. The Chernobyl nuclear explosion in the former Soviet Union on April 26, 1986 released large amounts of radioactive substances into the atmosphere. These substances included cesium-137, iodine-131, and strontium-90. Although the radioactive material covered many countries, the actual amount and intensity of the fallout varied greatly from country to country, due to vagaries of the weather and the winds. One area that was particularly hard hit was Lapland, where heavy rainfall occurred just when the Chernobyl cloud was overhead.
a) Many of the pastures in Lapland were contaminated with cesium-137, a radioactive substance
with a half-life of 33 years. If the amount of cesium-137 was found to be ten times the normal
level, how long would it take until the level returned to normal? (Hint: Let No be the amount
that is ten times the normal level. Then you want to find the time when N(t) = No/10.)
b) Assume that the amount of cesium-137 was 100 times the normal level, how long would it take
until the level returned to normal? (Remark: Several days after the explosion, it was reported
that the level of cesium-137 in the air over Sweden was 10,000 times the normal level.
Fortunately there was little or no rainfall.)
2. According to Newton’s Law of Cooling, T(t) = T + (T – T ) e can be used to compute the approximate time of death for a person found recently expired, where T is the body temperature when it was found, T is the room temperature, T is the normal body temperature of 98.6 F, and t is the number of hours since death.
a) If a body was discovered in a room at 2:00 pm with a temperature of 81.2 F in a room at 68 F,
at approximately what time did the person die?
b) At approximately what time was the body temperature 86 F ?
c) Graph T(t), state the equation of the horizontal asymptote, and give a physical explanation for the
3. Paris’s Eiffel Tower was constructed in 1889 to
commemorate the one hundredth anniversary of the
French Revolution. The right side of the Eiffel Tower
has a shape that can be approximated by the graph of
the function defined by f(x) = -301 ln .
a) Assuming that the Eiffel Tower is symmetric about
the y-axis and given that the short horizontal line at the
top of the figure has length 15.7488 feet, determine the
approximate height of the Eiffel Tower.
b) Approximately how far from the y-axis is the point P,
on the right side of the tower, that is 500 feet above
4. The weight of any object decreases as the distance from Earth’s surface increases. As the object rises, the effect of Earth’s gravitational pull on the object is reduced. The weightlessness that astronauts experience in the space shuttle as it orbits the earth is due to the distance of the shuttle above the Earth’s surface.
If an object weighs E kilograms at sea level on Earth’s surface, then the weight W (also in kilograms) of the object at a distance of h kilometers above sea level is given by the function W = .
a) If an astronaut weighs 70 kilograms (154 lbs) at sea level what is the weight of astronaut, in kilograms,
when the space shuttle reaches its orbiting altitude of 40,000 kilometers above sea level? Since
1 kilogram = 2.20 lbs, what is the astronaut’s weight in lbs?
b) At what altitude does the astronaut’s weight equal one-half of what it is at sea level?
c) Graph the weight function for 0 < h < 40,000 kilometers. 5. A Norman window is in the shape of a rectangle surmounted by a semicircle, as shown in the figure. Assume that the perimeter of the window (perimeter does NOT include the line from A to D) is 150 inches. Show that the area of window is a maximum when = r = inches. Show that this maximum area is square inches.