Problem 56 - College Algebra

Question

A falling object travels a distance given by the formula \(d=5 t+16 t^{2} \mathrm{ft}\) , where \(t\) is measured in seconds. How long will it take for the object to traveled 74 \(\mathrm{ft}\) ?

Step-by-Step Solution

Answer

The object will take 2 seconds to travel 74 ft.

Step 1: Identify the formula and known values

The distance formula given is \(d = 5t + 16t^2\). We know that \(d = 74\) ft since the object needs to travel 74 feet. Our goal is to find the time \(t\) in seconds.

Step 2: Set up the equation

Using the known values, set up the equation \(74 = 5t + 16t^2\). This is a quadratic equation in the form \(ax^2 + bx + c = 0\), with \(a = 16\), \(b = 5\), and \(c = -74\).

Step 3: Use the quadratic formula

The solution to a quadratic equation \(ax^2 + bx + c = 0\) is given by the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 16\), \(b = 5\), and \(c = -74\) into this formula.

Step 4: Calculate the discriminant

Compute the discriminant, \(b^2 - 4ac\): \(5^2 - 4 \times 16 \times (-74) = 25 + 4736 = 4761\).

Step 5: Solve for time using the quadratic formula

Substitute the discriminant back into the quadratic formula: \(t = \frac{-5 \pm \sqrt{4761}}{32}\). Since \(\sqrt{4761} = 69\), this becomes \(t = \frac{-5 \pm 69}{32}\).

Step 6: Find the possible values for t

Calculate the two possible solutions for \(t\):1. \(t = \frac{-5 + 69}{32}\) which simplifies to \(t = \frac{64}{32} = 2\).2. \(t = \frac{-5 - 69}{32}\) which simplifies to \(t = \frac{-74}{32} = -2.3125\).

Step 7: Choose the valid solution

Since time cannot be negative, the valid solution is \(t = 2\) seconds.

Key Concepts

Understanding the Distance Formula

The distance formula used in this exercise helps us determine how far an object travels over time under certain conditions. Given by the equation \( d = 5t + 16t^2 \), it combines linear and quadratic terms to model the motion of a falling object. Here:

• \( d \) is the distance the object travels, measured in feet.
• \( t \) represents the time in seconds.

The formula captures the effect of initial velocity and acceleration due to gravity. The linear term \( 5t \) handles initial conditions, and the quadratic term \( 16t^2 \) represents the increase in velocity as the object accelerates downward. In our problem, we know the object has traveled 74 feet, so \( d = 74 \). The goal is to find \( t \), the time it takes to reach this distance.

Applying the Quadratic Formula

Quadratic equations like \( 74 = 5t + 16t^2 \) are common in physics to represent changing quantities. These equations take the form \( ax^2 + bx + c = 0 \). For this problem:

• \( a = 16 \)
• \( b = 5 \)
• \( c = -74 \)

To solve for \( t \), we use the quadratic formula:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]It provides the solutions for \( t \) by calculating the expression under the square root, known as the discriminant, and determining possible values for \( t \). This formula is invaluable for solving quadratic equations, letting us find solutions even when they involve complex numbers or scenarios.

Exploring the Discriminant

The discriminant \( b^2 - 4ac \) is a key part of the quadratic formula. It helps determine the nature of the roots we can expect from the equation. For our case:

• The given values were \( a = 16 \), \( b = 5 \), \( c = -74 \).
• So, the discriminant calculation is \( 5^2 - 4 \times 16 \times (-74) = 4761 \).

A positive discriminant, like our result of 4761, indicates there are two distinct real roots. This implies two possible solutions for \( t \), although not all may be feasible in context—negative time, for instance, won't make sense here. Understanding the discriminant lets us check how many solutions are expected and what kinds they will be—real or imaginary.

Finding the Time of Travel

Finally, with the calculated discriminant and quadratic formula, we solve for time, \( t \). Substituting the values:

• The discriminant \( \sqrt{4761} = 69 \).
• Then, \( t = \frac{-5 \pm 69}{32} \).

Calculating these expressions gives two potential values for \( t \):

• \( t = \frac{-5 + 69}{32} = 2 \).
• \( t = \frac{-5 - 69}{32} = -2.3125 \).

Since time cannot be negative in this physical context, the meaningful solution is \( t = 2 \) seconds. So, it takes 2 seconds for the object to travel 74 feet. This method of discerning the plausible answer highlights the practical reasoning involved in interpreting mathematical solutions.