x 18
Area of the rectangle = 144 square meters
The area of the triangle will be calculated by the formula written as,
Area of triangle = 1/2 x B x H
Area of triangle = 1/2 x ( 18 - 6 ) x 8
Area of triangle = 1 / 2 x 12 x 8
Area of triangle = 48 square meters
Total area = 144 + 48
Total area = 192 square meters
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In a network of 39 computers, 4 have a copy of a particularly critical file to sustain an organization's regular operations. Suppose that 8 computers at random fail. What is the probability that all 4 computers with the critical file fail in this incident? (Round to four decimal places.)
Mathematical models are used as tools to describe reality. These models are supposed to characterize the important features of the analyzed phenomena and provide insight. The normal distribution is an example of a random variable that is widely used by researchers to model real data.
Researchers often model real observations using the normal distribution, but sometimes the real distribution is a bit different from the perfect, normal distribution. List some reasons why researchers might make approximations like this and describe at least one situation when researchers should not use this approximation.
When forming your answer to this question you may give an example of a situation from your own field of interest for which a random variable can serve as a model.
There are several reasons why researchers might not use the normal distribution to model real data which are due to assumptions of normal distribution.
First, the normal distribution assumes that the data is symmetric and is distributed around the mean. However, in some cases, the data may be skewed, making the normal distribution an inappropriate model.
Second, the normal distribution assumes that the data is homoscedastic, meaning that the variance is constant across the sample. However, in some cases, the variance may be heteroscedastic, meaning that the variance changes across the sample. This makes the normal distribution an inappropriate model for this type of data.
One situation in which researchers should not use the normal distribution approximation is when modeling financial data. Financial data often follows a power-law distribution, which is not normal. Therefore, researchers should use a power-law distribution to model this type of data.
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Given
G(t) = 9 − 5t,
write
G(−5 + h) − G(−5)
in simplest form.
The value of the function G(-5 + h) - G(-5) in the simplest form -5h.
What is function?Function is a combination of different types of variable and constants in which for the different values of x the value of function y is unique.
The given function is,
G(t) = 9 - 5t (1)
To find the value of expression G(-5 + h) - G(-5),
First, find the value of G(-5 +h) by substituting t = -5+h in equation (1),
G(-5 + h) = 9 - 5(-5 + h)
= 9 +25 -5h
= 34 - 5h
The value of G(-5)
G(-5) = 9 - 5(-5)
= 9 + 25
= 34
The required value,
G(-5 + h) - G(-5) = 34 - 5h - 34 = -5h.
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How do I find the value of a + b + c?
The expression of (a + b + c) is equivalent to 16.5.
What is a mathematical function, equation and expression?Function : In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function
Expression : A mathematical expression is made up of terms (constants and variables) separated by mathematical operators
Equation : A mathematical equation is used to equate two expressions.
Given are two circles intersecting at each others centers at D and E.
We can write the area of the common region as -
[A] = 4 x {(θ/180) x (πr²/2) - (1/2)r²sinθcosθ}
Now, we can write θ as -
cos(θ) = (3/6) = 1/2
θ = 60°
[A] = 4 x {(60/180) x (π(6)²/2) - (1/2)(6)²sin(60)cos(60)}
[A] = 4 x {(1/3 x 36π/2 - (1/2) x 36 x [tex]\sqrt{\frac{3}{2} }[/tex] x 1/2}
[A] = 4 x {6π - 9 [tex]\sqrt{\frac{3}{2} }[/tex] }
[A] = 24π + (- 9 [tex]\sqrt{\frac{3}{2} }[/tex])
So, on comparing, with the expression below -
aπ + b[tex]\sqrt{c}[/tex]
We get -
a = 24
b = - 9
c = 3/2
So, we get -
a + b + c
24 - 9 + 3/2
24 - 9 + 1.5
24 - 7.5
16.5
Therefore, the expression of (a + b + c) is equivalent to 16.5.
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on monday therease went to the doctor and got an antibiotic for strep throat. The doctor told her take a dose of 4.8 ml every 12 hours for 7 days. If thereas took her first dose at 9:00 AM on Monday, what day and time should she take her 7th dose?
Therease should take her 7th dose on Friday at 9:00AM
What is time ?
Time can be described in mathematics as an ongoing and continuous series of events that take place one after another, from the past through the present, and into the future. The duration of events or the gaps between them can be measured, compared, or even ordered using time.
Time is the ongoing progression of existence and things that happen in what seems to be an irrevocable order from the past, present, and forward into the future.
Time is defined by physicists as the flow of events from the past through the present and into the future. In essence, a system is timeless if it is unchanging. When describing events that take place in three-dimensional space, time can be thought of as the fourth dimension of reality.
The doctor told her to take a dose of 4.8 ml every 12 hours for 7 days.
If Theresa took her first dose at 9:00 AM on Monday,
Simply add 12 hours every time up to the 7th dose,
First dose ----> 9:00 AM on Monday,
Second dose ----> 9:00 PM on Monday,
Third dose ----> 9:00 AM on Tuesday,
Fourth dose ----> 9:00 PM on Tuesday,
Fifth dose ----> 9:00 AM on Thursday,
Sixth dose ----> 9:00 PM on Thursday,
Seventh dose ----> 9:00 AM on Friday,
Hence, Theresa should take her 7th dose at 9:00 AM on Friday.
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Five years ago, you acquired a 30-year loan of $130,750, charging 6.6% annual interest, compounded monthly, and requiring monthly payments. At this time, interest rates on 15-year loans have dropped to 2.1% APR, compounded monthly, and you wish to refinance what you still owe with a new loan at this new rate. (a) How much (in dollars) will you be refinancing? Round your answer to the nearest dollar. (b) How much (in dollars) will your new monthly payment be after refinancing? Round your answer to the nearest cent.
Five years ago, you acquired a 30-year loan of $130,750, charging 6.6% annual interest, compounded monthly,
a) You will be refinancing $122536.
b) New monthly payment be after refinancing is
$794.1858 or 79 418.58 cents.
We have given that
Initial loan amount = $ 130,750
Number of year of loan = 30 year
Nomber of month of loan "n" = 30x12= 360
Annual interest rate = 6.6%
monthly rate "r" = 6.6% /12 =
monthly payment on the loan is
PMT= loan× r / [1 - (1+r)⁻ⁿ]
= $ 130750 X 12/(1 - (1+ 6.6%/12)⁻³⁶⁰)
= $719,125/(1- (1+0.0055) ⁻³⁶⁰
= $835.046
Monthly payment, $835.046 on initial loan.
Now after 5 years refinishing is done . So, amount and remaining balance is for 30-5 = 25 years and 25×12 = 300 months . The present value of unpaid monthly payment is
= PMT× ( (1 - (1+r)⁻ⁿ)/r)
= $835.046( 1 - (1+6.6%/12)⁻³⁰⁰/ 6.6%/12]
= $835.046× 196.74179
= $122536
Hence, amount refinanced is $122536.
b) New monthly payment will be on loan =$122536
Annual rate = 2.1%
monthly rate, r' = 2.1%/12
Number of years for refinancing = 15 years
Number of months , n' = 15× 12 = 180
Using the formula new monthly payment is
PMT = new loan× r'/ [1 - (1+r')⁻ⁿ´ ]
= $122536× 2.1%/12[ 1 - (1+2.1%/12)⁻¹⁸⁰]
= $214.43826/0.2700104
= $794.1858
Hence, new monthly payment is $794.1858.
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Find the equation perpendicular to y=-2x+10 through (10,7)
Answer:
y = [tex]\frac{1}{2}[/tex] x + 2
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
y = - 2x + 10 ← is in slope- intercept form
with slope m = - 2
given a line with slope m then the slope of a line perpendicular to it is
[tex]m_{perpendicular}[/tex] = - [tex]\frac{1}{m}[/tex] = - [tex]\frac{1}{-2}[/tex] = [tex]\frac{1}{2}[/tex] , then
y = [tex]\frac{1}{2}[/tex] x + c ← is the partial equation of the perpendicular line
to find c substitute (10, 7 ) into the partial equation
7 = 5 + c ⇒ c = 7 - 5 = 2
y = [tex]\frac{1}{2}[/tex] x + 2 ← equation of perpendicular line
If four vertices of a regular octagon are chosen at random then the probability that the quadrilateral formed by them is a rectangle is
A.1/8
B.2/21
C.1/32
D.1/35
The correct answer is option D. 1/35. To count the number of rectangles that can be formed, we need to consider the two cases where the diagonals of the rectangle are parallel to the sides of the octagon, and where the diagonals are perpendicular to the sides of the octagon.
In the first case, there are four choices for which side the diagonals are parallel to, and once that side is chosen, there are two choices for which pair of vertices lie on that side. Therefore, there are $4 \cdot 2 = 8$ rectangles of this type.
In the second case, there are four choices for which vertex the diagonals intersect, and once that vertex is chosen, there are two choices for which pair of vertices lie on the same side of the intersection point as the chosen vertex. Therefore, there are $4 \cdot 2 = 8$ rectangles of this type.
Altogether, there are $8 + 8 = 16$ rectangles. Therefore, the probability that the quadrilateral formed by the four vertices is a rectangle is
$\frac{16}{70} = \frac{16}{{8 \choose 4}}= \frac{1}{{8 \choose 4}/16}
= \frac{1}{{8 \choose 4}/2^4}
= \frac{1}{{8 \choose 4}/2^4}
= \boxed{\text{(D) } 1/35}$.
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To start dividing 126 by 23, Miranda used the estimate 120÷20 = 6. How could you tell six is too high?
Answer:
the product of 6 and 23 is more than 126
Step-by-step explanation:
You want to know how to tell that 6 is too high an estimate for the first digit of the quotient of 126 and 23.
Trial dividendWhen the trial quotient value of 6 is multiplied by the actual divisor of 23, we are computing 6(20 +3) = 120 +18. This is more than 126, so the trial quotient value is too large.
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Additional comment
Another way to tell is to consider the dual problem that 120/20 = 6 represents: 120/6 = 20. It is easy to see that 126/6 = 21, so we know that a divisor of 23 (larger than 21) will give a quotient less than 6.