Question

PROBLEM SOLVING A woodland jumping mouse hops along a parabolic path given by y = ?0.2x2 + 1.3x, where x is the mouse’s horizontal distance traveled (in feet) and y is the corresponding height (in feet). Can the mouse jump over a fence that is 3 feet high? Justify your answer.

Short answer

No, the mouse cannot jump over the fence. The maximum jump height the mouse can achieve according to the given equation is approximately 2.0875 feet, which is less than the height of the fence.

Step-by-step solution

Understand the problem and the equation

In the equation, y = -0.2x^2 + 1.3x, 'x' represents the horizontal distance traveled by the mouse (in feet) and 'y' is the corresponding height (in feet). Our task is to find if 'y', the height, ever reaches or exceeds 3 feet.

Determine the maximum height

The maximum height of a parabola given by the equation y = ax^2 + bx + c is given by the vertex of the parabola. The x-coordinate of the vertex is given by the formula -b/(2a). From the given equation, a = -0.2 and b = 1.3. Therefore, x = -1.3/(2*-0.2) = 3.25.

Calculate height at maximum point

To calculate the maximum height, substitute x = 3.25 in the given equation. y = -0.2*(3.25)^2 + 1.3*3.25 = 2.0875. Hence, the maximum height reached by the mouse is 2.0875 feet.

Analyze the result

Looking at the calculated maximum height, it is clear that this height (approx. 2.0875 feet) is less than 3 feet. Hence, it can be inferred that according to the given equation for its parabolic path, the mouse cannot jump over a fence that is 3 feet high.

Key concepts

Quadratic Functions

Understanding quadratic functions is essential for analyzing problems involving parabolic motion, like the trajectory of a woodland jumping mouse. A quadratic function is expressed in the standard form y = ax^2 + bx + c, where x represents an independent variable, y is the dependent variable, and a, b, and c are constants. The graph of a quadratic function forms a shape known as a parabola, which can open upwards or downwards depending on the sign of coefficient a.

If a is positive, the parabola opens up, and the vertex represents the minimum point. Conversely, if a is negative, as in our jumping mouse's parabolic path equation y = -0.2x^2 + 1.3x, the parabola opens downward, indicating that the trajectory has a maximum point—that is, the highest point the mouse can reach when hopping.

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction, representing either the maximum or minimum value of the quadratic function. The coordinates of the vertex can be determined using the formula (-b/(2a), f(-b/(2a))), where f(x) denotes the quadratic function.

In our exercise, the vertex reveals the highest point of the mouse's jump. Applying the formula, we calculate the x-coordinate of the vertex as x = -1.3/(2*-0.2) = 3.25. Knowing this value helps in understanding not just the specific point but also the nature of the motion. If plotted, the entire jump would be symmetric around this vertical line passing through the vertex, giving a visual representation of the mouse's parabolic leap.

Problem Solving with Quadratics

To solve real-world problems using quadratic functions, such as determining if the mouse can leap over a 3-feet high fence, we apply a systematic approach. First, we identify the maximum or minimum point by finding the vertex. In this case, we want the maximum height as the mouse jumps along a downward opening parabola.

After finding the vertex's x-coordinate, we plug it back into the original quadratic equation to find the corresponding y-coordinate, which gives us the maximum height. As shown in our step-by-step solution, substituting x = 3.25 into the mouse's trajectory equation results in a maximum height of approximately 2.0875 feet. This height, being less than 3 feet, allows us to deduce that the mouse cannot clear the fence. Problem-solving with quadratics often includes such practical applications, where identifying the vertex and evaluating the function at specific points lead us to the solution.