Problem 61 - Algebra 2

Question

PHYSICS If a stone is dropped from a cliff, the equation \(t=\frac{1}{4} \sqrt{d}\) represents the time \(t\) in seconds that it takes for the stone to reach the ground. If \(d\) represents the distance in feet that the stone falls, find how long it would take for a stone to hit the ground after falling from a 150 -foot cliff.

Step-by-Step Solution

Answer

It takes approximately 3.06 seconds for the stone to hit the ground.

Step 1: Identify the Known Values

We are given that the stone is dropped from a 150-foot cliff, so the distance \( d \) is 150 feet. This is the value we will use to find the time \( t \).

Step 2: Write Down the Equation

The equation given in the problem is \( t = \frac{1}{4} \sqrt{d} \). This equation relates the time \( t \) to the distance \( d \).

Step 3: Substitute the Known Value into the Equation

Insert the value of \( d = 150 \) feet into the equation: \( t = \frac{1}{4} \sqrt{150} \). This substitution allows us to calculate the time \( t \).

Step 4: Calculate the Square Root of the Distance

First, calculate the square root of 150: \( \sqrt{150} \approx 12.25 \). This value is an approximation of the square root of 150.

Step 5: Multiply the Result by the Coefficient

Now multiply the square root value by \( \frac{1}{4} \): \( t = \frac{1}{4} \times 12.25 \).

Step 6: Compute the Time

Perform the multiplication to find \( t \): \( t \approx 3.0625 \). This is the approximate time taken for the stone to reach the ground.

Key Concepts

Problem Solving

Problem-solving is a crucial skill in mathematics and physics, empowering you to tackle various problems systematically. In this context, we'll look at how to approach a problem where you need to find out how long it takes for a stone to hit the ground after being dropped from a cliff. The key steps start with identifying the known values given in the problem. Here, the distance from which the stone is dropped, 150 feet, is provided. Next, you'll write down the equation given in the problem. This equation, \( t = \frac{1}{4} \sqrt{d} \), relates the time \( t \) and distance \( d \).
Next, you substitute the known values into the equation. This process of substitution is crucial because it allows you to work with actual numbers rather than abstract symbols. Following this, you perform calculations such as finding the square root, and multiplying by constants to solve for \( t \), the time taken. These steps illustrate a structured approach to problem-solving:

• Identify known values
• Write down the relevant equation
• Perform calculations

Working through each step systematically enables you to break down and solve complex problems efficiently.

Distance-Time Relationship

The distance-time relationship is a fundamental concept in physics that describes how the distance an object travels relates to the time it takes. For the exercise at hand, this relationship is expressed through the formula \( t = \frac{1}{4} \sqrt{d} \). Here, \( d \) is the distance the stone falls, while \( t \) is the time taken to reach the ground.
This formula shows a non-linear relationship between distance and time. As the distance increases, time increases, but not proportionally. The square root function signals that time grows at a reducing rate concerning distance. This relationship is typical in physics when dealing with falling objects due to the effect of gravity.

• In the equation \( t = \frac{1}{4} \sqrt{d} \), the square root of the distance \( d \) is first calculated, showing time doesn't increase linearly with distance.
• Next, it's multiplied by \( \frac{1}{4} \), which adjusts the time to account for the specific conditions of the problem, such as gravitational acceleration.

Understanding this relationship allows one to predict how different distances affect the time it takes for objects to fall, a valuable insight in various physics applications.

Square Root Calculation

Square root calculation is essential in solving the given problem, as it helps find how time \( t \) relates to the distance \( d \) a stone falls. In this exercise, you need to determine \( \sqrt{150} \), the square root of 150.
The square root of a number is a value that, when multiplied by itself, gives the original number. While perfect squares, like 4 or 9, are straightforward, computing the square root of non-perfect squares involves approximation.
Thirty minutes typically involve using a calculator to approximate, finding that \( \sqrt{150} \approx 12.25 \). With this calculated value, you then proceed to the next calculation step.

• Calculate the square root to understand part of the relationship between time and distance.
• Convert this value into a practical measurement by multiplying by constants, such as \( \frac{1}{4} \) in this problem.

Successfully approximating square roots is an indispensable skill in both mathematics and science, providing important results even when dealing with imperfect numbers.