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grigory [225]
3 years ago
10

A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through a

n angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water?

Mathematics
1 answer:
sukhopar [10]3 years ago
5 0

Answer:

  θ = 60°

Step-by-step explanation:

The cross sectional area of the trapezoid shape will be that of a trapezoid with bases of 10 cm and (10 cm + 2·(10 cm)·cos(θ)) and height (10 cm)·sin(θ).

That area in cm² is ...

  A = (1/2)(b1 +b2)h = (1/2)(10 + (10 +20cos(θ))(10sin(θ)

  A = 100sin(θ)(1 +cos(θ))

A graphing calculator shows this area to be maximized when ...

  θ = π/3 radians = 60°

_____

<em>A</em> will be maximized when its derivative with respect to θ is zero. That derivative can be found to be 2cos(θ)² +cos(θ) -1, so the solution reduces to ...

  cos(θ) = 1/2

  θ = arccos(1/2) = π/3

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The lifetime of a certain type of TV tube has a normal distribution with a mean of 61 and a standard deviation of 6 months. What
Ad libitum [116K]

Answer:

P(57

P(-0.67

P(-0.67

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Solution to the problem

Let X the random variable that represent the lifetime for a TV of a population, and for this case we know the distribution for X is given by:

X \sim N(61,6)  

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We are interested on this probability

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And the best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\sigma}

If we apply this formula to our probability we got this:

P(57

And we can find this probability like this:

P(-0.67

And in order to find these probabilities we can use tables for the normal standard distribution, excel or a calculator.  

P(-0.67

6 0
3 years ago
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Romashka-Z-Leto [24]
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Answer:

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For every 7 votes for candidate a, there are 4 votes for candidate c
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Step-by-step explanation:

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