Mathematics Assignment: Quadratic Equations, Discriminants, etc.

This document provides a comprehensive solution to a mathematics assignment. It includes the calculation of the maximum height of a ball launched from a cliff, addressing a quadratic equation. The solution involves finding the derivative and applying it to determine the maximum height. The assignment also addresses the properties of a parabola, including its shape, axis of symmetry, and intercepts, with a graphical illustration provided. Finally, the solution calculates the discriminant of a quadratic expression, demonstrating the application of mathematical formulas and principles. The document references relevant mathematical texts to support the solutions.

Mathematics Assignment
Name
Institution Affiliation

Question 1
h=−4.9 t2+ 19.6t−9.6
We seek to find a maximum point, and therefore, we take the derivative of h with respect to t at
that point and equate it to zero.
dh
dt =−4.9× 2t +19.6=−9.8 t+19.6=0
9.8 t=19.6
t=2 is the time taken to reach maximum height.
Maximum height h=−4.9 ×22 +19.6 × 2−9.6=10 meters
ans : hmax =10
Question 2
At y-intercept, x=0 and y=2
At x-intercept, x=2 and y=0
y= 1
2 x2−2 and y=−1
2 x2−2 gives y-intercept of (0 ,−2) which is not desired.
y=−x2 +2 gives x-intercept of ( √2 , 0) which is not desired
y= 1
2 x2 +2 gives a U-shaped parabola which is not desired.
Therefore y=−1
2 x2 +2 is the quadratic equation that corresponds to the n-shaped parabola, with
x=0 as the axis of symmetry, y-intercept of ( 0,2 ) and x-intercept of ( 2,0 )
ans : y=−1
2 x2+2
Graphical illustration.
Question 10
The total area of rectangle ABCD =l ×w=v (v +4)
The area of the shaded part ABEFGD= Total area – area of Unshaded CEFG

Area of ABEFGD =v ( v+ 4 )−9 × 4=v ( v +4 ) −36
This can also be expressed as v2+ 4 v −36 by opening the brackets.
ans : v ( v +4 ) −36∧v2+4 v−36
Question 3
Considering a quadratic equation of the form a x2 +bx+ c=0
Discriminant, D=b2−4 ac
D=52− ( 4 ×2 ×−2 )=41
ans : 41
References
Burzynski, D. and Ellis, W. (1989). Fundamentals of Mathematics. Houston, Texas:
Rice University Press, USA.
Woodbury, G. (2006). Elementary and Intermediate Algebra. London, England:
College of the Sequoias, Pearson Publishers Ltd.