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Nikitich [7]
3 years ago
12

WILL GIVE THANKS, BRAINIEST, AND 5 STARS! PLEASE HELP!

Mathematics
1 answer:
Andrew [12]3 years ago
6 0

Answer:

a = \frac{1}{2} , b = 5, and  \frac{2b}{a} = 20

Step-by-step explanation:

f(x) = - x² + b, x ≤ 0     You would use this equation for f(-2). Since -2 is less than 0.

f(x) = - x² + b, x ≤ 0   ← You can ignore the x ≤ 0 part.

f(-2) = - (-2)² + b         Input the value -2 as x. The question states that f(-2) =1.

1 = - (-2)² + b              So switch f(-2) with 1 on the left side only.

1 = - (4) + b                Simplify. Do the exponents first, so (-2)² = 4.

1 = - 4 + b

<u>+4  +4   </u>                    Do inverse operations

5 = b

Next,

f(x) = 2ax +3, x > 0      You would use his equation for f(2). Since 2 is greater than 0.

f(x) = 2ax +3, x > 0   ← You can ignore the x > 0 part.

f(2) = 2a(2) + 3             Input the value 2 as x.

f(2) = 4a +3           Simplify. The equation states f(2) = 5. So switch f(2) with 5

5 = 4a +3              on the left side only.

<u>-3        - 3</u>                     Do inverse operations

2 = 4a                          

\frac{2}{4} = \frac{4a}{4}                           Divide 4 on both sides to isolate the variable a

\frac{2}{4} = a                            Simplify

\frac{1}{2} = a

Then,

The second part says to find \frac{2b}{a}

\frac{2b}{a}

\frac{2(5)}{(\frac{1}{2}) }                  Input both the values of a and b

\frac{10}{\frac{1}{2} }                    Simplify

20.

The answer for a is \frac{1}{2}, the answer for b is 5, and the answer for \frac{2b}{a} is 20.

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Answer:

a) 0.0486 = 4.86% probability that exactly two of the four components last longer than 1000 hours.

b) 0.9996 = 99.96% probability that the subsystem operates longer than 1000 hours.

Step-by-step explanation:

For each component, there are only two possible outcomes. Either they last more than 1,000 hours, or they do not. Components operate independently, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

In which C_{n,x} is the number of different combinations of x objects from a set of n elements, given by the following formula.

C_{n,x} = \frac{n!}{x!(n-x)!}

And p is the probability of X happening.

One subsystem has eight identical components, each with a probability of 0.1 of failing in less than 1,000 hours.

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P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

P(X = 2) = C_{4,2}.(0.9)^{2}.(0.1)^{2} = 0.0486

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b. the subsystem operates longer than 1000 hours.

The subsystem has 8 components, which means that n = 8

It will operate if at least 4 components are working correctly, so we want:

P(X \geq 4) = 1 - P(X < 4)

In which

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

So

P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}

P(X = 0) = C_{8,0}.(0.9)^{0}.(0.1)^{8} \approx 0

P(X = 1) = C_{8,1}.(0.9)^{1}.(0.1)^{7} \approx 0tex][tex]P(X = 2) = C_{8,2}.(0.9)^{2}.(0.1)^{6} \approx 0

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Then

P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0 + 0 + 0 + 0.0004 = 0.0004

P(X \geq 4) = 1 - P(X < 4) = 1 - 0.0004 = 0.9996

0.9996 = 99.96% probability that the subsystem operates longer than 1000 hours.

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