Right Circular Cone - Sphere

Category: Differential Calculus, Solid Geometry, Algebra

"Published in Newark, California, USA"

Find the radius r of the right circular cone of maximum volume which can be inscribed in a sphere of radius R.

Solution:

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

Photo by Math Principles in Everyday Life

We know that the volume of a right circular cone is 

Next, we need another equation in order to eliminate h at the above equation. Apply Pythagorean Theorem at the right triangle inside the right circular cone, we have

Substitute the value of h to the first equation, we have

Take the derivative on both sides of the equation with respect to r. Consider R as a constant because a right circular cone is inscribed in a sphere.

 

 

Equate dV/dr = 0 because we want to maximize the volume of a right circular cone

Divide both sides of the equation by ⅓ πr, we have

Divide both sides of the equation by 2R, we have

Square on both sides of the equation to remove the radical sign, we have

Therefore,

Maximum Minimum Problem, 6

Category: Differential Calculus, Plane Geometry, Algebra

"Published in Suisun City, California, USA"

If three sides of a trapezoid are 6 in. long, how long must the fourth side be if the area is maximum.

Solution:

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

Photo by Math Principles in Everyday Life

Since the altitude of the trapezoid is not given in the problem, then label further the above figure as follows

Photo by Math Principles in Everyday Life

Apply Pythagorean Theorem at the two right triangles of a trapezoid in order to get the value of h, we have

We know that the area of a trapezoid is

Substitute the values of h, b₁, and b₂ to the above equation, we have

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

Set dA/dx = 0 because we want to maximize the area of a trapezoid

Equate each factor to zero and solve for the value of x:

If 

Since the value of x is negative, then it is not accepted as a part of a length of a base of a trapezoid.

If

Since the value of x is positive, then it is accepted as a part of a length of a base of a trapezoid.

Therefore, the length of the fourth side of a trapezoid is

Stoichiometry Problem - Material Balance, 3

Category: Chemical Engineering Math, Algebra

"Published in Suisun City, California, USA"

A single effect evaporator is fed with 10,000 kg/hr of weak liquor containing 15% caustic by weight and is concentrated to get thick liquor containing 40% by weight caustic. Calculate (a) kg/hr of water evaporated, and (b) kg/hr of thick liquor obtained. 

Solution:  

The given word problem is about the evaporation of caustic liquid from weak liquor to thick liquor which involves the principles of Stoichiometry. The total amount of a substance in the reactants or incoming ingredients must be equal to the total amount of a substance in the final products. In short, the Law of Conservation of Mass must be followed all the time. To illustrate the problem, it is better to draw the flow diagram as follows

Photo by Math Principles in Everyday Life

Basis: 10,000 kg/hr of weak liquor

Let x = be the amount of thick liquor
      y = be the amount of water evaporated

Overall Material Balance of Evaporator:

Material Balance of NaOH:

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

Therefore,

Amount of Thick Liquor = 3,750 kg/hr
Amount of Water Evaporated = 6,250 kg/hr