The measure of the angles obtained using trigonometric identities are;
sin(2·θ) = -(4·√5)/9cos(2·θ) = 1/9tan(2·θ) = -4·√5What are trigonometric identities?Trigonometric identities are mathematical equations that consists of the trigonometric functions and which are correct for the values of the angles entered into the equations.
The value of sin(2·θ) can be obtained by making use of the Pythagorean identity as follows;
cos²(θ) + sin²(θ) = 1
sin²(θ) = 1 - cos²(θ)
sin(θ) = √(1 - cos²(θ))
3·cos(θ) = √5
cos(θ) = √5/3
sin(θ) = √(1 - (√5/3)²) = 2/3
180° ≤ θ ≤ 360°, therefore, sin(θ) is negative, which indicates;
sin(θ) = -2/3
sin(2·θ) = 2·sin(θ)·cos(θ)
sin(2·θ) = 2×(-2/3) × (√5)/3 = -(4·√5)/9
sin(2·θ) = -(4·√5)/9The double angle formula for cosines, indicates that we get;
cos(2·θ) = cos²(θ) - sin²(θ)
Therefore;
cos(2·θ) = ((√5)/3)² - (-2/3)² = 5/9 - 4/9 = 1/9
cos(2·θ) = 1/9tan(2·θ) = sin(2·θ)/cos(2·θ)
Therefore;
tan(2·θ) = ((4·√5)/9)/(1/9) = 4·√5
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To purchase $13200 worth of machinery for her business, Nicole made a down payment of 1200 and took out a business loan for the rest. After 3 years of paying monthly payments of 365.07, she finally paid off the loan.
(a) What was the total amount Nicole ended up paying for the machinery (including the down payment and monthly payments)?
(b) How much interest did Nicole pay on the loan?
The answer of the given question based on the compound interest to find total amount Nicole ended up paying for the machinery and the interest did Nicole pay on the loan is (A) the total amount Nicole ended up paying for the machinery is $14,342.52. (B) Nicole paid $2,342.52 in interest on the loan.
What is Compound interest?Compound interest is type of interest that is calculated not only on initial principal amount but also on accumulated interest from previous periods. In other words, interest earned in each period is added to principal amount, and interest for the next period is calculated on new, larger principal amount.
Compound interest can be thought of as "interest on interest" and is used in many financial transactions, like loans, investments, and savings accounts.
(a) The total amount Nicole ended up paying for the machinery is the sum of her down payment and all of her monthly loan payments over the 3-year period. We can calculate this as follows:
Total amount paid = Down payment + (Monthly payment x Number of payments)
Total amount paid = 1200 + (365.07 x 36)
Total amount paid = 1200 + 13142.52
Total amount paid = $14,342.52
Therefore, the total amount Nicole ended up paying for the machinery is $14,342.52.
(b) To calculate how much interest Nicole paid on the loan, we first need to calculate the total amount of the loan. We can do this by subtracting her down payment from the total cost of the machinery:
Total loan amount = Total cost of machinery - Down payment
Total loan amount = $13,200 - $1,200
Total loan amount = $12,000
Next, we can calculate the total amount of interest paid over the 3-year period by subtracting the total loan amount from the total amount paid:
Total interest paid = Total amount paid - Total loan amount
Total interest paid = $14,342.52 - $12,000
Total interest paid = $2,342.52
Therefore, Nicole paid $2,342.52 in interest on the loan.
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n january 1,2020 flounder corportaion purchased 325 of the $1000 face value, 11%, 10-year bonds of Walters Inc. The bonds mature on January 1,2030 and pay interest annually beginning January 1, 2021. Flounder purchased bonds to yield 11%. How much did Flounder pay for the bonds?
Answer:
Step-by-step explanation:
To calculate how much Flounder paid for the bonds, we need to use the present value formula for a bond:
PV = C/(1+r)^1 + C/(1+r)^2 + ... + C/(1+r)^n + F/(1+r)^n
where PV is the present value, C is the annual coupon payment, r is the yield, n is the number of years, and F is the face value.
In this case, Flounder purchased 325 bonds with a face value of $1000 each, so the total face value of the bonds is:
325 * $1000 = $325,000
The coupon rate is 11%, which means that the annual coupon payment is:
0.11 * $1000 = $110
The bonds mature in 10 years, so n = 10. The yield is also 11%, so r = 0.11.
Using these values, we can calculate the present value of the bond:
PV = $110/(1+0.11)^1 + $110/(1+0.11)^2 + ... + $110/(1+0.11)^10 + $1000/(1+0.11)^10
PV = $110/(1.11)^1 + $110/(1.11)^2 + ... + $110/(1.11)^10 + $1000/(1.11)^10
PV = $110/1.11 + $110/(1.11)^2 + ... + $110/(1.11)^10 + $1000/(1.11)^10
PV = $110*(1-(1.11)^-10)/0.11 + $1000/(1.11)^10
PV = $750.98 + $314.23
PV = $1,065.21
Therefore, Flounder paid $1,065.21 for the 325 bonds of Walters Inc.
Participant A did 120 jumping jacks in 10 minutes. Participant B did 140 jumping jacks in 14 minutes. Which participant had the greater jumping jack rate?
Answer: Participant A
Step-by-step explanation:
if you divide 120 by 10 you would get 12 jumping jacks per minute and if you divide 140 by 14 you would get 10 jumping jacks per minute
cyryl hikes a distance of 0.75 kilomiters in going to school every day draw a number line to show the distance
Answer:
Step-by-step explanation:
Sure! Here's a number line showing the distance of 0.75 kilometers:
0 -------------|-------------|------------- 0.75 km
The "0" on the left represents the starting point (such as home), and the "|---|" in the middle represents the distance of 0.75 kilometers to the destination (such as school).
Scenario #3:
Imagine you find a map with a scale of 1:63,360. On that map, you see your hiking destination is
seven inches from your current location.
(a) How far away is that in reality (in miles)?
(b) Explain how you arrived at this decision.
SHOW YOUR WORK! This includes the potential for partial value, if incorrect.
Answer:
7 miles
Step-by-step explanation:
Note there are 63,360 inches in a mile.
So, rewrite scale as 1 inch: 1 mile, where inch replaces the unit.
Therefore 7 inches: 7 miles.
A random variable X follows the uniform distribution with a lower limit of 670 and an upper limit of 770.
a. Calculate the mean and the standard deviation for the distribution. (Round intermediate calculation for Standard deviation to 4 decimal places and final answer to 2 decimal places.)
Mean:
Standard deviation:
b. What is the probability that X is less than 710? (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)
Probability:
a) The mean is 720 and the standard deviation for the distribution is approximately 28.87
b. The probability that X is less than 710 is 33%
a. To calculate the mean and standard deviation, we use the following formulas:
Mean (μ) = (lower limit + upper limit) / 2
Standard deviation (σ) = (upper limit - lower limit) / √12
Substituting the values, we get:
Mean (μ) = (670 + 770) / 2 = 720
Standard deviation (σ) = (770 - 670) / √12 ≈ 28.87
b. To find the probability that X is less than 710, we use the following formula:
Probability (X < 710) = (710 - lower limit) / (upper limit - lower limit)
Substituting the values, we get:
Probability (X < 710) = (710 - 670) / (770 - 670) ≈ 0.33
This means that there is a 33% chance that X will be less than 710.
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can anyone help with this triangle question
The triangle's other leg, side B, measures 12 cm in length.
Are there 180 right triangles in all?When one of the interior angles is 90 degrees, or a right angle, the triangle is said to be a right triangle. The three internal angles of a triangle add up to 180 degrees in a right triangle because one angle must always be 90 degrees and the other two must always total to 90 degrees (they are complementary).
We can observe that the given triangle is a right triangle because angle A's measure is 90 degrees. The hypotenuse, which is represented by the letter "c," is the side that is opposite the right angle. The legs are the other two sides, and they are indicated by "a" and "b".
We are told that the hypotenuse (side c) is 13 cm long and that one leg (side a) is 5 cm long. The length of the other leg must be determined (side b).
The Pythagorean theorem, which asserts that in a right triangle, can be used.
a² + b² = c²
Inputting the values provided yields:
5² + b² = 13²
25 + b² = 169
b² = 144
b = 12
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6TH GRADE MATH, PLS HELP. FIND SLOPE OF THE TABLE, TY
bug s is slow and bug f is fast. both bugs start at 0 on a number line and move in the positive direction. the bugs leave 0 at the same time and move at constant speeds. four seconds later, f is at 12 and s is at 8. how many units apart are bugs f and s thirty seconds after leaving point 0.
After 30 seconds the bugs will be apart by (x -y) = 30 units
What is speed?The speed of an object, aIso known as "v" in everyday speech and kinematics, is a scaIar quantity that measures how much its position changes over time or how much it changes per unit of time.
The distance traveIIed by an object during a time intervaI is equaI to the duration of the intervaI divided by the object's speed, with the instantaneous speed being the upper Iimit of the average speed as the intervaI's duration gets cIoser to zero. VeIocity differs from speed. Distance divided by time is the formuIa for measuring speed. A metre per second (m/s) is the metric unit of speed.
a) The speed of F is 12 units in 4 seconds, or 3 units per second.
x = 3t
The speed of S is 8 units in 4 seconds, or 2 units per second.
y = 2t
b) x-y = 3t -2t = t
After 30 seconds the bugs will be apart by (x -y) = 30 units
c) x - y = 100 = t
The bugs will be 100 units apart after 100 seconds.
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The distance between bug f and bug s after 30 seconds is: distance = 90 - 60 = 30 units.
what is algebra?
Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.
When bug s and bug f leave point 0, they are initially at the same location, which is at position 0 on the number line. After four seconds, bug s has traveled a distance of 4v_s units from point 0, and bug f has traveled a distance of 4v_f units from point 0. We know that bug f has traveled 12 units in this time, so:
4v_f = 12
Solving for v_f, we get:
v_f = 3 units per second
Similarly, we know that bug s has traveled a distance of 8 units in four seconds, so:
4v_s = 8
Solving for v_s, we get:
v_s = 2 units per second
Now we can find out how far apart the bugs are after 30 seconds. Bug f has traveled a distance of 30v_f units from point 0, which is:
30v_f = 30 x 3 = 90 units
Similarly, bug s has traveled a distance of 30v_s units from point 0, which is:
30v_s = 30 x 2 = 60 units
Therefore, the distance between bug f and bug s after 30 seconds is:
distance = 90 - 60 = 30 units.
So the bugs are 30 units apart after 30 seconds.
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Alexander and Rhiannon left school at the same time. Alexander travelled 14 km home at an average speed of 20 km/h. Rhiannon travelled 10 km home at an average speed of 24 km/h. a) Who arrived home earlier? b) How much earlier did this person arrive at home? Give your answer to the nearest minute.
Rhiannon arrived home approximately 17 minutes earlier than Alexander.
What is the average?This is the arithmetic mean and is calculated by adding a group of numbers and then dividing by the count of those numbers. For example, the average of 2, 3, 3, 5, 7, and 10 is 30 divided by 6, which is 5.
According to the given information:To solve this problem, we can use the formula:
time = distance / speed
a) The time it took Alexander to get home is:
time_Alexander = 14 km / 20 km/h = 0.7 hours
The time it took Rhiannon to get home is:
time_Rhiannon = 10 km / 24 km/h = 0.41667 hours
Since Rhiannon's time is smaller than Alexander's, Rhiannon arrived home earlier.
b) The time difference between their arrivals is:
time_difference = time_Alexander - time_Rhiannon = 0.7 hours - 0.41667 hours = 0.28333 hours
To convert this to minutes, we can multiply by 60:
time_difference_in_minutes = 0.28333 hours x 60 minutes/hour ≈ 17 minutes
Therefore, Rhiannon arrived home approximately 17 minutes earlier than Alexander.
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Graph the linear equation.
42 + 6y = -12
Plot two points on the line to graph the line.
The graph of the linear function 4x + 6y = -12 is given by the image presented at the end of the answer.
How to define a linear function?The slope-intercept representation of a linear function is given by the equation presented as follows:
y = mx + b
The coefficients of the function and their meaning are described as follows:
m is the slope of the function, representing the change in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, which is the initial value of the function, i.e., the numeric value of the function when the input variable x assumes a value of 0. On a graph, it is the value of y when the graph of the function crosses the y-axis.The function for this problem is given as follows:
4x + 6y = -12.
In slope-intercept form, the function is given as follows:
6y = -4x - 12.
y = -2x/3 - 2.
The slope and the intercept are given as follows:
Intercept of b = -2, meaning that when x = 0, y = -2.Slope of -2/3, meaning that when x decays by 3, y increases by two, hence the graph also passes through point (-3,0).More can be learned about linear functions at https://brainly.com/question/24808124
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He has 2 pens. His friend gives him 2 more pens. How many pens he has?
Step-by-step explanation:
4 i guess... sry i m not good at maths
Suppose that Y1,Y2,Y3 denote a random sample from an exponential distribution with density funtion:
f(x) = (1/θ)e^-y/θf, y>0
Consider the following five estimators of θ:
θ(hat)1=Y1 ;
θ(hat)2= (Y1+Y2)/2
θ(hat)3= (Y1+2Y2)/3
θ(hat)4 = min(Y1,Y2,Y3)
θ(hat)5 = Y (bar)
a)Which of these estimators are unbiased and why?
b) Among the unbiased estimators, which has the smallest varienace?
Y1,Y2,Y3 denote a random sample from an exponential distribution with density funtion where θ(hat)1, θ(hat)2, θ(hat)3 and θ(hat)5 are unbiased. θ(hat)2 has the smallest variance among the unbiased estimators
To check for unbiasedness, we need to find the expected value of each estimator and see if it equals θ.
θ(hat)1 = Y1
E(θ(hat)1) = E(Y1) = θ
Thus, θ(hat)1 is an unbiased estimator.
θ(hat)2 = (Y1+Y2)/2
E(θ(hat)2) = E[(Y1+Y2)/2] = (1/2)[E(Y1) + E(Y2)] = θ
Thus, θ(hat)2 is an unbiased estimator.
θ(hat)3 = (Y1+2Y2)/3
E(θ(hat)3) = E[(Y1+2Y2)/3] = (1/3)[E(Y1) + 2E(Y2)] = θ
Thus, θ(hat)3 is an unbiased estimator.
θ(hat)4 = min(Y1,Y2,Y3)
For this estimator, we need to find the probability density function (pdf) of the minimum of the three random variables.
Let F_Y1_Y2_Y3 be the joint cumulative distribution function (cdf) of Y1,Y2,Y3, then the pdf of the minimum is given by:
f(θ(hat)4) = d/dy [F_Y1_Y2_Y3(y,y,y)] = 3(1/θ)e^-y/θf, y>0
E(θ(hat)4) = ∫0^∞ y(3/θ)e^-y/θ dy
= (3/θ) ∫0^∞ ye^-y/θ dy
Using integration by parts, we get:
= (3/θ) [θ + θ^2]
= 3θ + 3θ^2
Thus, θ(hat)4 is biased.
θ(hat)5 = Y (bar)
E(θ(hat)5) = E(Y (bar)) = E[(Y1+Y2+Y3)/3] = (1/3)[E(Y1) + E(Y2) + E(Y3)] = θ
Thus, θ(hat)5 is an unbiased estimator.
To compare the variances of the unbiased estimators, we need to compute their variances.
Var(θ(hat)1) = Var(Y1) = θ^2
Var(θ(hat)2) = Var[(Y1+Y2)/2] = (1/4)Var(Y1) + (1/4)Var(Y2) + (1/2)Cov(Y1,Y2)
Since Y1,Y2 are independent and identically distributed, Cov(Y1,Y2) = 0.
Thus, Var(θ(hat)2) = (1/4)θ^2 + (1/4)θ^2 = (1/2)θ^2
Var(θ(hat)3) = Var[(Y1+2Y2)/3] = (1/9)Var(Y1) + (4/9)Var(Y2) + (4/9)Cov(Y1,Y2)
Again, Cov(Y1,Y2) = 0, so Var(θ(hat)3) = (1/9)θ^2 + (4/9)θ^2 = (5/9)θ^2
Var(θ(hat)4) = Var(min(Y1,Y2,Y3))
Let X = min(Y1,Y2,Y3). Then,
F_X(x) = P(X ≤ x) = P(min(Y1,Y2,Y3) ≤ x)
= 1 - P(Y1 > x and Y2 > x and Y3 > x)
Since Y1,Y2,Y3 are independent,
= 1 - P(Y1 > x)P(Y2 > x)P(Y3 > x)
= 1 - (e^-x/θ)^3
Thus, the pdf of X is given by:
f_X(x) = d/dx[F_X(x)] = 3(e^-x/θ)^2 (1/θ)e^-x/θf, x>0
Using the formula for the variance of a continuous random variable, we get:
Var(θ(hat)4) = ∫0^∞ [x - E(θ(hat)4)]^2 f_X(x) dx
= ∫0^∞ [x - (3θ + 3θ^2)]^2 3(e^-x/θ)^2 (1/θ)e^-x/θf dx
= 6θ^2
Thus, Var(θ(hat)4) = 6θ^2.
Var(θ(hat)5) = Var(Y (bar)) = Var[(Y1+Y2+Y3)/3]
= (1/9)Var(Y1) + (1/9)Var(Y2) + (1/9)Var(Y3) + (2/9)Cov(Y1,Y2) + (2/9)Cov(Y1,Y3) + (2/9)Cov(Y2,Y3)
Since Y1,Y2,Y3 are independent, Cov(Yi,Yj) = 0 for i ≠ j.
Thus, Var(θ(hat)5) = (1/9)θ^2 + (1/9)θ^2 + (1/9)θ^2 = (1/3)θ^2.
Therefore, among the unbiased estimators, θ(hat)1, θ(hat)2, θ(hat)3, and θ(hat)5, the one with the smallest variance is θ(hat)2.
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Find the difference. 2.1 0.25 = ?
Answer: 1.85
Step-by-step explanation:
Find the measures of angles 1 through 5 in the figure shown !
Answer:
55 degrees angles on a rights angle triangle. 1 and 3 they are equal cause they are vertical opp angles 55 degrees
Michelle is baking chocolate chip cookies and she wants to cut the recipe in half. If the recipe calls for 3 cups of flour, how much flour should Michelle
add to her cookie dough?
0 1 cups of flour
O 13 cups of flour
0 1² cups of flour
0 1 cups of flour
Answer:
1 1/2 or 3/2
Step-by-step explanation:
Cutting the recipe in half means multiplying each measurement by 1/2.
3 x 1/2 = 3/2 or 1 1/2
please help I know its 9:35 PM I Just need help what this question2.1 × 1.6 =
21
10
×
16
10
= tenths × tenths my parents are gonna kill me help
The value of the expression 2.1 × 1.6 = 3.36.
What are decimals?Decimals are a collection of numbers falling between integers on a number line. They are only an additional mathematical representation of fractions. Decimals allow us to express quantifiable quantities like length, weight, distance, money, etc. with more accuracy. Integers, also known as whole numbers, are represented to the left of the decimal point, while decimal fractions are shown to the right of the decimal point.
Given that the expression is: 2.1 × 1.6.
2.1 × 1.6 can be written as:
2.1 × 1.6 = 21/10 × 16/10
Multiply the numerator and denominator:
21/10 × 16/10 = 336/100
Covert the fraction into decimal:
336/100 = 3.36
Hence, the value of the expression 2.1 × 1.6 = 3.36.
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For the given matrix A. find a 3 × 2 nonzero matrix B such that AB that any such matrix B must have rank 1. (Hint: The columns of B belong to the nullspace of A.] 0, Prove A= 13.421
The nullspace of matrix A found by solving the equation Ax=0. It is the nullspace of A is spanned by the vector [-2, 1, 0] and [0, -1, 1]. Any 3x2 matrix B that satisfies AB=0 belongs to the nullspace of A.
To find a 3×2 nonzero matrix B such that AB = 0, we need to find the nullspace of matrix A. The nullspace of a matrix A is the set of all vectors x such that Ax = 0.
Let's first write matrix A in row-echelon form:
1 2 1
0 -1 0
From this, we can see that the pivot variables are x1 and x2, and the free variable is x3. Thus, the general solution to Ax = 0 is given by:
x1 = -2x2 - x3
x2 = x2
x3 = x3
We can now write the columns of matrix B as:
[1, 0]
[-2, 1]
[0, -1]
To show that any such matrix B must have rank 1, we need to show that the columns of B are linearly dependent. Let's assume that B has rank 2. Then, the columns of B are linearly independent, and we can write:
c1[1, -2, 0] + c2[0, 1, -1] = [0, 0, 0]
where c1 and c2 are constants. This gives us the system of equations:
c1 = 0
-2c1 + c2 = 0
c2 = 0
which has only the trivial solution c1 = c2 = 0. This means that the columns of B are linearly dependent, and hence, any such matrix B must have rank 1.
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_____The given question is incomplete, the complete question is given below:
For the given matrix A.= [1 2 1 1 1 1] find a 3 × 2 nonzero matrix B such that AB = 0 that any such matrix B must have rank 1. (Hint: The columns of B belong to the nullspace of A.] 0, Prove A= 13.421
Find the standard normal area for each of the following (LAB)Round answers to 4 decimals
The answer of the standard normal area for each of the following questions are given below respectively.
What is standard normal area?Standard normal area refers to the area under the standard normal distribution curve, which is a normal distribution with a mean of 0 and a standard deviation of 1.
a. P(1.24<Z<2.14) = 0.0912
b. P(2.03 <Z<3.03) = 0.0484
c. P(-2.03 <Z<2.03) = 0.9542
d. P(Z > 0.53) = 0.2977
Note: The standard normal distribution is a continuous probability distribution with mean 0 and standard deviation 1. The area under the curve represents probabilities and can be calculated using a standard normal distribution table or a calculator with a normal distribution function.
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I need help with this
Answer: D
Step-by-step explanation:
i think so
Suppose that the cost C, in dollars, of processing the exhaust gases at an industrial site to ensure that only p percent of the particulate pollution escapes is given by the equation shown below. C(p) = 7000(100− p) p (a) Find the rate of change of cost C with respect to the percent of particulate pollution that escapes when p = 5 (percent). (b) Write a sentence that explains the meaning of your answer in part (a).the cost would increase by the absolute value of this amount if 6% instead of 5% of the particulate pollution were allowed to escape.The cost would decrease by this amount if 4% instead of 5% of the particulate pollution were allowed to escape.The cost would decrease by the absolute value of this amount if 6% instead of 5% of the particulate pollution were allowed to escape.The cost would increase by this amount for each percent of the particulate pollution over 5% allowed to escape.
The rate of change of cost C with respect to the percent of particulate pollution that escapes when p = 5 (percent) is -190000.
We know that the cost C, in dollars, of processing the exhaust gases at an industrial site to ensure that only p percent of the particulate pollution escapes is given by the equation C(p) = 7000(100− p) p
we have to find the rate of change of cost C with respect to the percent of particulate pollution that escapes when p = 5 (percent)
a) c (p) = 7600 (100-p) / p
c (p) = 7600 [p(-1) - (100-p) (1)] / 2p
= 7600/2p = 760000/4
= -190000
therefore, the rate of change of cost C with respect to the percent of particulate pollution that escapes when p = 5 (percent) is -190000.
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Work out the probability of scoring a total of 4
Answer: 1/12
Step-by-step explanation:
A piece of wire 18cm long is bent to form a rectangle. If its length is x cm, obtain an expression for its area in terms of
* and hence calculate the dimensions of the rectangle with maximum area
Answer:
To form a rectangle, the piece of wire will have two sides of length x and two sides of length (18 - 2x)/2 = 9 - x. Therefore, the perimeter of the rectangle is given by:
2x + 2(9 - x) = 18 - 2x
The area of the rectangle is given by:
A = x(9 - x)
Expanding this expression, we get:
A = 9x - x^2
To find the dimensions of the rectangle with maximum area, we can differentiate the area expression with respect to x:
dA/dx = 9 - 2x
Setting this equal to zero to find the maximum:
9 - 2x = 0
x = 4.5
So, one side of the rectangle is x = 4.5 cm and the other side is (18 - 2x)/2 = 4.5 cm. Therefore, the dimensions of the rectangle with maximum area are 4.5 cm by 4.5 cm.
To calculate the maximum area, we can substitute x = 4.5 into the area expression:
A = 9(4.5) - (4.5)^2 = 20.25 cm^2
Step-by-step explanation:
Let D be the disk with center the origin and radius a. What is the average distance from points in D to the origin?
Consider a disk D having center as origin and radius 'a', the average distance from point say ( r, θ) to the origin D to the origin is equals to the 2a/3.
The distance of a point to the origin is given by the function d(x, y) = √x²+ y². Denote the disk by D. To compute the average we need to evaluate the integral, for which we use polar coordinates, f(r, θ) = (1/area of region R)∬ f(r,θ)dA.
We have a disk D with center the origin (0,0) and radius 'a'. We have to determine the average distance from points in D to the origin. Since r is defined as the distance from the origin, the distance of any point (r,θ) from the origin is f(r,θ) = r
Integrating f over the region D, with limits r = 0 to a, and 0≤ θ≤2π , dA = r×dr×dθ
[tex]∬f(r,θ)dA = \int_{0}^{a} \int_{0}^{2π} r (rdr)dθ [/tex]
[tex]= \int_{0}^{2\pi} \int_{0}^{a} r^{2} drdθ = \int_{0}^{2\pi}[\frac{{r}^{3}}{3}]_{0}^{a}dθ[/tex]
[tex]= \int_{0}^{2π}[ \frac{a³}{3} ] dθ = \frac{{a}^{3}}{3} \int_{0}^{2π}dθ[/tex]
[tex]= \frac{a³}{3} [θ]_{0}^{2π} = 2\pi(\frac{a³}{3})[/tex]
Area of disk D with radius 'a' = πa²
So, the average distance from point ( r,θ) in D to the origin is = (1/area of region D)∬f(r,θ)dA
= (1/πa²)( 2πa³/3)
= 2a/3
Hence, required distance is 2a/3.
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Curious about people's recycling behaviors, Sandra put on some gloves and sifted through some recycling and trash bins. She kept count of the plastic type of each bottle and which bottles are properly dispensed.
What is the probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle? Please show your work.
The probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is 0.25 or 25%
What is Conditional probability?
Conditional probability is the probability of an event occurring given that another event has occurred or is known to have occurred. It is denoted by P(A|B), which reads as "the probability of A given B."
The formula for conditional probability is:
P(A|B) = P(A and B) / P(B)
where P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.
The total number of Plastic #2 bottles is 8 (correctly placed) + 5 (incorrectly placed) = 13.
The total number of Plastic #4 bottles is 5 (correctly placed) + 2 (incorrectly placed) = 7.
The probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is given by:
(number of Plastic #4 bottles correctly placed) / (total number of bottles)
So the probability is:
5/20 = 1/4 = 0.25
Therefore, the probability that a randomly selected bottle is correctly placed AND is a Plastic #4 bottle is 0.25 or 25%.
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ANSWER NEEDED IN 10 SEC
write the equation of a circle whose center is at latex: \left(-11,\:15\right) and whose radius is latex: 9 9 .
The equation of the circle with center at (-11, 15) and radius 9 is found to be x² + y² + 22x -30y + 337 = 0.
The center of the circle is located at the point (-11, 15). The radius of the circle is given to be 9.
Now, using the standard equation of the circle that has point (a, b) as center and r as the radius, we can write,
(x-a)² + (y-b)² = r²
Now, putting all the values,
(x-(-11))² + (y-15)² = (3)²
Solving further,
x² + 121 + 22x + y² + 225 - 30y = 9
x² + y² + 22x -30y + 337 = 0
So, the equation of the circle is found to be x² + y² + 22x -30y + 337 = 0.
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b. A kennel has 90 dogs in total, some are puppies and some are adult dogs. The ratio of puppies to adult dogs in a kennel is 1:4. What fraction of dogs are adults? Use this to work out how many adult dogs there are. See the examples on the formula sheet for this assessment.
Answer:4/5 are adults. There are 72 adult dogs.
Step-by-step explanation:
What is the area of the rhombus?
Answer:
36 units²------------------------
As per given coordinates we can observe that the diagonals of the rhombus are horizontal and vertical.
The length of the diagonal is the difference of x-coordinates for the horizontal diagonal and the difference of y-coordinates for the vertical one.
The diagonals have the length:
4 - (-8) = 12 units, and-1 - (-7) = 6 unitsThe area of a rhombus is half the product of the diagonals, therefore the area of the given rhombus is:
A = 12*6/2 = 36 units²The area οf the given rhοmbus is 22.5 square units. Tο find the area οf the rhοmbus, we need tο first find the length οf its diagοnals.
what is a rhοmbus?A rhοmbus is a type οf quadrilateral that has fοur sides οf equal length. It is alsο knοwn as a diamοnd οr a lοzenge. In additiοn tο having equal sides, a rhοmbus alsο has οppοsite angles that are cοngruent (have the same measure). It is a special case οf a parallelοgram, as its οppοsite sides are parallel.
Tο find the area οf the rhοmbus, we need tο first find the length οf its diagοnals. We can dο this using the distance fοrmula:
[tex]\rm d_1 = \sqrt{((4 - (-2))^2 + (-4 - (-1))^2)} = \sqrt{(6^2 + (-3)^2)} = \sqrt{(45)[/tex]
[tex]\rm d_2 = \sqrt{((-8 - (-2))^2 + (-4 - (-7))^2)} = \sqrt{((-6)^2 + 3^2)} = \sqrt{(45)[/tex]
Now that we have the lengths of the diagonals, we can use the formula for the area of a rhombus:
Area = [tex]\rm (d_1 \times d_2) / 2 =\dfrac{ (\sqrt{(45)} \times \sqrt{(45)}}{2}[/tex]
= (45/2) square units
Therefore, the area of the given rhombus is 22.5 square units.
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Question 13 (2 points)
Suppose you flip a coin and then roll a die. You record your result. What is the
probability you flip heads or roll a 3?
1/2
3/4
7/12
1
Step-by-step explanation:
a probability is always the ratio
desired cases / totally possible cases
we have 2 possible cases for the coin and 6 possible cases for the die.
so, we have 2×6 = 12 combined possible cases :
heads, 1
heads, 2
heads, 3
heads, 4
heads, 5
heads, 6
tails, 1
tails, 2
tails, 3
tails, 4
tails, 5
tails, 6
out of these 12 cases, which ones (how many) are desired ?
all first 6 plus (tails, 3) = 7 cases
so, the correct probability is
7/12
formally that is calculated :
1/2 × 6/6 + 1/2 × 1/6 = 6/12 + 1/12 = 7/12
the probability to get heads combined with the probability to roll anything on the die, plus the probability to get tails combined with the probability to roll 3.