# 035 Review Problem - Amount of concrete in a caisson

**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?

# 034 Review Problem - Sphere dropped into a cone

# 033 Review Problem - Finding which one is the better bargain

# 032 Review Problem - How many cups of coffee a coffee pot can hold?

# Calculator for a triangle of given three sides

For any triangle of given three sides, this calculator will compute the following: area, perimeter, radius of inscribed circle, radius of circumscribing circle, radii of escribed circles, length of medians, length of angular bisectors, and length of altitudes.

# 11 - Area inside a circle but outside three other externally tangent circles

**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.

# 02 - Time to dissipate 90% of certain radioactive substance

# 01 - Find how long would it take for half amount af radium to decompose

**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.

# Simple Chemical Conversion

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.

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

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

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

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

$\ln x = \ln Ce^{-kt}$