Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, use a calculator to find the answer. The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function \(h(x)=\frac{-32}{(80)^{2}} x^{2}+x\) where \(x\) is the horizontal distance traveled and \(h(x)\) is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

Step-by-Step Solution

Answer

The height is 50 feet when the object is 100 feet away horizontally.

Step 1: Understanding the Problem

We have a quadratic function \(h(x) = \frac{-32}{6400} x^2 + x\) that represents the height \(h(x)\) of an object launched at a 45-degree angle with an initial velocity of 80 feet per second. We need to find the height when the object has traveled 100 feet horizontally, meaning we substitute \(x = 100\) into the function.

Step 2: Substitute the Known Value

Substitute \(x = 100\) into the function:\[ h(100) = \frac{-32}{6400} (100)^2 + 100 \]

Step 3: Calculate the Value of the Quadratic Term

First, calculate the quadratic term. Compute \( (100)^2 = 10000 \) and then multiply by \(\frac{-32}{6400}\): \[ \frac{-32}{6400} \times 10000 = -50 \]

Step 4: Calculate the Height

Add the result of the quadratic term to 100:\[ h(100) = -50 + 100 = 50 \]

Step 5: Conclusion

The height of the object when it has traveled 100 feet horizontally is 50 feet. Therefore, using a calculator to confirm by substituting into the function and calculating when needed helps confirm this result.

Key Concepts

Projectile Motion

Projectile Motion describes the motion of an object that is thrown or projected into the air, subject to only the acceleration of gravity. This type of motion can be seen in sports, for example, when someone throws a ball or shoots an arrow.
Projectile Motion consists of two components: horizontal and vertical motion. These components are independent of each other. The horizontal motion is consistent because no extra forces are acting in that direction — only if air resistance is negligible.
The vertical motion, however, is affected by gravity, causing the object to accelerate downward. This results in a curved path, which is a signature of projectile motion.

Parabolic Path

A Parabolic Path is the curve traced by a projectile in motion under the influence of gravity (without air resistance). This distinct path is a mathematical parabola. Parabolas are U-shaped, symmetrical curves defined by quadratic functions.
The quadratic equation that we are dealing with here has the general form of \( ax^2 + bx + c \), resulting from projecting an object at an angle with an initial velocity. The negative coefficient (\( -\frac{32}{6400} \) in the exercise) affects how narrow or wide the parabola opens.
When solving problems involving parabolas, always pay attention to the vertex of the parabola, as it represents the maximum height if the projectile path opens downwards or the minimum height if the path opens upwards. In the given function, since it's opening downwards (the \( a \) value is negative), the vertex would signify the peak point of our object's trajectory.

Calculator Usage

When dealing with quadratic equations, like the one given in the exercise, using a calculator is very helpful. Specifically, graphing calculators can quickly solve these equations and find specific values like maximum height or the height at a certain distance.
The TRACE feature on many calculators can be used to pinpoint precise values at specified points along the curve, simplifying the process. This is particularly useful when trying to solve for the height at a specific horizontal distance, such as in this exercise.
To use the TRACE function, enter the quadratic function into the calculator, input the specific x-value you want to check (like 100 feet in this case), and the calculator will give you the corresponding height, saving time on manual calculations.

Initial Velocity

Initial Velocity is the speed at which an object is launched. In the context of projectile motion, it plays a significant role in determining how high and how far the projectile will travel.
In the given problem, the object is projected at an initial velocity of 80 feet per second at a 45-degree angle. This means that the initial speed is being used equally for both the horizontal and vertical components of the motion.
Initial velocity can often be broken down into these orthogonal components using trigonometric functions. For instance, if an object is projected at an angle \( \theta \), the horizontal velocity component would be \( v_i \cos(\theta) \), and the vertical component would be \( v_i \sin(\theta) \). So, understanding initial velocity is essential for predicting and calculating different outcomes in projectile motion.