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zaharov [31]
3 years ago
12

The Cartesian coordinates of a point are given. (a) (−5, 5) (i) Find polar coordinates (r, θ) of the point, where r > 0 and 0

≤ θ < 2π. (r, θ) = (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) = (b) (3, 3 3 ) (i) Find polar coordinates (r, θ) of the point, where r > 0 and 0 ≤ θ < 2π. (r, θ) = (ii) Find polar coordinates (r, θ) of the point, where r < 0 and 0 ≤ θ < 2π. (r, θ) =
Mathematics
1 answer:
Alex73 [517]3 years ago
6 0

Answer:

a)

(i) The coordinates of the point in polar form is (5√2 , 7π/4)

(ii) The coordinates of the point in polar form is (-5√2 , 3π/4)

b)

(i) The coordinates of the point in polar form is (6 , π/3)

(ii) The coordinates of the point in polar form is (-6 , 4π/3)

Step-by-step explanation:

* Lets study the meaning of polar form

- To convert from Cartesian Coordinates (x,y) to Polar Coordinates (r,θ):

1. r = √( x2 + y2 )

2. θ = tan^-1 (y/x)

- In Cartesian coordinates there is exactly one set of coordinates for any

 given point

- In polar coordinates there is literally an infinite number of coordinates

 for a given point

- Example:

- The following four points are all coordinates for the same point.

# (5 , π/3) ⇒ 1st quadrant

# (5 , −5π/3) ⇒ 4th quadrant

# (−5 , 4π/3) ⇒ 3rd quadrant

# (−5 , −2π/3) ⇒ 2nd quadrant

- So we can find the points in polar form by using these rules:

 [r , θ + 2πn] , [−r , θ + (2n + 1) π] , where n is any integer

 (more than 1 turn)

* Lets solve the problem

(a)

∵ The point in the Cartesian plane is (-5 , 5)

∵ r = √x² + y²

∴ r = √[(5)² + (-5)²] = √[25 + 25] = √50 = ±5√2

∵ Ф = tan^-1 (y/x)

∴ Ф = tan^-1 (5/-5) = tan^-1 (-1)

- Tan is negative in the second and fourth quadrant

∵ 0 ≤ Ф < 2π

∴ Ф = 2π - tan^-1(1) ⇒ in fourth quadrant r > 0

∴ Ф = 2π - π/4 = 7π/4

OR

∴ Ф = π - tan^-1(1) ⇒ in second quadrant r < 0

∴ Ф = π - π/4 = 3π/4

(i) ∵ r > 0

∴ r = 5√2

∴ Ф = 7π/4 ⇒ 4th quadrant

∴ The coordinates of the point in polar form is (5√2 , 7π/4)

(ii) r < 0

∴ r = -5√2

∵ Ф = 3π/4 ⇒ 2nd quadrant

∴ The coordinates of the point in polar form is (-5√2 , 3π/4)

(b)

∵ The point in the Cartesian plane is (3 , 3√3)

∵ r = √x² + y²

∴ r = √[(3)² + (3√3)²] = √[9 + 27] = √36 = ±6

∵ Ф = tan^-1 (y/x)

∴ Ф = tan^-1 (3√3/3) = tan^-1 (√3)

- Tan is positive in the first and third quadrant

∵ 0 ≤ Ф < 2π

∴ Ф = tan^-1 (√3) ⇒ in first quadrant r > 0

∴ Ф = π/3

OR

∴ Ф = π + tan^-1 (√3) ⇒ in third quadrant r < 0

∴ Ф = π + π/3 = 4π/3

(i) ∵ r > 0

∴ r = 6

∴ Ф = π/3 ⇒ 1st quadrant

∴ The coordinates of the point in polar form is (6 , π/3)

(ii) r < 0

∴ r = -6

∵ Ф = 4π/3 ⇒ 3rd quadrant

∴ The coordinates of the point in polar form is (-6 , 4π/3)

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Answer:

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Step-by-step explanation:

When a line is perpendicular to another, their slopes will be opposite reciprocal. For example, 1 would be -1, -3 would be \frac{1}{3} , and \frac{5}{1} would be -5. The equation is written in slope-intercept form:

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m is the slope, so find the equation with the opposite reciprocal of 3, -\frac{1}{3}:

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3 years ago
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3 years ago
Bob rolls a fair six-sided die each morning. If Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime numb
hammer [34]

Bob eats a 171 days sweetened cereal and 194 days he eat unsweetened cereal.

According to the statement

we have given that the Bob rolls a composite number, he eats sweetened cereal. If he rolls a prime number, he eats unsweetened cereal.

and we have to find that the expected value of the difference between the number of days.

So, the total number of days in a year is 365 days.

Chance to get a prime number on a die = 3/6

Chance to get a prime number on a die = 1/2.

and chances to get a composite number = 2/6

chances to get a composite number = 1/3

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Then the expected value =

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then after calculating

Expected value = 1+2+1

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So, Bob eats a 171 days sweetened cereal and 194 days he eat unsweetened cereal.

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3 0
1 year ago
The following shows the monthly sales in units of six salespersons before and after a bonus plan was introduced. At 95% confiden
Anarel [89]

Answer:

Since the calculated value of t= 2.249 does not  falls in the rejection region we therefore accept  the null hypothesis at 5 % significance level . On the basis of this we conclude that the  bonus plan has not  increased sales significantly.

Step-by-step explanation:

The null and alternative hypotheses as

H0: μd=0    Ha: μd≠0

Significance level is set at ∝= 0.05

n= 6

degrees of freedom = df = 6-1 = 5

The critical region is t  ( base alpha by 2 with df=5) ≥ ± 2.571

The test statistic under H0 is

t = d/ sd/ √n

Which has t distribution with n-1 degrees of freedom

Sales                                      Difference

Person      After      Before   d = after - before      d²

1                   94          90                4                      16

2                  82          84               -2                       4

3                  90          84               6                       36

4                  76           70               6                       36

5                  79           80               -1                        1

<u>6                  85          80                 5                       25   </u>

∑                                                       18                   118

d`= ∑d/n= 18/6= 3

sd²= 1/6( 118- 18²/6) = 1/6 ( 118 - 54) = 10.67

sd= 3.266

t= 3/ 3.266/ √6

t= 2.249

Since the calculated value of t= 2.249 does not  falls in the rejection region we therefore accept  the null hypothesis at 5 % significance level . On the basis of this we conclude that the  bonus plan has not  increased sales significantly.

7 0
3 years ago
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Angelina_Jolie [31]

Answer: 5-13i

Step-by-step explanation:

5 0
2 years ago
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