Higher Education Math Presentation: Remainder, Factor & Trigonometry

This presentation delves into several key mathematical concepts, providing examples and applications for each. It begins with the Remainder and Factor Theorems, illustrating their use in comparing costs (oranges example) and solving polynomial problems. The presentation then moves on to trigonometric concepts, including AC circuit analysis, with examples of impedance diagrams and waveform calculations. Further, it explores the solving of polynomial problems related to cake dimensions, and includes trigonometric identities and their verification. The presentation concludes by emphasizing the relevance of these concepts in various real-world scenarios and includes references to related academic texts. This assignment is a presentation that provides a detailed explanation of mathematical concepts.

Presentation
Assignment
Name

Lesson 4.8: Remainder and Factor
Theorom
Example 1
Factor Theorem
This concept aids in situations that calls for comparisons for exmpale
price comparisons for example if 12 oranges cost $48 while while 8
oranges costs $40. Using Factor theory,
Cost/orange for 12 oranges=48/12=$4 per orange while the for later
arrangement
Cost/orange=40/8=$5 per orange hence buying 12 oranges is cheaper
Example 2

Example 2
The total cost of power generation of a company is expressed in the
form C(x)=4.5x^4-3.1x^3-1.01x^2+5x-0.91
Find the cost of generating 10 and 51 units
Solution

Lesson 4.9 : Roots and Zeroes
• Applicable in solving polynomial problems
• Example
• A new hotel offers special cakes for weddings. The volume of the cake is to be 351 cubic cm. The cake is rectangular and the desired
length s to be 4 times longer than the width while the height is to be a third of the width. The bakery wants to know the dimentions
• Solution
• V=lwh
• But l=w+4 while h=1/3w
• The volume is therefore
• V=(w+4)*w*1/3w^2
• From which
• V=1/3w^3+4/3w^2=351
• Multipying all through by 3,
• 1053=w^3+4w^2
• 0=w^3+4w^2-1053
• From the rational zero theorem the possible roots are

Continuation

Example 2
Optimization
Example
A rectangular field is to be fenced. The cost of the vertical side is $10/m, cost of the bottom is $2
while the top is $7. If 7000 is available, determine the dimensions of the field that minimizes the
cost.
The cost C=700=10y+7x +2x=20y+9x. We seek to minimize A=xy
Substituting for x and rearranging, we have
• At minimum vale
• From which x=350/9
• And y=35/2

Lesson 9.5: Trigonometric Graphs
AC circuit analysis
Example 1
An AC circuit has an impedance of 10kΩ and a current of phase shift of
65 degrees. Sketch the AC waveform. From the impedance diagram
shown. If the current is expressed in the form of i(t)=3.2sinwt mA
where w is the angular frequency, find v(t)
V(t)=i(t)*10*10^3=3.2*sin(wt_65) *10=32sin(wt+65)

Example 2
• A low frequency audio system has frequency of 30Hz. Find its period .
Given that its amplitude is 2, write an expression for the waveform
• Period(T)=1/f=1/30
• Waveform
• Assuming the waveform is a cosine wave,
• It will be expressed in the form of
W(t)=2sin(60πt)

Lesson 10.1: Trigonometric
identities
Can be Applied in Instrumentation
Exampel 1
The analogue ammeter is calibrated such that its deflection x is related to
the ammeter’s current density through equation
j
• The current when the deflection θ=24
• From the relationship,
• j
• j

Example 2
• The deflections when the current j=3A/m^2
• From the relationship, we have
• Dividing all through by 0.735 and squaring, we have
• From which x=10.91 degrees

Lesson 10.2: Verifying Trigonometric identities
• A student approximated the height of cliff expressed in terms of the
angle (x) to be
• From the literature, the same cliff’s height was expressed in the form
of
• Prove whether the students were right.