Problem 89 - 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. Penny tossing. If a penny is thrown downward at \(30 \mathrm{ft} / \mathrm{sec}\) from the bridge at Royal Gorge, Colorado, how long does it take to reach the Arkansas River 1000 ft below?

Step-by-Step Solution

Answer

It takes approximately 7.0 seconds for the penny to reach the river.

Step 1: Define the equation of motion

Use the equation of motion for an object thrown downward under gravity: \[ s = v_0t + \frac{1}{2}gt^2 \]Where: - \( s \) is the distance traveled (1000 feet below)- \( v_0 \) is the initial velocity (30 ft/sec downward)- \( g \) is the acceleration due to gravity (32 ft/sec² on Earth)- \( t \) is the time in seconds

Step 2: Substitute and set up the equation

Substitute the known values into the equation: \[ 1000 = 30t + \frac{1}{2}(32)t^2 \] Simplify this to: \[ 1000 = 30t + 16t^2 \] This can be written in the standard quadratic form: \[ 16t^2 + 30t - 1000 = 0 \]

Step 3: Solve the quadratic equation

Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 16 \), \( b = 30 \), and \( c = -1000 \).Find the discriminant: \[ b^2 - 4ac = 30^2 - 4(16)(-1000) = 900 + 64000 = 64900 \]Evaluate the square root of the discriminant: \[ \sqrt{64900} \approx 254.8 \]

Step 4: Calculate the solutions for time

Substitute back into the quadratic formula to find the time values: \[ t = \frac{-30 \pm 254.8}{32} \]This gives two potential solutions: \[ t = \frac{-30 + 254.8}{32} \approx 7.0 \] and\[ t = \frac{-30 - 254.8}{32} \approx -8.9 \]

Step 5: Interpret the physical solution

Given that time cannot be negative, the only realistic solution is:\[ t \approx 7.0 \] seconds.

Key Concepts

quadratic formula

Quadratic equations can be tricky to solve, but the quadratic formula makes it easier. The quadratic formula is given by: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps you find the roots (solutions) for any quadratic equation of the form: \[ ax^2 + bx + c = 0 \]. Here's how to use the quadratic formula:

• Identify the coefficients \(a\), \(b\), and \(c\).
• Calculate the discriminant \(b^2 - 4ac\).
• Find the square root of the discriminant.
• Substitute the values back into the quadratic formula to solve for \(t\).

In our problem, we have the equation: \[ 16t^2 + 30t - 1000 = 0 \]By recognizing \(a = 16\), \(b = 30\), and \(c = -1000\), we plug these into the quadratic formula and solve for \(t\). Understanding this universal equation can help solve a wide range of quadratic problems.

equation of motion

The equation of motion describes the relationship between an object's displacement, initial velocity, acceleration, and time. For an object thrown downward under gravity, the equation is:\[ s = v_0t + \frac{1}{2}gt^2 \]Here:

• \(s\) is the distance traveled.
• \(v_0\) is the initial velocity.
• \(g\) is the acceleration due to gravity.
• \(t\) is the time.

For our problem, a penny is thrown downward from a bridge, so we use: \[ 1000 = 30t + \frac{1}{2}(32)t^2 \]We substitute the values and solve for \(t\), revealing how long it takes for the penny to hit the river below.

discriminant

When solving quadratic equations, the discriminant is crucial as it determines the nature of the roots. The discriminant \(D\) is part of the quadratic formula under the square root and is given by: \[ D = b^2 - 4ac \] Depending on the value of the discriminant, we can infer the following:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is one real, repeated solution.
• If \(D < 0\), there are no real solutions, but two complex solutions.

In our example, the discriminant is: \[ 30^2 - 4(16)(-1000) = 64900 \]Since \(D\) is greater than zero, this tells us there are two distinct real solutions. However, practical constraints like time being non-negative help refine which solution we consider (7.0 seconds). Understanding the discriminant helps in analyzing the possible solutions for any quadratic equation.