Which statement is true?
1 answer:
You might be interested in
<h3>Given</h3>
- ΔABC
- A(-3, -1), B(0, 3), C(1, 2)
- the length of the perimeter of ΔABC to the nearest tenth
The perimeter of a triangle is the sum of the lengths of its sides. The length of each side can be found using the Pythagorean theorem. Effectively, each pair of points is treated as the end-points of the hypotenuse of a right triangle with legs parallel to the x- and y-axes. The leg lengths are the differences betweeen the x- and y- coordinates of the points.
The difference of the x-coordinates of segment AB are 0-(-3) = 3. The y-coordinate difference is 3-(-1) = 4. So, the leg lengths of the right triangle whose hypotenuse is segment AB are 3 and 4. The Pythagorean theorem tells us
... AB² = 3² +4² = 9 +16 = 25
... AB = √25 = 5
You may recognize this as the 3-4-5 triangle often introduced as one of the first ones you play with when you learn the Pythagorean theorem.
LIkewise, segment AC has coordinate differences of ...
... C - A = (1, 2) -(-3, -1) = (4, 3)
These are the same leg lengths (in the other order) as for segment AB, so its length is also 5.
Segment BC has coordinate differences ...
... C - B = (1, 2) -(0, 3) = (1, -1)
The length of the line segment is figured as the root of the sum of squares, even though one of the coordinate differences is negative. The leg lengths of the right triangle used for finding the length of BC are the absolute value of these differences, or 1 and 1. Then the length BC is
... BC = √(1² +1²) = √2 ≈ 1.4
So the perimeter of the triangle ABC is
... AB + BC + AC = 5 + 1.4 + 5 = 11.4 . . . . perimeter of ∆ABC in units
_____
Please be aware that the advice to "round each step" is <em>bad advice,</em> in general. For real-world math problems, you only round the final result. You always carry at least enough precision in the numbers to ensure that there will not be any error in the final rounding.
In this problem, the only number that is not an integer is √2, so it doesn't really matter.
Answer:
(a) P(M) = 155/254
P(C) = 76/127
P(M ∩ C) = 55/127
(b) P(M U C) = 197/254
(c) P(Neither of the perks) = 57/254
(d) Probability tree drawn.
(e) P(C'|M) = 9/31
(f) P(M'|C') = 57/102
Step-by-step explanation:
The question states that:
Total executives = 254
Executives with mobile phones = 155
Executives with club memberships = 152
Executives with both mobile phones and club memberships = 110
(a) P(M) = No. of executives with mobile phones/Total no. of executives
= 155/254
P(M) = 155/254
P(C) = No. of executives with club memberships/Total no. of executives
= 152/254
P(C) = 76/127
P(M ∩ C) = No. of executives with both mobile phones and club memberships/Total no. of executives
= 110/254
P(M ∩ C) = 55/127
(b) We are asked to find the probability that a corporate has at least one of the two perks i.e. either they have a mobile phone or a club membership which means we need to find P(M U C).
P(M U C) = P(M) + P(C) - P(M ∩ C)
= 155/254 + 152/254 - 110/254
P(M U C) = 197/254
(c) The probability that a corporate executive does not have either of these perks can be calculated by subtracting the probability that a corporate executive has at least one of these perks from the total probability (i.e. 1). So,
P(Neither of the perks) = 1 - P (M U C)
= 1 - 197/254
P(Neither of the perks) = 57/254
(d) Probability tree can be drawn in two stages where the first stage represents the ownership of mobile phone and the second stage represents the ownership of club membership.
M = having a mobile phone
M' = not having a mobile phone
C = having a club membership
C' = not having a club membership
I have drawn the probability tree and attached it as an image.
(e) We will use the conditional probability formula here to calculate the probability that a corporate executive does not have club membership given that that executive has a mobile phone
P(C'|M) = P(C' ∩ M) / P(M)
P(C' ∩ M) is the number of executives who do not have a club membership but only have a mobile phone. We can calculate the no. of executives with only mobile phones as:
Executives with mobile phones - Executives with both mobile phones and club memberships
= 155 - 110 = 45 executives with only mobile phones
So, P(C' ∩ M) = 45/254
P(C'|M) = (45/254)/(155/254)
P(C'|M) = 9/31
(f) We will again use the conditional probability formula here. We need P(M'|C'). So,
P(M'|C') = P(M' ∩ C')/(P(C')
P(M' ∩ C') represents the number of people who do not have a mobile phone nor a club membership. i.e. the number of corporate executives who have neither of these perks. We calculated this probability in part (c).
P(C') is the number of people who do not have a club membership. These include the number of people who have only a mobile phone and the people who have neither of these things. So,
P(C') = P(C' ∩ M) + P(M' U C')
= 45/254 + 57/254
P(C') = 102/254
So, P(M'|C') = P(M' ∩ C')/(P(C')
= (57/254)/(102/254)
P(M'|C') = 57/102
