Right Circular Cylinder - Right Circular Cone

Category: Solid Geometry

"Published in Newark, California, USA"

The solid shown consists of a right circular cylinder of diameter d and altitude h, surmounted by a cone of diameter  of base d and altitude h/2. 

(a) Write a formula for the volume.

(b) Find the volume, given d = 4.42 in. and h = 5.17 in.

Photo by Math Principles in Everyday Life

Solution:

Consider the volume of right circular cylinder

Consider the volume of right circular cone

If h = h/2, then

The combination of the volume of right circular cylinder and right circular cone is

If r = d/2, then the equation becomes

If d = 4.42 in. and h = 5.17., then the volume is

Word Problem - Finding Unknown Fraction

Category: Algebra

"Published in Newark, California, USA"

Find the value of a fraction whose numerator is 4 less than its denominator and which when added to ½ the value of the fraction gives a sum of ³/₁₀. 

Solution:

The first thing that we have to do is to analyze the problem very well because we will interpret a word statement into a working equation. 

Let     x = be the denominator of the unknown fraction

     x - 4 = be the numerator of the unknown fraction

From the word statement that "when added to ½ the value of the fraction gives a sum of ³/₁₀", the working equation will be  

Multiply the both sides of the equation by their Least Common Denominator (LCD) which is 10x as follows

The numerator of the unknown fraction is 5 - 4 = 1.

The denominator of the unknown fraction is 5.

Therefore, the unknown fraction is ¹/₅. 

Check: 

Differentiation - Lifting Rate Problem

Category: Differential Calculus, Plane Geometry

"Published in Newark, California, USA"

A weight is attached to one end of a 33-foot rope which passes over a pulley 18 feet above the ground. The other end is attached to a truck at a point 3 feet above the ground. If the truck moves away at a rate of 2 feet per second, how fast is the weight rising when the truck is 8 feet from the spot directly under the pulley?

Solution: 

To visualize the problem, let's draw the figure as follows

Photo by Math Principles in Everyday Life

As a truck moves away from the weight, a right triangle is formed between the weight, pulley, and a truck 

Photo by Math Principles in Everyday Life

By Pythagorean Theorem

In this problem, y is a constant because the height of a pulley from the ground is already fixed. As a truck moves away from the weight, the length of a rope from the truck to the pulley increases. Take the derivative of the above equation with respect to time t as follows

In order to get the value of ^dc/_dt, we need to solve for the value of c first because it is not given in the problem. Since the distance of a truck from the spot directly under the pulley, the height of a truck from the ground, and the height of a pulley from the ground are given, we can solve for the value of c which is the length of a rope from the truck to the pulley as follows

Photo by Math Principles in Everyday Life

By Pythagorean Theorem

The rate of increasing the length of a rope as a truck moves away from the spot directly under the pulley is also the same as the rate of rising the weight from the ground. Therefore, the rate of rising the weight from the ground is