Problem 434 - Elementary Algebra

Question

In the following exercises, solve. Round approximations to one decimal place. Gravity A hang glider dropped his cell phone from a height of 350 feet. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the cell phone to reach the ground.

Step-by-Step Solution

Answer

4.7 seconds

Step 1 - Understand the Formula

The formula given is \[ t = \frac{\sqrt{h}}{4}\] where \( t \) is the time in seconds and \( h \) is the height from which the object is dropped. In this case, \( h = 350 \) feet.

Step 2 - Substitute the Given Height

Substitute \( h = 350 \) feet into the formula:\[ t = \frac{\sqrt{350}}{4} \]

Step 3 - Calculate the Square Root

Calculate the square root of 350:\[ \sqrt{350} \approx 18.7 \]

Step 4 - Divide by 4

Now divide the result by 4:\[ t = \frac{18.7}{4} \approx 4.675 \]

Step 5 - Round to One Decimal Place

Round the result to one decimal place:\[ t \approx 4.7 \]

Key Concepts

Understanding Algebra in the Context of the Gravity Formula

In this exercise, algebra plays a key role in manipulating and solving the given formula. Algebra involves working with variables and constants to solve equations. Here, our main equation is \ t = \frac{\sqrt{h}}{4} \, where \( t \) (time in seconds) depends on \( h \) (height in feet).

To solve for \( t \), we replace the variable \( h \) with its actual value. This substitution is fundamental in algebra, where variables are placeholders that can be replaced by known values.

By understanding how to manipulate algebraic equations, you can rearrange and solve for different variables, facilitating the calculation of various scenarios in science and math.

Calculating and Understanding the Square Root

The square root is a fundamental concept in mathematics, denoted as \( \sqrt{ } \). It implies finding a number that, when multiplied by itself, gives the original number.

In our formula, we need to calculate \( \sqrt{350} \). The square root of 350 is approximately 18.7, meaning that 18.7 * 18.7 is close to 350.

Understanding the square root helps in simplifying problems, especially in physics and engineering, where you often deal with quadratic relationships.

The Importance of Rounding Numbers

Rounding numbers simplifies complex calculations and makes results more understandable. In this problem, we round the final answer to one decimal place as instructed.

After calculating \( t = \frac{18.7}{4} \approx 4.675 \), we round it to 4.7 seconds.

Rounding follows certain rules: if the digit after the rounding place is 5 or more, you round up. If it's less than 5, you round down. It's a useful skill for ensuring that numbers are easy to read and communicate.

Time Calculation from Height

Time calculation is essential in many physics problems. Here, we used the height-distance relationship in free fall to calculate the time. The formula \( t = \frac{\sqrt{h}}{4} \) embodies the principles of physics under gravity.

By substituting the height \( h = 350 \) feet, taking the square root, and then dividing by 4, we calculated the time it took for the cell phone to hit the ground.

This type of calculation is vital for understanding how different forces and distances affect time, an essential concept in kinematics and dynamics.