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Lyrx [107]
3 years ago
14

A homeowner has two smoke detector alarms installed, one in the dining room (adjacent to the kitchen) and one in an upstairs bed

room (above the kitchen). If cooking produces smoke in the kitchen, the probability of setting off the dining room alarm (D) is .95. The probability of setting off the bedroom alarm (B) is .40. The two alarms detect smoke independently of each other. If there is smoke in the kitchen, what is the probability that the smoke will be detected and will set off an alarm
Mathematics
1 answer:
cestrela7 [59]3 years ago
7 0

Answer:

0.97

Step-by-step explanation:

Given that a homeowner has two smoke detector alarms installed, one in the dining room (adjacent to the kitchen) and one in an upstairs bedroom (above the kitchen). If cooking produces smoke in the kitchen, the probability of setting off the dining room alarm (D) is .95. The probability of setting off the bedroom alarm (B) is .40.

Both alarms are independent of each other.

Probability for smoke detection=P(any one alarm rings)

=P(Kitchen alarm sets off)+P(bed room alarm sets off)-P(Both sets off)

Since both are independent

P(Both) = 0.95*0.40\\=0.38

Probability for smoke detection

= 0.95+0.40-0.38

=0.97

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A radio station signal covers a circular area out from the signal tower. It gets weaker the farther away your are. The local sta
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1. We would hear better at Dunkin's since its distance from the radio station is the shortest.

2. McDonald's is out of range of the radio station since its distance is greater than the range of the radio station.

Step-by-step explanation:

1) Will we hear it better at Zaxby's (2,-4), McDonalds (4, -4), or Dunkin' (-3,3). Show work to support your decision and explain your reasoning.

We need to find the distance of each point from the tower.

The shorter the distance from the radio station the better the signal.

The distance between two points (x₁, y₁) and (x₂, y₂) is given by d = √[(x₂ - x₁)² + (y₂ - y₁)²].

With (x₁, y₁) = (0, 0) the location of local station WDHS

For Zaxby's distance located at (2, -4) = (x₂, y₂)

So, d₁ = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(2 - 0)² + (-4 - 0)²]

= √[(2)² + (-4)²]

= √[4 + 16]

= √20

= √(5 × 4)

= √5 × √4

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For McDonald's distance located at (4, -4) = (x₂, y₂)

So, d₂ = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(4 - 0)² + (-4 - 0)²]

= √[(4)² + (-4)²]

= √[16 + 16]

= √32

= √(16 × 2)

= √16 × √2

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= 5.66 miles

For Dunkin's distance located at (-3, 3) = (x₂, y₂)

So, d₃ = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(-3 - 0)² + (-3 - 0)²]

= √[(-3)² + (-3)²]

= √[9 + 9]

= √18

= √(9 × 2)

= √9 × √2

= 3√2

= 4.24 miles

<u>Since Dunkin's distance, d₃ = 4.24 is the least distance from the radio station and is within range of 5 miles, we would hear better at Dunkin's.</u>

2) Are any of these location outside the range of the radio station? If so, which one(s). Defend your decision mathematically.

The radio station is outside of range if its distance is greater than 5 miles. That is, d > 5 miles.

So, we calculate the distance of each point from the radio station.

With (x₁, y₁) = (0, 0) the location of local station WDHS

For Zaxby's distance located at (2, -4) = (x₂, y₂)

So, d₁ = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(2 - 0)² + (-4 - 0)²]

= √[(2)² + (-4)²]

= √[4 + 16]

= √20

= √(5 × 4)

= √5 × √4

= 2√5

= 4.47 miles

For McDonald's distance located at (4, -4) = (x₂, y₂)

So, d₂ = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(4 - 0)² + (-4 - 0)²]

= √[(4)² + (-4)²]

= √[16 + 16]

= √32

= √(16 × 2)

= √16 × √2

= 4√2

= 5.66 miles

For Dunkin's distance located at (-3, 3) = (x₂, y₂)

So, d₃ = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(-3 - 0)² + (-3 - 0)²]

= √[(-3)² + (-3)²]

= √[9 + 9]

= √18

= √(9 × 2)

= √9 × √2

= 3√2

= 4.24 miles

<u>So, for each of the distances calculated above, only McDonald's distance, d₂ = 5.66 miles > 5 miles. </u>

<u>So, McDonald's is out of range of the radio station.</u>

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