Problem 435 - Elementary Algebra

Question

In the following exercises, solve. Round approximations to one decimal place. Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the hammer to reach the river.

Step-by-Step Solution

Answer

Approximately 15.8 seconds.

Step 1: Identify the given values

The height from which the hammer is dropped, denoted as h, is 4000 feet.

Step 2: Substitute the height into the formula

The formula to determine the time is \[ t = \frac{\sqrt{h}}{4} \]. Substitute the value of h:\[ t = \frac{\sqrt{4000}}{4} \]

Step 3: Compute the square root

Find the square root of 4000. \[ \sqrt{4000} \approx 63.2 \] (rounded to one decimal place)

Step 4: Divide by 4

Now divide the result by 4:\[ t = \frac{63.2}{4} = 15.8 \]

Step 5: Interpret the result

It takes approximately 15.8 seconds for the hammer to reach the river.

Key Concepts

Gravity Calculations

Gravity is an invisible force that pulls objects toward the center of the Earth. When an object is dropped, gravity accelerates it downward. This acceleration due to gravity is approximately 32 feet per second squared (ft/s²) on Earth. For our exercise, we are dealing with a construction worker dropping a hammer from 4000 feet above the Colorado River.

The gravitational force acting on the hammer pulls it down, and as a result, we can calculate how long it takes to hit the ground using specific formulas related to gravity and motion. In this case, we use the formula for time:

• This formula accounts for gravity's role in the object’s velocity as it falls.
• By understanding gravity calculations, you apply basic principles of physics to real-world scenarios like this one.

Time Formula

To find out how long it takes for an object to fall from a certain height, we use specific time formulas. The formula we used in this exercise is:
\( t = \frac{\sqrt{h}}{4} \)
This formula helps to estimate the time, t, in seconds, for an object to fall a certain height, h, in feet.

Here's a step-by-step breakdown:

• Firstly, insert the given height (4000 feet) into the formula: \( t = \frac{\sqrt{4000}}{4} \)
• Secondly, calculate the square root of the height.
• Lastly, divide the result by 4 to determine the time.

This simplified time formula is particularly useful for quick calculations, although it is derived from more complex physics principles.

Square Root Approximation

When solving equations involving square roots, it's often necessary to approximate the square root to make the calculation easier. In this exercise, we approximated the square root of 4000.

• The exact square root of 4000 is a long decimal but we approximate it to one decimal place as 63.2.
• Approximating helps to simplify the following steps and is sufficient for most practical purposes.
• Most calculators or software tools can compute this quickly, saving time.

After approximating, you can easily perform further arithmetic. For example, division by 4 to find the time it takes for the hammer to fall.