The equation c=−g2x2+v0x+h0 gives the instantaneous height, c, of a projectile based on a given real-world scenario. If h0=0, for what values of c is there exactly one solution for x? Explain.(1 point)
a ) There is exactly one solution for x only when c is zero. Since the equation is quadratic, there will always be two solutions except for the point where the projectile strikes the ground.
b ) There are no values of c where there is exactly one solution for x. Since the equation is quadratic, there will always be either two solutions or no solutions.
c )There is exactly one solution for x only when c is equal to the maximum height. Since the initial height is zero, the projectile reaches all heights in its trajectory (except the peak) twice—once on the way up and once on the way down.
d) There is exactly one solution for x only when c is zero or when c is equal to the maximum height. Since the equation is quadratic, there will always be two solutions except for the points where the projectile reaches its peak and where it strikes the ground.
if h0=0, then
c=−g2x2+v0x = -x(g2x-v0)
clearly there are two solutions:
x=0
x = v0/g2
so what im getting is... There are no values of c where there is exactly one solution for x. Since the equation is quadratic, there will always be either two solutions or no solutions.
The correct answer is d) There is exactly one solution for x only when c is zero or when c is equal to the maximum height.
To understand why this is the case, let's break down the equation: c = -g/2 * x^2 + v0x + h0.
The equation represents the instantaneous height, c, of a projectile at a given value of x. Here, g represents the acceleration due to gravity, v0 is the initial velocity, x is the horizontal distance travelled by the projectile, and h0 is the initial height.
Since the equation is quadratic (because of the x^2 term), it will generally have two solutions for x, unless it reaches its highest point (the peak) or hits the ground. These are the points where the projectile changes direction, so there will only be one solution for x at those specific heights.
When c is zero, it means the projectile is at ground level. At this point, it strikes the ground, and there is only one possible value of x.
Similarly, when c is equal to the maximum height of the projectile, it means the projectile has reached its peak. At this point, it changes direction and begins to descend. This is another case where there is only one possible value of x.
Therefore, the equation has exactly one solution for x only when c is zero or when c is equal to the maximum height of the projectile.
Option d) correctly explains this property of the equation.