Find the distance between the points with polar coordinates (4, π/3) and (8, 2π/3).
1 answer:
A suitable vector calculator can perform this arithmetic directly, as can many graphing or scientific calculators that are capable of handling complex numbers. Mine says
(8, 2π/3) - (4, π/3) = (4√3, 5π/6)
The distance is 4√3 ≈ 6.9282.
If you cannot use your calculator for this purpose, you can use the Law of Cosines. You have two sides of the triangle (4 and 8) and the angle between them (2π/3 - π/3), so you have all the information needed.
c² = a² + b² -2ab·cos(C)
c² = 4² + 8² - 2·4·8·cos(π/3)
c² = 16 + 64 - 32 = 48
c = √48 = 4√3
Or, you can simply recognize that the two vectors have the ratio 1:2 and the angle between them is 60°. That is, they are one leg and the hypotenuse of a 30°-60°-90° triangle, so the other leg is 4√3.
(8, 2π/3) - (4, π/3) = (4√3, 5π/6)
The distance is 4√3 ≈ 6.9282.
If you cannot use your calculator for this purpose, you can use the Law of Cosines. You have two sides of the triangle (4 and 8) and the angle between them (2π/3 - π/3), so you have all the information needed.
c² = a² + b² -2ab·cos(C)
c² = 4² + 8² - 2·4·8·cos(π/3)
c² = 16 + 64 - 32 = 48
c = √48 = 4√3
Or, you can simply recognize that the two vectors have the ratio 1:2 and the angle between them is 60°. That is, they are one leg and the hypotenuse of a 30°-60°-90° triangle, so the other leg is 4√3.
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Answer:
The equation that represents the relationship between 'y' the total points, and 'x' the number of movies is option A;
A y = 3·x
Step-by-step explanation:
The information required in the question is an equation that represents the the relationship between the number of movie attended and the total number of points Marcy has given that she earns a fixed amount of points for attending each movie
The number of points she will earn for attending each movie = 3 points
The total number of points she will earn 'y' when she attends 'x' number of movies is given as follows;
y = 3 + 3 + 3 + ... + 3, 'x' number of times, and we can write
Therefore the equation that represents the relationship between 'y' the total points, and 'x' the number of movies is y = 3·x.
Answer:
For the pairs that satisfy that the sum is less than 7 we have:
(1,1) (2,1) (1,2) (3,1) (2,2) (1,3) (4,1) (3,2) (2,3) (1,4) (5,1) (4.2) (3,3) (2,4) (1,5)
So we have a total of 15 pairs where the sum is less than 7
And for an odd sum we have the following pairs
(2,1) (1,2) (4,1) (3,2) (2,3) (1,4) (6,2) (5,2) (4,3) (3,4) (2,5) (1,6) (6,3) (5,4) (4,5) (3,6) (6,5) (5,6)
A total of 18 pairs and we have 6 pairs who are in both cases for the sum less than 7 and the sum an odd number that represent the intersection and using the total probability rule we got:
Step-by-step explanation:
For this case when a pair of dice is rolled we have the following sample pace for the outcomes:
(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3,2) (3,3) (3,4) (3,5) (3,6)
(4,1) (4,2) (4,3) (4,4) (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6)
(6,1) (6,2) (6,3) (6,4) (6,5) (6,6)
We see 36 possible outcomes and we want to find how many of these pairs we got rolling an odd sum or a sum less than 7
For the pairs that satisfy that the sum is less than 7 we have:
(1,1) (2,1) (1,2) (3,1) (2,2) (1,3) (4,1) (3,2) (2,3) (1,4) (5,1) (4.2) (3,3) (2,4) (1,5)
So we have a total of 15 pairs where the sum is less than 7
And for an odd sum we have the following pairs
(2,1) (1,2) (4,1) (3,2) (2,3) (1,4) (6,2) (5,2) (4,3) (3,4) (2,5) (1,6) (6,3) (5,4) (4,5) (3,6) (6,5) (5,6)
A total of 18 pairs and we have 6 pairs who are in both cases for the sum less than 7 and the sum an odd number that represent the intersection and using the total probability rule we got: