Problem 433 - Elementary Algebra

Question

In the following exercises, solve. Round approximations to one decimal place. Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the flare to reach the water.

Step-by-Step Solution

Answer

8.0 seconds

Step 1: Understand the Problem

The problem asks to find the time it takes for a flare to fall from a height of 1024 feet to the ground. Use the given formula: \[ t = \frac{\sqrt{h}}{4} \] where \(h\) is the height in feet and \(t\) is the time in seconds.

Step 2: Identify Given Values

Identify the height \(h\) from which the flare is dropped. Here, \(h = 1024\) feet.

Step 3: Plug in the Given Values

Substitute \(h = 1024\) into the formula: \[ t = \frac{\sqrt{1024}}{4} \]

Step 4: Calculate the Square Root

Find the square root of 1024: \[ \sqrt{1024} = 32 \]

Step 5: Perform Division

Divide the result by 4 to find the time \(t\): \[ t = \frac{32}{4} = 8 \]

Step 6: Round the Answer

Since the answer must be accurate to one decimal place, confirm the division result. Here, the result is already in whole number form, so we have:\[ t = 8.0 \]

Key Concepts

gravity effect

When discussing how an object falls from a height, like a flare from an airplane, we have to consider the effect of gravity. Gravity is a force that pulls objects toward the center of the Earth. This pull accelerates the object, causing it to increase speed as it falls.
The formula used in this exercise, \[ t = \frac{\sqrt{h}}{4} \], is derived from the equations of motion under gravity. Here, \(h\) represents height in feet, and \(t\) is the time in seconds.
Without gravity, objects would not fall to the ground, but float indefinitely. It’s important to remember that this formula assumes there is no air resistance, which in reality can affect the falling time.

square root calculation

The square root calculation is crucial for this problem. To recall, the square root of a number is a value that, when multiplied by itself, gives the original number. In this problem, you are given the height \(h = 1024\) feet.
First, you need to find the square root of 1024. The square root of 1024 is 32. We reach this value because \(32 \times 32\) equals 1024.

• This calculation simplifies to: \( \sqrt{1024} = 32 \)

Knowing how to find square roots is essential, as it allows us to progress with our formula and determine the falling time of the flare.

time distance formula

The given problem involves a specific time-distance formula, which tells us how long it takes for an object to fall to the ground. This unique formula uses the height from which the object falls and calculates the time based on the square root of this height.
The formula provided, \[ t = \frac{\sqrt{h}}{4} \], simplifies the complex process of free fall calculation. By substituting \(h = 1024\) into the formula, you perform the following steps:

• First, find the square root of 1024, which is 32.
• Then divide this result by 4. So, \(t = \frac{32}{4} = 8\).

The flare takes 8 seconds to reach the water when dropped from a height of 1024 feet.
This approach is very handy as opposed to more complicated physics equations.