Problem 432 - Elementary Algebra

Question

In the following exercises, solve. Round approximations to one decimal place. Gravity While putting up holiday decorations, \(\quad\) Renee dropped a light bulb from the top of a 64 foot tall tree. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the light bulb to reach the ground.

Step-by-Step Solution

Answer

It took 2 seconds for the light bulb to reach the ground.

Step 1: Identify the Given Values

The height of the tree is given as 64 feet.

Step 2: Use the Given Formula

The formula to determine the time it takes for the light bulb to reach the ground is given by \(t = \frac{\text{sqrt}{h}}{4}\)

Step 3: Substitute the Given Height into the Formula

Substitute the height, 64 feet, into the formula: \( t = \frac{\text{sqrt}{64}}{4} \)

Step 4: Simplify the Expression Inside the Square Root

\( \text{sqrt}{64} = 8 \)

Step 5: Calculate the Time

Now divide by 4: \( t = \frac{8}{4} = 2 \)

Step 6: Conclusion

It took 2 seconds for the light bulb to reach the ground.

Key Concepts

Algebraic Formulas

Algebraic formulas are equations that express relationships between different quantities. In this exercise, we used the formula:

\( t = \frac{\text{sqrt}(h)}{4} \)

This equation helps us find the time, \(t\), it takes for an object to fall from a certain height, \(h\), due to gravity. Knowing this formula allows us to quickly calculate the falling time without having to perform more complicated physics calculations.

To use an algebraic formula, you simply need to identify the values for the variables and substitute them into the formula. In our exercise, the variable \(h\) represents the height of the tree, which is 64 feet.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 64 is 8 because 8 multiplied by itself (8 * 8) equals 64.
In mathematical notation, this is written as:
\[ \text{sqrt}(64) = 8 \]

In our exercise, the height of 64 feet was under the square root symbol in the formula. The simplified form helped us to easily calculate the time, showing how important understanding square roots is for such problems. Knowing how to simplify square roots will help make calculations quicker and easier.

Problem-Solving Steps

When solving any problem, it's helpful to break it down into clear, manageable steps. Here's how we approached the gravity calculation problem:

• Step 1: Identify the given value (height of the tree = 64 feet).
• Step 2: Use the given formula: \(t = \frac{\text{sqrt}(h)}{4}\).
• Step 3: Substitute the given value into the formula: \(t = \frac{\text{sqrt}(64)}{4}\).
• Step 4: Simplify the square root: \(\text{sqrt}(64) = 8\).
• Step 5: Calculate the time by performing the division: \(t = \frac{8}{4} = 2\).
• Step 6: Conclude that it took 2 seconds for the light bulb to reach the ground.

By following these steps, we can systematically solve the problem and ensure we don't miss any important parts.

Approximation and Rounding

Sometimes, when solving problems, you need to round your answer to make it more understandable or practical. Rounding helps us express numbers in a simpler form that is still close to the exact value.

In this exercise, we were asked to round any approximations to one decimal place. However, since our exact calculated time was already a whole number (2 seconds), rounding was not necessary.

Here are the basic rules for rounding:

• If the digit you are rounding is less than 5, round down.
• If the digit you are rounding is 5 or more, round up.

Practicing these rules regularly will help you become skilled at making approximations in different types of math problems.