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Verizon [17]
3 years ago
9

A lighthouse is located on a small island 5 km away from the nearest point P on a straight shoreline and its light makes seven r

evolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?

Mathematics
1 answer:
Vaselesa [24]3 years ago
6 0

Answer:

Light is moving at the speed = 77.78 km/min

Step-by-step explanation:

Given:

Distance between island and point P = 5 km

Moving along the the shoreline when it is 1 km from P.

So, x = 1

From the above statement light makes seven revolutions per minute.

Therefore,              

\frac{d\theta}{dt}=\frac{7\times 2\pi\ rad}{1\ min}  = 14\pi\ rad/min  

Solution:

From the given figure.

tan\theta=\frac{x}{5}

Substitute x = 1

tan\theta=\frac{1}{5}  -------------------(1)

Now, first we differentiate both side with respect to t.

\frac{d(tan\theta)}{dt} = \frac{d}{dt}.(\frac{x}{3})

Using chain rule.

\frac{d(tan\theta)}{dt}.\frac{d\theta}{dt}  = (\frac{1}{5})\frac{dx}{dt}

sec^{2} \theta.\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

(1+tan^{2} \theta).\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

Substitute tan\theta=\frac{1}{5} from equation 1.

(1+(\frac{1}{3})^{2} ).\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

(1+\frac{1}{9}).\frac{d\theta}{dt} = \frac{1}{5}.\frac{dx}{dt}

Substitute \frac{d\theta}{dt}=14\pi

\frac{10}{9}.14\pi  = \frac{1}{5}.\frac{dx}{dt}

Applying cross multiplication rule.

\frac{dx}{dt}= \frac{700\pi }{9}

\frac{dx}{dt}= 77.78\pi

Therefore, light is moving along the shore at the speed of 77.78 km/min

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Step-by-step explanation:

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Read 2 more answers
Illinois license plates used to consist of either three letters followed by three digits or two letters followed by four digits.
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Answer:

The probability that a randomly chosen plate contains the number 2222 is 0.000028 approximately.

The probability that a randomly chosen plate contains the sub-string HI is 0.002548  approximately.

Step-by-step explanation:

Consider the provided information.

Illinois license plates used to consist of either three letters followed by three digits or two letters followed by four digits.

Part (A)

Let A is the ways in which plates consist of three letters followed by three digits and B is the ways in which two letters followed by four digits.

Here repetition is allow. The number of alphabets are 26 and the number of distinct digits are 10.

The numbers of ways in which three letters followed by three digits can be chosen is: 26\times 26\times 26 \times 10 \times 10 \times10

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The numbers of ways in which two letters followed by four digits can be chosen is: 26\times 26\times 10 \times 10 \times 10 \times10

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Randomly chosen plate contains the number 2222 that means the first two letter can be any alphabets but the rest of the digit should be 2222.

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The probability that a randomly chosen plate contains the number 2222 is:

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Part (B)

The number of ways in which chosen plate contains the sub-string HI:

If three letters followed by three digits plate contains the sub-string HI, then the number of possible ways are:

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If two letters followed by four digits plate contains the sub-string HI, then the number of possible ways are:

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Thus, the total number of ways that a randomly chosen plate contains the sub-string HI are:

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The probability that a randomly chosen plate contains the sub-string HI is:

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