Question

Refer to the falling object of Example 1 on page 457 , where we found that the total distance, in feet, that an object falls in \(n\) seconds is given by \(S_{n}=16 n^{2}\). The formula for \(S_{n}\) is defined for positive integers but also can be written as a function \(f(n)\), where \(n \geq 0\). Find and interpret \(f(3.5)\) and \(f(7.9)\).

Short answer

Question: Using the formula \(S_n = 16n^2\), find the position of an object after 3.5 seconds and 7.9 seconds. Answer: After 3.5 seconds, the object has fallen a total distance of 196 feet. After 7.9 seconds, the object has fallen a total distance of 998.56 feet.

Step-by-step solution

Understand the formula and the function

The given formula \(S_n = 16n^2\) represents the total distance fallen by the object after \(n\) seconds. It is valid only for \(n \geq 0\). This formula can also be written in the form of a function, \(f(n) = 16n^2\), where \(n \geq 0\).

Check the validity of the given time values

We are given two time values, \(n_1 = 3.5\) seconds and \(n_2 = 7.9\) seconds. We need to ensure that these time values satisfy the condition \(n \geq 0\). For \(n_1 = 3.5\): \(3.5 \geq 0\), so it is a valid value. For \(n_2 = 7.9\): \(7.9 \geq 0\), so it is also a valid value.

Applying the formula

Now that we have validated the given time values, we will apply the function \(f(n)\) to find the total distance fallen by the object at times \(n_1 = 3.5\) and \(n_2 = 7.9\). For \(n_1 = 3.5\): \(f(3.5) = 16(3.5)^2 = 16(12.25) = 196\) feet For \(n_2 = 7.9\): \(f(7.9) = 16(7.9)^2 = 16(62.41) = 998.56\) feet

Interpretation

For \(n_1 = 3.5\) seconds, the object has fallen a total distance of 196 feet. This means that after 3.5 seconds, the object is located 196 feet below its starting position. For \(n_2 = 7.9\) seconds, the object has fallen a total distance of 998.56 feet. This means that after 7.9 seconds, the object is located 998.56 feet below its starting position.

Key concepts

Distance Formula

In the context of this problem, the distance formula allows us to calculate how far an object has fallen over time under the influence of gravity. The formula used here is a quadratic function:

• \( S_n = 16n^2 \), where \( S_n \) represents the total distance fallen, measured in feet, after \( n \) seconds.

Quadratic functions like this one describe a parabolic relationship between two variables—in this case, time and distance. The key part of the formula is \( n^2 \), which emphasizes how the distance increases at a faster rate as time progresses. By using this formula, you can predict how far an object will fall after any given time of freefall.

Time Values

Time values are crucial in our distance formula because they determine at which point you want to measure the fallen distance. In this scenario:

• The values given are \( n_1 = 3.5 \) seconds and \( n_2 = 7.9 \) seconds.
• Each time value should satisfy \( n \geq 0 \), meaning the calculation applies only to positive time periods.

Both \( n_1 \) and \( n_2 \) are valid because they are greater than or equal to zero. Using these values within the formula, you can compute how much distance the object will have fallen after 3.5 seconds and 7.9 seconds. Simply substitute the time values into the function:

• At 3.5 seconds, find \( f(3.5) \).
• At 7.9 seconds, find \( f(7.9) \).

Interpretation of Results

Once you have calculated the distances using the time values, interpreting these results helps in understanding the real-world implications of the numerical answers. Let's assess these results:

• When \( f(3.5) = 196 \) feet, it means the object is 196 feet below its starting point after 3.5 seconds of falling.
• Similarly, \( f(7.9) = 998.56 \) feet indicates that after 7.9 seconds, the object has fallen 998.56 feet.

These distances demonstrate how the object's speed increases over time. The object falls further during each successive second, highlighting the quadratic nature of the function. Interpreting these values sheds light on the motion's acceleration caused by gravity, which can be applied in various physics and engineering problems.