Problem 85 - Algebra for College Students

Question

Find the exact solution \((s)\) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. Projected pine cone. If a pine cone is projected upward at a velocity of \(16 \mathrm{ft} / \mathrm{sec}\) from the top of a 96 -foot pine tree, then how long does it take to reach the earth?

Step-by-Step Solution

Answer

The pine cone takes 3 seconds to reach the earth.

Step 1 - Set up the equation

Use the formula for the height of an object in free fall: \[ h(t) = h_0 + v_0 t - \frac{1}{2} g t^2 \].Here, \(h_0 = 96\) feet (initial height), \(v_0 = 16\) feet/sec (initial velocity), and \(g = -32\) feet/sec² (acceleration due to gravity). The pine cone will hit the ground when \(h(t) = 0\). So, set up the equation: \[ 0 = 96 + 16t - 16t^2. \]

Step 2 - Simplify the equation

Simplify the quadratic equation by combining like terms and factoring if necessary: \[ 0 = 96 + 16t - 16t^2. \].Rearrange to the standard form: \[ 16t^2 - 16t - 96 = 0. \].Divide the entire equation by 16: \[ t^2 - t - 6 = 0. \].

Step 3 - Solve the quadratic equation

Solve the quadratic equation \( t^2 - t - 6 = 0 \) by factoring:\[ (t - 3)(t + 2) = 0. \]So, the potential solutions are: \[ t - 3 = 0 \implies t = 3 \].\[ t + 2 = 0 \implies t = -2 \].Since time cannot be negative, the physical solution is \( t = 3 \) seconds.

Key Concepts

Free Fall Motion

Free fall motion occurs when an object is moving only under the influence of gravity. When a pine cone is projected upward from the top of a tree, its motion can be described using a specific height equation. This equation accounts for the initial height, initial velocity, and gravitational acceleration.

Factoring Quadratic Equations

Factoring quadratic equations is a method used to find the roots of the equation. In this exercise, we started with the quadratic equation in standard form \(16t^2 - 16t - 96 = 0\). By dividing every term by 16, we simplified the equation to \(t^2 - t - 6 = 0\). We then factored this into two binomial expressions: \((t - 3)(t + 2) = 0\). Solving these gives us the potential solutions for \(t\).

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. The height of the pine cone over time is given by \[ h(t) = h_0 + v_0 t - \frac{1}{2} g t^2 \], where * \(h_0 \) is the initial height (96 feet) * \(v_0\) is the initial velocity (16 feet/sec) * \(g\) is the acceleration due to gravity (32 feet/sec²). This equation helped us set up a quadratic equation to find the time (\(t\)) it takes for the pine cone to hit the ground.