Question

The path of a basketball thrown at an angle of \(45^{\circ} \mathrm{can}\) be modeled by \(y=-0.02 x^2+x+6\).

Short answer

The maximum height reached by the basketball is 18.75 units.

Step-by-step solution

- Identifying the components of the given equation

The equation given is a standard quadratic equation represented as \( y = ax^2 + bx + c \), where 'a' is -0.02, 'b' is 1 and 'c' is 6 in this case, which respectively represents the acceleration, initial velocity and initial height of the basketball.

- Finding the derivative for maximal height

To find the maximum height reached by a ball, differentiation comes into play. Let's differentiate the given quadratic equation with respect to 'x', which gives \( dy/dx = 2ax + b\). On substituting 'a' and 'b' values, \(dy/dx = -0.04x + 1\)

- Finding the time of maximal height

Set the derivative equal to zero and solve for 'x'. This will tell when the ball reaches its maximum height. Hence, solving \( -0.04x + 1 = 0 \) gives 'x' as 25 units.

- Calculating the maximum height

Substitute 'x' from step 3 into the original equation \( y = -0.02x^2 + x + 6 \) to get the maximum height 'y'. On substituting 'x' as 25, we get 'y' as 18.75 units. Hence, the maximum height reached by the ball is 18.75 units

Key concepts

Derivatives

In the realm of calculus, derivatives measure how a function changes as its input changes. By understanding derivatives, we can gain insights into the behavior of various real-world phenomena.
For quadratic functions like the basketball path modeled by the equation \(y = -0.02x^2 + x + 6\), the derivative helps us find key characteristics, such as points of maximum height.
The derivative of a function, represented as \(dy/dx\), tells us the rate at which 'y' changes with respect to 'x'. This helps identify the slope or steepness of the curve at any point.

• For our basketball path, the derivative is found by differentiating \(y = -0.02x^2 + x + 6\), resulting in \(dy/dx = -0.04x + 1\).
• By setting the derivative equal to zero, we find the point where the slope of our curve is flat, indicating a peak or a trough. In quadratic equations of this type, this almost always indicates a maximum or minimum point due to the nature of parabolas.

Maximum Height

Achieving maximum height for a projectile like a basketball is a pivotal point of interest. It tells us how high the ball can go. The maximum height occurs at the 'vertex' of the parabola represented by our equation.
In our basketball example, the path is represented by \(y = -0.02x^2 + x + 6\). To find the time when the basketball reaches its highest point, we use the derivative of the path function.

• Derived from the quadratic function, \(dy/dx = -0.04x + 1\), set this to zero to find the 'x' at maximum height.
• Solve \(-0.04x + 1 = 0\) to get \(x = 25\).

Now that we have the 'x' value, which is the time in terms of the distance along the horizontal path, substitute it back into the original equation to find the maximum height.

• By substituting \(x = 25\) into \(y = -0.02x^2 + x + 6\), the maximum height \(y\) is calculated as 18.75 units.

This is the highest point that the basketball will reach before descending back towards the ground. Understanding how to find and interpret maximum height has practical applications beyond sports, such as physics and engineering.

Mathematical Modeling

Mathematical modeling allows us to describe real-world scenarios through mathematics, making complex phenomena easier to analyze and understand. By using mathematical models like the quadratic equation in our problem, we convert a physical action, such as a thrown basketball, into a form that can be manipulated mathematically.

• In our basketball problem, the model is \(y = -0.02x^2 + x + 6\), where 'x' is the horizontal distance and 'y' is the height of the ball.
• The coefficients (-0.02, 1, and 6) help describe the specific trajectory based on initial conditions like initial velocity and height.

This transformation from a physical event to a mathematical equation enables us to predict future behavior, calculate significant points like maximum height, and make informed decisions based on this information.
Mathematical models are powerful tools, providing clarity and insights across various fields, including sciences, engineering, and even economics.