Indeterminate Form - Zero Over Zero, 4

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given equation above

Substitute the value of x to the above equation, we have

Since the answer is 0/0, then it is Indeterminate Form that is not accepted as a final answer in Mathematics. We have to do something first in the equation above so that the final answer is a real number and not indeterminate form.

There are two ways in solving the value of the limit for the above equation. Let's consider the two ways in solving the limits.  

Method 1:

Since the indexes of the radicals at the above equation are different, then we have to convert them the same indexes as follows:

Let 

If x = 1, then y will be equal to

Substitute the value of y to the above equation, we have 

Since the answer is 0/0 again, then it is Indeterminate Form also that is not accepted as a final answer in Mathematics. We have to do something in the equation above so that the final answer is a real number and not indeterminate form. 

Since the numerator and the denominator can now be factored, then the above equation becomes

Substitute the value of y to the above equation, we have

Therefore,

Method 2:

There's another way in solving the limits which is called the L'Hopital's Rule. L'Hopital's Rule is applicable if the Indeterminate Form is either 0/0 or  ∞/∞. Consider again the given equation, we have

 
Apply the L'Hopital's Rule to the above equation, we have

Substitute the value of x to the above equation, we have

Therefore,

Spherical Zone Problems, 3

Category: Solid Geometry

"Published in Newark, California, USA"

Find the area illuminated by a candle h feet from the surface of a ball R feet in radius. How much surface is illuminated when a candle is 10 ft. away from a ball 5 ft. in radius?

Photo by Math Principles in Everyday Life

Solution:

To understand more the problem, it is better to label further the given figure above as follows

Photo by Math Principles in Everyday Life

A spherical zone or zone, in short is a portion of the surface of a sphere from its circular cross section to its end (for one base) or between two parallel circular planes (for two bases). The above figure is a zone of one base.

In this problem, we will solve for the area of spherical zone of one base. Since we want to solve for the area of the ball that is illuminated by a candle, then we have to solve for the area of the spherical zone of one base at the left side. The area of the spherical zone of one base is given by the formula

where R is the radius of a sphere and H is the height or distance of a circular base to the surface of a sphere. Since H is not given in the problem but the distance of a candle to the ball which is h is given, then we have to use the principles of solving similar triangles in order to solve for the value of H in terms of h and R. 

Since the light rays of a candle are tangent to the surface of a ball as shown in the figure, then ∆ABD is a right triangle. 

Since the radius of a ball is perpendicular to its surface which is illuminated by a candle as shown in the figure, then ∆BDO is also a right triangle. 

∆ABD is similar to ∆BDO because if a rt∆ABO is cut by a line segment BD which is perpendicular to line segment AO, then the two right triangles formed are similar to each other. 

Since ∆ABD ~ ∆BDO, then we can solve for the value of H in terms of h, r, and R as follows

Next, we need to eliminate r at the above equation because we want to solve for the value of H in terms of h and R. Apply Pythagorean Theorem for ∆BDO in order to get the value of r, we have

Hence, the above equation becomes

Therefore, the area of the spherical zone is

If h = 10ft. and R = 5 ft., then the area of the ball which is illuminated by a candle is

or

 

Spherical Zone Problems, 2

Category: Solid Geometry

"Published in Newark, California, USA"

A wooden ball 11.15 in. in diameter sinks to a depth of 9.37 in. in water. Find the area of the wet surface.

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

A spherical zone or zone, in short is a portion of the surface of a sphere from its circular cross section to its end (for one base) or between two parallel circular planes (for two bases). The above figure is a zone of one base.

In this problem, we will solve for the area of spherical zone of one base. Since we want to solve for the area of the wooden ball at the wet surface, then we have to solve for the area of the spherical zone of one base at the bottom part. The area of the spherical zone of one base is given by the formula

Substitute the values of R and H to the above equation, we have

Therefore,

or