Problem 35
In the foundation work of the Woolworth Building, a 55-story building in New York City, it was necessary, in order to reach the bedrock, to penetrate the sand and quicksand to a depth, in some instances, of 131 ft. If the largest circular caisson, 19 ft. in diameter, was 130 ft. deep and was filled with concrete to within 30 ft, of the surface, how many cubic yards of concrete were required?
 

The inside of a vase is an inverted cone 2.983 in. across the top and 5.016 in. deep. If a heavy sphere 2.498 in. in diameter is dropped into it when the vase is full of water, how much water will overflow?
 

Problem 33
Disregarding quality, and considering oranges as spheres, determine which is the better bargain, oranges 2-3/4 in. in diameter at 15 cents per dozen, or oranges averaging 3-1/2 in. in diameter at 30 cents per dozen.
 

Problem 32
A coffee pot is 5 in. deep, 4-1/2 in. in diameter at the top, and 5-3/4 in. in diameter at the bottom. How many cups of coffee will it hold if 6 cups equal 1 quart? Answer to the nearest whole number.
 

Problem 11
Three identical circles of radius 30 cm are tangent to each other externally. A fourth circle of the same radius was drawn so that its center is coincidence with the center of the space bounded by the three tangent circles. Find the area of the region inside the fourth circle but outside the first three circles. It is the shaded region shown in the figure below.
 

Problem 02
A certain radioactive substance has a half-life of 38 hour. Find how long it takes for 90% of the radioactivity to be dissipated.
 

Problem 01
Radium decomposes at a rate proportional to the quantity of radium present. Suppose it is found that in 25 years approximately 1.1% of certain quantity of radium has decomposed. Determine how long (in years) it will take for one-half of the original amount of radium to decompose.
 

From the results of chemical experimentation of substance converted into another substance, it was found that the rate of change of unconverted substance is proportional to the amount of unconverted substance.
 

If x is the amount of unconverted substance, then

with a condition that x = x_o when t = 0.
 

$\dfrac{dx}{dt} = -kx$

$\dfrac{dx}{x} = -k \, dt$

$\ln x = -kt + \ln C$

$\ln x = \ln e^{-kt} + \ln C$

Problem 01
A thermometer which has been at the reading of 70°F inside a house is placed outside where the air temperature is 10°F. Three minutes later it is found that the thermometer reading is 25°F. Find the thermometer reading after 6 minutes.
 

Newton's Law of Cooling states that the temperature of a body changes at a rate proportional to the difference in temperature between its own temperature and the temperature of its surroundings.
 

We can therefore write

where,
T = temperature of the body at any time, t
T_s = temperature of the surroundings (also called ambient temperature)
T_o = initial temperature of the body
k = constant of proportionality