MATHEMATICS COLLEGE

Sketch the region R defined by 1 ≤ x ≤ 2 and 0 ≤ y ≤ 1/x^3 .a. Find (exactly) the number a such that the line x = a divides R into two parts of equal area.
b. Then find (to 3 decimal places) the number b such that the line y = b divides R into two parts of equal area.

Answers

Answer 1
Answer: For part (a), you're looking to find a such that

\displaystyle\int_1^a(\mathrm dx)/(x^3)=\int_a^2(\mathrm dx)/(x^3)

You have

\displaystyle\int_1^a(\mathrm dx)/(x^3)=-\frac1{2x^2}\bigg|_(x=1)^(x=a)=-\frac12\left(\frac1{a^2}-1\right)

and

\displaystyle\int_a^2(\mathrm dx)/(x^3)=-\frac1{2x^2}\bigg|_(x=a)^(x=2)=-\frac12\left(\frac14-\frac1{a^2}\right)

Setting these equal, you get

\displaystyle-\frac12\left(\frac1{a^2}-1\right)=-\frac12\left(\frac14-\frac1{a^2}\right)\implies a=2√(\frac25)

For part (b), you have

y=\frac1{x^3}\implies x=\frac1{\sqrt[3]y}

and you want to find b such that

\displaystyle\int_0^(1/8)\mathrm dy+\int_(1/8)^b(\mathrm dy)/(\sqrt[3]y)=\int_b^1(\mathrm dy)/(\sqrt[3]y)

You have

\displaystyle\int_0^(1/8)\mathrm dy+\int_(1/8)^b(\mathrm dy)/(y^(1/3))=\frac18+\frac32y^(2/3)\bigg|_(y=1/8)^(y=b)=-frac14+\frac32b^(2/3)

and

\displaystyle\int_b^1(\mathrm dy)/(y^(1/3))=\frac32y^(2/3)\bigg|_(y=b)^(y=1)=\frac32-\frac32b^(2/3)

Setting them equal gives

-\frac14+\frac32b^(2/3)=\frac32-\frac32b^(2/3)\implies b=\frac7{24}√(\frac73)\approx0.446

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An installation technician for a specialized communication system is dispatched to a city only when 3 or more orders have been placed. Suppose the orders follow a Poisson distribution with a mean of 0.25 per week for a city with a population of 100,000 and suppose your city contains 800,000.a. What is the probability that a technician is required after a one-week period?
b. If you are the first one in the city to place an order, what is the probability that you have to wait more than two weeks from the time you place your order until a technician is dispatched?

Answers

Answer:

0.3233

0.09

Step-by-step explanation:

Given that :

Mean, λ = 0.25 for a 100,000 per week

For a population of 800,000 :

λ = 800,000 / 100,000 * 0.25 = 8 * 0.25 = 2 orders per week

Probability that technician is required after one week ;

After one week, order is beyond 2 ; hence, order, x ≥ 3

P(x ≥ 3) = 1 - [p(x=0) + p(x= 1) + p(x =2)]

P(x ≥ 3) = 1 - e^-λ(1 + 2¹/1! + 2²/2!)

P(x ≥ 3) = 1 - e^-2(1+2+2) = 1 - e^-2*5 = 1 - e^-10

P(x ≥ 3) = 1 - e^-2 * 10

P(x ≥ 3) = 1 - 0.6766764

P(x ≥ 3) = 0.3233

B.)

Mean, λ for more than 2 weeks = 2 * 2 = 4

P(x < 2) = p(x = 0) + p(x = 1)

P(x < 2) = e^-4(0 + 4^1/1!)

P(x < 2) = e^-4(0 + 4) = e^-4(5)

P(x < 2) = e^-4(5) = 0.0183156 * 5 = 0.0915

P(x < 2) = 0.09

What is the solution set for 4x - 6 < 6? {x:x > 3}

{x:x < 3}

{x:x < -1}

{x:x >_ 3}

Answers

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⇒ 4x < 6 + 6

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A binomial experiment consists of 16 trials. The probability of success on trial 9 is 0.48. What is the probability of success on trial 13?

Answers

Answer:

p = 0.48

Step-by-step explanation:

A binomial experiment is and experiment with n trials, every trial is identical and independent and every trial has the same probability p of success and 1-p of fail.

Then, we have a binomial experiment of 16 trials. it means that every trial has the same conditions. So, if the probability of success on trial 9 is 0.48, the probability of success on trial 13 is also 0.48.

Please answer correctly will mark brainliest

Answers

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Answers

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2. Anne needs to know how much of her back yard will be used by her newcircular pool. *
1 point
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Answers

Answer:

see below

Step-by-step explanation:   6  13  8  09

area = π r²    is the equation to calculate the area of the pool   r = radius

I don't no if the 1.11 ft is the diameter of the radius, so I will use the 1.11 ft as the diameter

diameter = 2×radius

diameter / 2 = radius

area = π (d/2)²   = T (d/2)²              d = diameter      π  = T      

        = 3.14(1.11 / 2)²              

        = 3.14 × (0.555)²

        = 0.9677 ft²    which seems like a small pool!
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