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Math Help - applications of integration

  1. #1
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    applications of integration

    1. Radiation Exposure- there is a very strong radiation source in the center of a large 120 x 120 foot room. A person must run through this room to get to safety. Her escape route takes her the full length along one wall of the room. she is not exposed to any radiation outside the room. the strength of the radiation is inversely proportional to the square of her distance from the source. at 50 feet, her body is absorbing raditaion at the rate of 200 units per secon. if she runs at 12 feet per second, what is the total radiation dose she receives during her dash to safety?

    2. Cylinder Energy Absorption - A horizontally oriented cylindrical pipe absorbs energy from the sun. this is a common structure in solar heat-exchange systems. consider a small region on the surface of the pipe. let \theta be the angle between the plane tangent at this location and a ray of the sun. the rate at which energy is absorbed at this region is directly proportional to sin( \theta) and to the area of the region. for a 1mm by 1mm. region receiving direct sunlight ( \theta=\pi/2, energy is absorbed at a rate of 0.035 units per minute. Find a definite integral that expresses the rate(units per minute) at which the entire pipe absorbs energy from the sum if the pipe has a radius 2 centimeters and length 1 meter. assume that the sun is directly over head.

    3. the number of hours H of daylight in Madrid as a function of the date can be approximated by the function H(t) - 12 + 2.4 sin(0.0172(t-80)), where t = the number of days since the start of the year. Find the average number number of hours in Madrid in (a) January, (b) June and (c) over the whole year.

    I understand for #1 the girl will be exposed to at least 2000 units of radiation assuming that she runs along the side of the room for 10 seconds.
    the other problems I don't really have much of an idea...thanks for the help
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  2. #2
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    1. Radiation Exposure- there is a very strong radiation source in the center of a large 120 x 120 foot room. A person must run through this room to get to safety. Her escape route takes her the full length along one wall of the room. she is not exposed to any radiation outside the room. the strength of the radiation is inversely proportional to the square of her distance from the source. at 50 feet, her body is absorbing raditaion at the rate of 200 units per secon. if she runs at 12 feet per second, what is the total radiation dose she receives during her dash to safety?
    let E = total exposure

    r = distance from the source

    \frac{dE}{dt} = \frac{k}{r^2}

    200 = \frac{k}{50^2} ... k = 200 \cdot 50^2

    let x = distance from the midpoint of the wall to the exit corner of the room

    consider her exposure running from the wall midpoint to the exit corner. because of the symmetry of the room, she will receive half of the total exposure during that time. calculate that exposure and double the result.

    r^2 = 60^2 + x^2

    \frac{dE}{dt} = \frac{k}{60^2 + x^2}

    dE = \frac{k}{60^2 + x^2} \, dt

    since \frac{dx}{dt} = 12 ... dt = \frac{dx}{12}

    dE = \frac{k}{12(60^2 + x^2)} \, dx

    total exposure ...

    E = 2 \int_0^{60} \frac{k}{12(60^2 + x^2)} \, dx
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  3. #3
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    ^^Thanks.
    Does anybody know how to start off #2 or 3? I'm assuming for #2 I'll be finding the changes in the arc length of the cylinder.
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