In the 1940s, the human cannonball stunt was performedregularly by Emmanuel Zacc

Question

In the 1940s, the human cannonball stunt was performedregularly by Emmanuel Zacchini for the Ringling Brothers and Barnum& Bailey Circus. The tip of the cannon rose 15 feet offthe ground, and the total horizontal distance traveled was 175feet. When the cannon is aimed at an angle of 45 degrees, andequation of the parabolic flight has the form y=ax^2+x+c A) Use the given information to find an equation of theflight. B) Find the maximum height attained by the humancannonball. In the 1940s, the human cannonball stunt was performedregularly by Emmanuel Zacchini for the Ringling Brothers and Barnum& Bailey Circus. The tip of the cannon rose 15 feet offthe ground, and the total horizontal distance traveled was 175feet. When the cannon is aimed at an angle of 45 degrees, andequation of the parabolic flight has the form y=ax^2+x+c A) Use the given information to find an equation of theflight. B) Find the maximum height attained by the humancannonball.

Explanation / Answer

With the boundary conditions given to be: @ x = 0, y = 15ft we have: 15 = a(0)^2 + (0) + C C = 15
@ x = 175, y = 0 because presumably he is on the ground at the endof the flight and not in a tree or something. 0 = a(175)^2 + 175 + 15 a = -(175+15)/(175^2) = -0.006204081 So the equation becomes: (A) y = -0.006204081x^2 + x + 15 For the second part, we want to know where the top of the arc(parabola) occurs, so we can make use of the fact that at the topof the arc, the slope of the tangent line is equal to zero. Thus, we take the derivative and set y' = 0 and solve for x y' = (2*-0.006204081)x +1 0 = -0.012408162x + 1 x = 80.5921 ft This means that at 80.5921 feet from the cannon, the stuntperson reaches the max height, so now we just plug this number backinto the orginal equation for the parabola. y = -0.006204081(80.5921)^2 + 80.5921 + 15 (B) y = 55.296 ft If you have not learned derivatives (as this is a pre-calforum) you can plot the parabola and determine where the top of thearc occurs and find the max height that way.

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