Jose bautista hits a baseball that travels for 142m before it lands. The flight of the ball can be modelled by a quadratic function in which x is the Horizontal distance the ball has travelled away from jose, and h(x) is the height Vertical distance of the ball at that distance.
Assume that the ball was between 0.6m and 1.5m above the ground when it was hit.
1. Explain why each function is not a good model of the situation and support your answer with a labelled sketch.
a) h(x)= -0.5x (x-142)
b) -0.5x^2+ 71x + 1
c) -0.0015x^2 + 0.213x + 1.2
2. Determine an equation that models the path of the ball.
no answers?
1. To determine whether each function is a good model of the situation, we need to analyze their behavior and compare it to the given information. We should also consider the range of possible values for the height of the ball.
a) The function h(x) = -0.5x(x-142) is a quadratic function. It suggests that the height of the ball increases as the horizontal distance from Jose increases, and then decreases as it gets closer to the landing point. However, this function does not account for the initial height of the ball because it does not have a constant term. In other words, it assumes the ball was hit from ground level (h = 0). This assumption is not correct because the ball was already between 0.6m and 1.5m above the ground when it was hit. Therefore, this function is not a good model for the situation.
To illustrate this, we can sketch a graph by plotting points with various values of x, considering the given vertical range. The graph will not intersect the y-axis at a non-zero value, indicating that it does not account for the initial height.
b) The function h(x) = -0.5x^2 + 71x + 1 is also a quadratic function. It considers a constant term (1), which means it accounts for the initial height of the ball. This is an improvement over the previous function. However, we still need to verify if this equation satisfies the given constraints. We can consider the range of possible values for x from 0 to 142, and check if the resulting y-values (heights) fall within the range of 0.6m to 1.5m.
c) The function h(x) = -0.0015x^2 + 0.213x + 1.2 is another quadratic function. It includes both a constant term (1.2) and a coefficient for the linear term (0.213x), indicating that it accounts for the initial height and the relationship between horizontal and vertical distance. Similar to the previous function, we need to verify if the resulting heights fall within the given range.
To compare these functions and determine the better model, we can plot their graphs and consider whether they satisfy the given constraints on the initial height and the range of possible heights.
2. To determine an equation that models the path of the ball, we need to find a quadratic function that satisfies the given constraints. We can do this by following these steps:
Step 1: Determine the vertex of the parabola
- The vertex point represents the highest point of the ball's trajectory (maximum height).
- The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of x^2 and x, respectively, in the quadratic equation.
- In this case, a = -0.0015 and b = 0.213.
- Calculate x = -0.213 / (2 * -0.0015) = 71.33 (rounded to two decimal places).
Step 2: Substitute the x-coordinate of the vertex into the equation to find the corresponding y-coordinate (height).
- Plug x = 71.33 into any of the given quadratic equations, such as h(x) = -0.0015x^2 + 0.213x + 1.2.
- Calculate y = -0.0015 * (71.33)^2 + 0.213 * 71.33 + 1.2 = approximately 11.95 (rounded to two decimal places).
Step 3: Construct the quadratic function for the ball's path.
- The equation will have the form h(x) = ax^2 + bx + c, where a, b, and c are determined by the vertex coordinates.
- In this case, the vertex is (71.33, 11.95), which means the equation will be h(x) = -0.0015(x - 71.33)^2 + 11.95.
This quadratic equation models the path of the ball, taking into account the initial height and the relationship between horizontal and vertical distance.