Frustum - Pyramid

Category: Solid Geometry

"Published in Newark, California, USA"

A baking pan has a rectangular base 12 in. by 8 in.; the sides and ends of the pan slope outward, so that the upper edges measure respectively 13 ½ in. by 9 in. Find the volume of a baking pan if the depth of the pan is 2 in. Find the amount of cake batter required to fill the pan to one-half its depth.

Solution:

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

Photo by Math Principles in Everyday Life

The area of the upper base is calculated as follows

The area of the lower base is calculated as follows

The volume of a baking pan or a frustum of a pyramid is calculated as follows

If a baking pan is filled with a cake batter one-half of its depth which is 1 in., the volume of a cake batter is calculated as follows

Photo by Math Principles in Everyday Life

The area of the bottom of a baking pan is the same as the bottom of a cake batter which is A₂ = 96 in².

The length of the top of a cake batter is calculated using similar triangles

Photo by Math Principles in Everyday Life

The width of the top of a cake batter is calculated using similar triangles

Photo by Math Principles in Everyday Life

The area of the top of a cake batter is

Therefore, the volume of a cake batter is

Implicit Differentiation, 2

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find dy/dx by implicit differentiation for

Solution:

Consider the given equation above

Since the given equation above is not a function, then we have to differentiate it by implicit differentiation as follows

Multiply both sides of the equation by their Least Common Denominator (LCD), we have

 
Therefore, the final answer is

Integration - Algebraic Substitution, 3

Category: Integral Calculus, Algebra

"Published in Suisun City, California, USA"

Evaluate

Solution:

Consider the given equation above

In this type of equation, we cannot integrate it by a simple integration because both the numerator and the denominator have radical equations. If you will rationalize the denominator in order to eliminate the radical sign, then the numerator still has radical equations and the above equation will be more complicated. To eliminate the radical signs at the numerator and the denominator, we have to use the Algebraic Substitution as follows

If 

then

 

 
Substitute the values of √x , √x + 1, and dx to the above equation, we have

but

Hence, the above equation becomes

Therefore, the final answer is