Problem 70 - Elementary Algebra

Question

The height of the building at 225 South Sixth in Minneapolis is \(776 \mathrm{ft}\). Ignoring air resistance, use the formula \(d=\left(\frac{16 \mathrm{ft}}{1 \mathrm{~s}^{2}}\right) t^{2}\) to find the time for an object to fall this distance. Round to the nearest hundredth. (Source: www.emporis.com)

Step-by-Step Solution

Answer

The time for an object to fall from the building is approximately 6.96 seconds.

Step 1: Write down the given information

The height of the building is given as 776 feet. The formula to calculate the distance an object falls due to gravity is provided as: \[ d = \frac{16 \text{ ft}}{1 \text{ s}^2} t^2 \]

Step 2: Set up the equation

In this context, the distance \(d\) is the height of the building. Substitute the given height into the formula: \[ 776 \text{ ft} = 16 \text{ ft/s}^2 \times t^2 \]

Step 3: Solve for \(t^2\)

To solve for \(t^2\), divide both sides of the equation by 16 \( \text{ft/s}^2 \) as follows: \[ t^2 = \frac{776 \text{ ft}}{16 \text{ ft/s}^2} \]

Step 4: Calculate the value

Now, perform the division:\[ t^2 = \frac{776}{16} = 48.5 \]

Step 5: Find \(t\)

To find \(t\), take the square root of both sides of the equation:\[ t = \frac{\theta}{\theta} = \frac{\theta}{\theta} = \frac{\theta}{\theta} = \frac{6.96 \]

Step 6: Round to the nearest hundredth

Round the calculated time to the nearest hundredth:\[ t \ approximately 6.96 \text{ seconds} \]

Key Concepts

Gravity Calculations

Gravity is a force that pulls objects toward the center of the Earth. When we talk about how objects fall, gravity plays a key role. The formula we use to calculate the distance an object falls because of gravity is:

• d = 16 t^2

This formula applies when we ignore air resistance. In this formula: d stands for the distance the object falls in feet. t stands for the time it takes in seconds. The number 16 is derived from the acceleration due to Earth's gravity, which is approximately 32 feet per second squared, divided by 2. Let's break down an example. If an object falls from a height of 776 feet, we can plug that in as our distance (d) and solve for time (t).

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is a square. The standard form of a quadratic equation is:

• ax^2 + bx + c = 0

In our gravity calculation, the formula we used is a simplified version of a quadratic equation. The equation from our example: 776 = 16 t^2 looks a lot like a quadratic equation where 'a' is 16, 'b' is 0, and 'c' is -776. To solve for 't' in such equations, we perform a series of algebraic steps, which usually involve isolating the variable (t) and then taking the square root. This specific quadratic equation doesn't have a 'b' term (bx), so it's simpler. We just solve for 't' directly after isolating it. This can help students see the practical, simplified applications of more complex quadratic formulas.

Distance and Time Relations

The relationship between distance and time is essential for understanding motion and dynamics. In the formula we used (d = 16 t^2), distance (d) depends directly on the square of time (t). This tells us that as time increases, the distance an object falls goes up by the square of that time.Suppose we observe real-world falling objects like a ball dropped from a building. Initially, it drops a small distance, but soon, it covers more significant distances very rapidly as time goes on. This quadratic relationship is important for understanding not only free-fall under gravity but various other physical phenomena as well. By looking at the problem step-by-step, you can understand how changing one variable affects the other. If we know the distance, we can calculate the time, or if we know the time, we can determine how far the object falls.

Basic Algebra

Algebra is the branch of mathematics that deals with variables and the rules for manipulating those variables. It's the foundation for solving equations, which is central to this exercise.In our example, we start by writing down the given information and the formula. Using algebraic principles, we rearrange the formula to solve for the unknown variable, 't'.Here are the main steps we used:

• Substitute the given values into the equation: 776 = 16 t^2
• Isolate the variable t^2 by dividing both sides by 16.
• Solve for t by taking the square root of both sides.

These steps showcase algebraic manipulation in its simplest form, which is a fundamental skill in math. By practicing such problems, students can strengthen their understanding of how algebra is used to solve real-world problems, even those related to physics and engineering.