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: Determine the distance fallen by an object after 3.5 seconds and 7.9 seconds using the formula \(S_n = 16n^2\), and interpret the results. Answer: After 3.5 seconds, the object would have fallen a total of 196 feet. After 7.9 seconds, the object would have fallen a total of approximately 998.56 feet.

Step-by-step solution

Calculate \(f(3.5)\)

To find the distance fallen in 3.5 seconds, plug \(n=3.5\) into the formula: \(f(3.5)= 16(3.5)^2\). Calculate the square of 3.5 and then multiply by 16. $$f(3.5) = 16(3.5^2)$$ $$f(3.5) = 16(12.25)$$ $$f(3.5) = 196$$

Interpret \(f(3.5)\)

The result \(f(3.5) = 196\) means that after 3.5 seconds, the object would have fallen a total of 196 feet.

Calculate \(f(7.9)\)

To find the distance fallen in 7.9 seconds, plug \(n=7.9\) into the formula: \(f(7.9)= 16(7.9)^2\). Calculate the square of 7.9 and then multiply it by 16. $$f(7.9) = 16(7.9^2)$$ $$f(7.9) = 16(62.41)$$ $$f(7.9) = 998.56$$

Interpret \(f(7.9)\)

The result \(f(7.9) = 998.56\) means that after 7.9 seconds, the object would have fallen a total of approximately 998.56 feet.