Approximation - Error Problem

Category: Differential Calculus, Solid Geometry

"Published in Newark, California, USA"

The altitude of a certain right circular cone is the same as the radius of the base, and is measured as 5 inches with a possible error of 0.02 inch. Find approximately the percentage error in the calculated value of the volume.

Solution:

From the description of a problem, it is an approximation and error problem because it involves the difference of the dimensions of a right circular cone as well  as the difference of the volume. 
 

Photo by Math Principles in Everyday Life

 
The volume of a right circular cone is calculated as

but r = h

Take the differentials on both sides, we have

The original volume of the right circular cone is calculated as

Therefore, the percent of error is calculated as

% Error = (± 0.012)(100) = 1.2%

Maximum Minimum Problem

Category: Differential Calculus

"Published in Newark, California, USA"

A farmer estimates that if he digs his potatoes now, he will have 120 bushels, which he can sell at $1.75 per bushel. If he expects his crop to increase 8 bushels per week, but the price to drop 5 cents per bushel per week, in how many weeks should he sell to realize the maximum amount for his crop?

Solution:  

Let x be the number of bushels of potatoes per week

Let y be the price change of potatoes per bushels per week

Let t be the time/period of harvesting potatoes in weeks

Let C be the total cost of the potatoes

By analyzing the problem above, initially, there are 120 bushels of potatoes that the farmer can sell at $1.75 per bushel.

Total Cost of Potatoes = (Total Number of Potatoes in bushels)(Total Price of Potatoes per bushel)

                    C (initial) = (120)($1.75)

At the next statement, his crop will increase 8 bushels per week. The total number of potatoes in bushels can be written as

      Total Number of Potatoes = 120 + xt

and there will be a change of price of potatoes per bushels per week. The total price of potatoes per bushel can be written as

    Total Price of Potatoes per bushel = $1.75 + yt

Therefore, we can now write the working equation of Total Cost of Potatoes as follows:

Next we have to take the derivative of the above equation with respect to t, we have

You notice that we use the derivative of the product of two functions. Next, set dC/dt = 0 because we want to maximize the total cost of potatoes. 

   
Therefore, he should sell his potatoes in 10 weeks.

Graphical Sketch - Circle

Category: Analytic Geometry, Plane Geometry

"Published in Newark, California, USA"

Given the equation of a circle:

Find its center and radius. Sketch the graph.

Solution:

From the given equation,

This equation represents a circle because the coefficients of x² and y² are the same and no xy term in the general equation of a conic section. Circle is also a type of conic section. Our goal right now is to find its center and radius. Since x² and y² have their coefficients, we have to divide both sides of the equation by 4 in order to eliminate their coefficients, we have

Group the above equation according to their variables and transpose the coefficient to the right side of the equation,

Next, let's do the completing the square for x² and y²,

The above equation can be written as

In order to get the center and radius of a circle, the equation must be simplified into standard form. The above equation is now in standard form.

To get the center of a circle,

                    x - 1 = 0                  y + 2 = 0          

                         x = 1                        y = -2

Therefore, C(1, -2)

The radius of a circle is 3.

We can now sketch the graph using the above results.

Photo by Math Principles in Everyday Life