Concept explainers
Path of a Cannonball A cannon fires a cannonball as shown in the figure. The path of the cannonball is a parabola with vertex at the highest point of the path. If the cannonball lands 1600 ft from the cannon and the highest point it reaches is 3200 ft above the ground, find an equation for the path of the cannonball. Place the origin at the location of the cannon.
To find: The equation of the path of the cannon ball.
Answer to Problem 42E
The equation of the path of the cannon ball is
Explanation of Solution
Given:
The cannon ball lands 1600ft from the cannon and the highest point it reaches is 3200ft and the cannon located at the origin.
Definition used:
Definition 1:
“The equation of shifted parabola with vertex
Definition 2:
“The equation of the parabola vertex
Definition 3:
“If
(i) Replacing
(ii) Replacing
(iii) Replacing
(iv) Replacing
Calculation:
The given canon is placed at the origin, if the ball is fired the path of the ball is parabolic and the highest point is the vertex of the parabola.
Let the coordinate axes of the parabola meet at O as shown in Figure 1.
Also, the ball strikes the ground at 1600ft from the origin
Therefore, the coordinate of the point where the ball strikes the ground is
The midpoint of OA is
That is,
The highest point reached by the ball is 3200ft above the ground.
Therefore, vertex of the parabolic path is
By the definition 1, the equation of the path of the cannon ball is
From Figure 1, it is clear that the equation (1) passes through
Substitute
Thus, the equation of the path of the cannon ball is
Chapter 11 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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