Maximum Minimum Problem, 8

Category: Differential Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the shortest distance from the point P(2, 0) to a point on the curve y² - x² = 1, and find the point on the curve closest to P.

Solution:

To illustrate the problem, it is better to draw the figure and sketch the graph of the curve (hyperbola) as well as the given point in Rectangular Coordinate System as follows

Photo by Math Principles in Everyday Life

As you can see from the figure above that there will be two distances of a point to a curve because the curve is hyperbola and it is symmetrical with x axis and y axis. The other point on the curve is a point of tangency.  The distance of a point to the point of tangency is the perpendicular distance of a point to the tangent line. To understand more the problem, label further the above figure as follows

Photo by Math Principles in Everyday Life

The distance of two points is

If the first point is P(2, 0), then the above equation becomes

If the second point is on the curve, then the above equation becomes

Next, we need to eliminate y at the above equation. If the equation of a curve is

,then the value of y will be equal to 

Substitute the value of y to the first equation which is the distance of two points, we have

Take the derivative of the above equation with respect to x, we have

Set dd/dx = 0 because we want to minimize the distance of a point to a curve.

Substitute the value of x to the equation of the curve in order to get the value of y, we have

Therefore, the point on the curve is

The shortest distance of a point to a curve is

 

Maximum Minimum Problem, 7

Category: Differential Calculus, Analytic Geometry, Algebra

"Published in Newark, California, USA"

Find the area of the largest rectangle having one side on the x-axis and inscribed in the triangle formed by the lines y = x , y = 0, and 3x + y = 20.

Solution:

To illustrate the problem, it is better to draw the figure and sketch the three given lines in Rectangular Coordinate System as follows

Photo by Math Principles in Everyday Life

The first thing that we have to do is to find the length of a rectangle which is x and then find the width of a rectangle which is y. 

To get the value of x for a rectangle, we need to use the lines y = x and 3x + y = 20. 

The line 3x + y = 20 can be written as

Hence, the length of a rectangle is

To get the value of y for a rectangle, we need to use the lines y = x and y = 0.

Hence, the width of a rectangle is 

We know that the area of a rectangle is

Next, we need to eliminate x at the above equation. We can use one of the three lines to substitute the value of x. The best is y = x because it is the simplest equation of the three lines. Substitute x = y to the above equation, we have

Take the derivative on both sides of the equation with respect to y, we have

Set dA/dy = 0 because we want to maximize the area of a rectangle.

Therefore,

Finally, the area of a rectangle is