Problem 108 - Algebra A Combined Function

Question

A performer with the Moscow Circus is planning a stunt involving a free fall from the top of the Moscow State University building, which is 784 feet tall. (Source: Council on Tall Buildings and Urban Habitat) Neglecting air resistance, the performer's height above gigantic cushions positioned at ground level after \(t\) seconds is given by the expression \(784-16 t^{2}\) a. Find the performer's height after 2 seconds. b. Find the performer's height after 5 seconds. c. To the nearest whole second, estimate when the performer reaches the cushions positioned at ground level. d. Factor \(784-16 t^{2}\).

Step-by-Step Solution

Answer

a. 720 feet; b. 384 feet; c. 7 seconds; d. \((28 - 4t)(28 + 4t)\).

Step 1: Understand the Problem

The performer's height above the ground after a certain time \( t \) is given by the expression \( 784 - 16t^2 \). We need to find out the height at certain time values and when the height reaches zero. Finally, we'll factor the quadratic expression.

Step 2: Calculate Height after 2 Seconds

Substitute \( t = 2 \) into the height expression:\[784 - 16(2)^2 = 784 - 16(4) = 784 - 64 = 720\]Therefore, the performer's height after 2 seconds is 720 feet.

Step 3: Calculate Height after 5 Seconds

Substitute \( t = 5 \) into the height expression:\[784 - 16(5)^2 = 784 - 16(25) = 784 - 400 = 384\]Therefore, the performer's height after 5 seconds is 384 feet.

Step 4: Determine When Performer Reaches the Ground

The performer reaches the ground when their height is zero:\[784 - 16t^2 = 0\]Solve for \( t \):\[16t^2 = 784\]\[t^2 = 49\]\[t = \pm 7\]Since time cannot be negative, the performer reaches the ground at \( t = 7 \) seconds.

Step 5: Factor the Expression

The expression is \( 784 - 16t^2 \). Notice that this is a difference of squares:\[784 = 28^2 \, \text{and} \, 16t^2 = (4t)^2 \]Thus, the expression factors to:\[(28 - 4t)(28 + 4t)\]

Key Concepts

Free Fall Motion

Free fall motion refers to a type of movement where an object is only influenced by gravity. When an object falls towards the ground without any resistance, such as air or wind, it is said to be in free fall. This concept is essential in physics, as it helps illustrate how gravitational forces act on objects.In this exercise, the performer is jumping off a building which highlights the concept of free fall motion. The building is 784 feet tall and the formula given, \(784 - 16 t^2\), helps describe how his height changes over time while in free fall. Here, the constant 16 represents the acceleration due to gravity in feet per second squared, while the expression \(t^2\) accounts for the time duration of the fall. Thus, we see real-world physics in action through this simple algebraic expression.When solving problems of free fall, understanding that objects fall at an acceleration due to gravity helps in predicting outcomes like how long an object will take to reach the ground.

Factoring Quadratics

Factoring quadratic expressions is a common algebraic technique used to simplify polynomials. It involves rewriting a quadratic expression as a product of two simpler binomials.In our particular problem, the quadratic expression is \(784 - 16t^2\), which can be recognized as a difference of squares. The expression can be broken down into

• \(784 = 28^2\)
• \(16t^2 = (4t)^2\)

The difference of squares can thus be factored as \((28 - 4t)(28 + 4t) \).Factoring in this way offers a clearer understanding of how the terms in the expression relate to zero. This factorization process is especially important for simplifying complex expressions and solving quadratic equations.

Physics in Algebra

Physics often intersects with algebra in solving real-world problems, such as calculating motion trajectories and force impacts. By expressing physical concepts like free fall in algebraic terms, students can predict outcomes and solve practical problems systematically.The equation in the problem, \(784 - 16t^2\), is a perfect example of how physics and algebra can work together. It uses the principle of gravitational acceleration and allows us to calculate how quickly the performer descends towards the ground after starting from 784 feet. Algebra provides the tools to manipulate these expressions so that we can understand and predict the height at any given second.This synthesis of physics and algebra empowers learners to tackle a wide variety of physical scenarios, from simple motions like falling objects, to more complex topics such as projectile motion.

Time Calculation in Word Problems

Time plays a crucial role in many word problems, especially those involving motion or changes over time. Calculating time accurately requires setting the right parameters and using algebraic equations to find solutions.For instance, in our scenario, you calculate when the height of the performer reaches ground level by setting the equation \(784 - 16t^2 = 0\) and solving for \(t\). Through algebraic manipulation, we isolate the variable \(t\) to find that the performer reaches the ground at \(t = 7\) seconds. This step is crucial, as understanding time in equations helps in predicting certain events or intervals with precision.Mastering how to solve for time in word problems enhances problem-solving skills and opens more avenues for analyzing practical events.