April, 2008

Problem 11

Differentiate x = 2y / (y - 1) by Delta method.

Solution 11

Calculus was developed on 17th century by two mathematicians, Isaac Newton of England and Gottfried Wilhelm Leibniz of Germany. This branch of mathematics made the solution of a problem, that once thought to be very complex, easy to solve. This subject is a prerequisite of those who wish to study engineering, economics, physics, chemistry, biology, finance, and many more. Computers made Calculus solved very complex problems.

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The HubbleSite of Space Telescope Science Institute releases pictures to NASA that astound not only the general public but the space scientists themselves. Galaxies collide and merge and morph into new shapes. This spectacular galactic display was released to the public in April 24, 2008 09:00 AM (Eastern Daylight Time).

Problem

Four balls of radius 2 in are placed in a square box whose inside base edges is 10 in. In the space between these balls, a fifth ball is placed. Find the minimum depth of the box in order that the cover will just touch the fifth ball, if its radius is 3 in.

Solution

The figure below shows the formation of the spheres inside the box. The isometric view of these balls maintains its proportion but hides one of the small balls. I decided to turn it into another view so that all 5 balls are visible. The turn the figure into distorted proportion but show us the complete view of the five balls. This will make our analysis easy. If you don’t use a computer to illustrate the problem, do not present the balls in solid drawing, it is easier to manually draw the elevation and top views of the ball.

When the guestbook was first implemented it becomes a target of spammers. I spend time in deleting spam messages entering into the guestbook. Some spammers are even hard to notice. I decided to disable the guestbook to trace the pattern of incoming entry. I made a discovery that although the guestbook is disabled, a remote computer is trying to add an entry into the guestbook, this is a clear indication that it is not human but spambot.

Problem 6

Differentiate by Δ-method the function x = ½ t⁴ – 5t – 3.

Solution 6

Problem 1

Differentiate y = 4x² – 3x – 2 by Δ-method

Solution 1
y + Δy = 4(x + Δx)² – 3(x + Δx) – 2 » Step 1
Δy = 4(x + Δx)² – 3(x + Δx) – 2 - y » Step 2
Δy = 4(x² + 2x Δx + Δx²) – 3(x + Δx) – 2 – (4x² – 3x – 2)
Δy = 4x² + 8x Δx + 4 Δx² – 3x – 3 Δx – 2 – 4x² + 3x + 2 » Step 3
Δy = 8x Δx + 4 Δx² – 3 Δx
Δy / Δx = 8x + 4 Δx - 3 » Step 4
y' = 8x - 3 answer » Step 5

The Derivative

Derivative of a function is the limit of the ratio of the incremental change of dependent variable to the incremental change of independent variable as change of independent variable approaches zero. For the function y = f(x), the derivative is symbolized by y’ or dy/dx, where y is the dependent variable and x the independent variable.

Problem 37

A ship sails east 20 miles and then turns N 30° W. If the ship’s speed is 10 mi/hr, find how fast it will be leaving the starting point 6 hr after the start.

Solution 37

Problem 33

From a car traveling east at 40 miles per hour, an airplane traveling horizontally north at 100 miles per hour is visible 1 mile east, 2 miles south, and 2 miles up. Find when this two will be nearest together.

Solution 33

This is just for fun, no offense. I'm a woman too. This is not my idea, I just found this picture from Flickr.

Note that this discussion is limited only to my case, this is not applicable in all places and all soil profile. In our place, the soil profile is, from top, 10 ft clay, 5 ft sand, 5 ft gravel then bedrock. The water table is 8 ft below the ground surface. You may not follow exactly what I did; at least you have now an idea of a hand dug well.

I’ve been busy this summer for our small home improvements. I tiled the floor with a ceramic tile and after it I dug a deep well. Water is one of the problems in our place. The developer is trying too many alternatives but of no avail. I am living in a hill; our place is called Regina Hills Subdivision. The problem of water is not only in our subdivision but a city-wide problem. So I created this move, dug a water well for my self.

The Cost of Digging

I used the word dig because we literally dig the well, drilling is not in our list. When I open a conversation to some workers that I plan to dig a well, many workers come into my house and offer their service. They are at different rate and of different level of knowledge and experience in well digging. The following are the rates of digging in our place.

500 pesos per foot (P500/ft)
There are three contractors offer me their service with this rate. It includes the digging, pipe fittings, and pump installation. These workers are no doubt knowledgeable of the job, and they can do work without my instructions. They have the full set of digging tools.

300 pesos per foot (P300/ft)
Most contractors have this rate; in fact some contractors in 500 pesos per foot are willing to drop to this rate just to get the job. It also includes the digging, pipe fittings, and pump installation but they are not as knowledgeable as the 500 pesos per foot contactors and their equipments are not that complete.

250 pesos per foot (P250/ft)
This is the rate of almost all diggers in our place.

None of the above rate I choose. I offered to one laborer 3,000 pesos (P3,000) for the first 20 feet and I will increase the payment if he can go deeper than 20 ft. He agreed and we start digging. Of course, he needs all my instructions; all he knows is to dig.

Problem 30

Two railroad tracks intersect at right angles, at noon there is a train on each track approaching the crossing at 40 mi/hr, one being 100 mi, the other 200 mi distant. Find (a) when they will be nearest together, and ( what will be their minimum distance apart.

Solution 30

Problem 26

A kite is 40 ft high with 50 ft cord out. If the kite moves horizontally at 5 miles per hour directly away from the boy flying it, how fast is the cord being paid out?

Solution 26

Problem 22

One city C, is 30 miles north and 35 miles east from another city, D. At noon, a car starts north from C at 40 miles per hour, at 12:10 PM, another car starts east from D at 60 miles per hour. Find when the cars will be nearest together.

Solution 22

Problem 19

One city A, is 30 mi north and 55 mi east of another city, B. At noon, a car starts west from A at 40 mi/hr, at 12:10 PM, another car starts east from B at 60 mi/hr. Find, in two ways, when the cars will be nearest together.

Solution 19
1^st Solution (Specific):

Problem 15

A light at eye level stands 20 ft from a house and 15 ft from the path leading from the house to the street. A man walks along the path at 6 ft per sec. How fast does his shadow move along the wall when he is 5 ft from the house?

Solution 15

Problem 10

A boy on a bike rides north 5 mi, then turns east (Fig. 47). If he rides 10 mi/hr, at what rate does his distance to the starting point S changing 2 hour after he left that point?

Solution 10

Problem 6

A ladder 20 ft long leans against a vertical wall. If the top slides downward at the rate of 2 ft/sec, find how fast the lower end is moving when it is 16 ft from the wall.

Solution 6

Problem 1

Water is flowing into a vertical cylindrical tank at the rate of 24 ft³/min. If the radius of the tank is 4 ft, how fast is the surface rising?

Solution 1

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If a quantity x is a function of time t, the time rate of change of x is given by dx/dt.

When two or more quantities, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating both sides of the equation with respect to t.

Problem 72

A light is to be placed above the center of a circular area of radius a. What height gives the best illumination on a circular walk surrounding the area? (When light from a point source strikes a surface obliquely, the intensity of illumination is I = k sin θ / d², where θ is the angle of incidence and d the distance
from the source.)

Solution 72

Problem 69

A man on an island 12 miles south of a straight beach wishes to reach a point on shore 20 miles east. If a motorboat, making 20 miles per hour, can be hired at the rate of $2.00 per hour for the time it is actually used, and the cost of land transportation is $0.06 per mile, how much must he pay for the trip?

Solution 69