Problem 13 - Applied Calculus

Question

A grapefruit is tossed straight up with an initial velocity of \(50 \mathrm{ft} / \mathrm{sec}\). The grapefruit is 5 feet above the ground when it is released. Its height at time \(t\) is given by $$y=-16 t^{2}+50 t+5$$ How high does it go before returning to the ground?

Step-by-Step Solution

Answer

The grapefruit reaches a maximum height of 44.0625 feet.

Step 1: Identify and Express the Problem

To find the maximum height reached by the grapefruit, we need to determine the vertex of the parabolic equation given for the height: \(y=-16t^2+50t+5\). The quadratic formula is in the form \(y=at^2+bt+c\), where \(a=-16\), \(b=50\), and \(c=5\).

Step 2: Find the Vertex's Time \(t\)

The time \(t\) at which the maximum height is reached for a parabola \(y=at^2+bt+c\) can be calculated using the formula \(t=-\frac{b}{2a}\). Here, \(b=50\) and \(a=-16\). Plug these values in to find \(t\):\[t=-\frac{50}{2 \times -16} = \frac{50}{32} = 1.5625\] seconds.

Step 3: Calculate Maximum Height \(y\)

Use the time \(t=1.5625\) seconds to find the height by substituting it back into the height equation:\[y=-16(1.5625)^2+50(1.5625)+5\]Calculate step-by-step:1. \((1.5625)^2 = 2.44140625\)2. \(-16 \times 2.44140625 = -39.0625\)3. \(50 \times 1.5625 = 78.125\)4. \(y = -39.0625 + 78.125 + 5 = 44.0625\)Thus, the maximum height is \(44.0625\) feet.

Key Concepts

Quadratic Function

A quadratic function is a mathematical expression of the form \( y = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). This type of function results in a parabolic graph that can open upwards or downwards depending on the sign of \( a \).

The quadratic equation represents any type of behavior where the rate of change is not linear but occurs in a curved motion. In the context of projectile motion, such as our grapefruit, the quadratic function defines the vertical motion affected by gravity.

The coefficients in the function, particularly \( a \), \( b \), and \( c \), provide important information about the motion:

• \( a \): Influences the direction and steepness of the parabola. In our example, \( a = -16 \) indicating the parabola opens downwards.
• \( b \): Contributes to the position of the vertex (the highest point in our case), and affects the slope at \( t = 0 \).
• \( c \): Represents the initial height of the projectile, which is 5 feet for our grapefruit.

Understanding the quadratic function is crucial for solving projectile motion problems, as it helps in predicting the trajectory and maximum height achieved by the object.

Vertex Formula

The vertex of a parabola is a fundamental concept when analyzing quadratic equations, especially when determining the maximum or minimum point. For a quadratic function \( y = ax^2 + bx + c \), the vertex can either be the peak (maximum) or the trough (minimum) of the parabola.

To find the time at which the maximum height occurs, since our parabola opens downwards, we use the vertex formula for time \( t \):

• \( t = -\frac{b}{2a} \)

For our projectile problem:

• \( b = 50 \)
• \( a = -16 \)

Substituting these values yields:
\[ t = -\frac{50}{2(-16)} = \frac{50}{32} = 1.5625 \text{ seconds} \]
This formula is essential for finding the precise moment when the projectile reaches its maximum height. Once we have this time, we can substitute it back into the original quadratic function to find the maximum height.

By understanding and applying the vertex formula, one can easily locate the highest point in any parabolic motion, allowing for accurate predictions and calculations.

Parabolic Motion

Parabolic motion describes the trajectory path that follows a symmetrical curve, commonly known as a parabola. This type of motion is typical for objects under projectile motion, like a ball being thrown or a grapefruit tossed upwards, as gravity affects their path.

In our specific scenario, the grapefruit follows a parabolic motion described by the equation \( y = -16t^2 + 50t + 5 \). The path of this motion is characterized by an upward climb followed by a descent.

Key features of parabolic motion include:

• The initial direction is determined by the velocity, here \( 50 \text{ ft/s} \), which propels the grapefruit upwards.
• Gravity acts as a constant downward force, represented by the \( -16t^2 \) term in the equation, creating the downward curve of the parabola.
• The maximum height point is the vertex of the parabola, found using the vertex formula. This point signifies the turning point where the upward motion ends and descent begins.

Understanding parabolic motion helps us comprehend real-world phenomena where objects are influenced by gravity, enabling predictions about distances, heights, and landing points. This makes it a critical concept in fields like physics and engineering as well as in various sports and everyday activities.