The gravity on the moon differs from that on Earth. The function f(x)=-2x^2+30x+32 models an objects height, in feet, from the moon’s surface x seconds after being launched.
If x>0, which statement is true?
If x>0, it means that the object has been launched and some time has passed since then. Therefore, the statement "the object is in motion above the moon's surface" is true.
If x>0, then the object has already been launched and is in motion. Therefore, the statement "an object's height from the moon's surface x seconds after being launched" would be relevant.
However, in order to determine which statement is true, we would need to know the specific statements given as options.
To determine which statement is true when x > 0, we need to analyze the given function f(x) = -2x^2 + 30x + 32. This function models the height, in feet, of an object from the moon's surface x seconds after being launched.
To solve this problem, we can consider the properties of the function. The given function is a quadratic function, where the highest power of the variable x is 2. When we have a quadratic function in the form f(x) = ax^2 + bx + c, certain characteristics can be derived from the coefficients a, b, and c.
In our case, the coefficient of x^2 is -2, the coefficient of x is 30, and the constant term is 32. We can analyze these coefficients to determine the properties of the function.
1. Coefficient of x^2 (-2): Since this coefficient is negative, the parabola representing the function opens downward. This means the function has a maximum value.
2. Discriminant (b^2 - 4ac): In this case, the discriminant is b^2 - 4ac = (30)^2 - 4(-2)(32) = 900 + 256 = 1156. Since the discriminant is positive, we know the function has two real roots.
Now, let's consider the function for values of x greater than 0. When x > 0, the object has already been launched, and we are looking at its height above the moon's surface.
Based on the properties discussed above, there are two possibilities:
1. The object reaches a maximum height before falling back to the moon's surface:
In this case, the graph of the function will have a maximum point, and the statement "The object reaches a maximum height and then falls back to the moon's surface" would be true.
2. The object continues to go up indefinitely:
In this case, the graph of the function will not have a maximum point, and the statement "The object continues to rise indefinitely" would be true.
To determine which statement is true in our specific case, we can calculate the x-coordinate of the vertex (maximum point) using the formula x = -b / (2a).
For our function f(x) = -2x^2 + 30x + 32, the x-coordinate of the vertex is x = -30 / (2 * -2) = 15 / 2 = 7.5 seconds.
Since x is greater than 0, which means the object has already been launched, and the object reaches a maximum height before falling back to the moon's surface, we can conclude that the statement "The object reaches a maximum height and then falls back to the moon's surface" is true when x > 0.