Problem 53 - Algebra 2

Question

DIVING For Exercises \(49-51\) , use the following information. The distance of a diver above the water \(d(t)\) (in feet) \(t\) seconds after diving off a platform is modeled by the equation \(d(t)=-16 t^{2}+8 t+30\) Find the time it would take for the diver to hit the water from this new height.

Step-by-Step Solution

Answer

The diver hits the water at approximately 1.645 seconds after diving.

Step 1: Set the Equation to Zero

To find the time when the diver hits the water, we need to determine when the distance above the water is zero. This is done by setting the equation for distance equal to zero:\[d(t) = -16t^2 + 8t + 30 = 0\]

Step 2: Identify Coefficients for the Quadratic Formula

The equation \(-16t^2 + 8t + 30 = 0\) is a quadratic equation of the form \(at^2 + bt + c = 0\). From this equation, identify the coefficients: \(a = -16\), \(b = 8\), and \(c = 30\).

Step 3: Apply the Quadratic Formula

The quadratic formula to find the roots is given by:\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the coefficients into the formula:\[t = \frac{-8 \pm \sqrt{8^2 - 4(-16)(30)}}{2(-16)}\]

Step 4: Calculate the Discriminant

Compute the discriminant, \(b^2 - 4ac\):\[8^2 - 4(-16)(30) = 64 + 1920 = 1984\]Since the discriminant is positive, there are two real solutions.

Step 5: Solve for Time (t)

Using the quadratic formula with the calculated discriminant, solve for \(t\):\[t = \frac{-8 \pm \sqrt{1984}}{-32}\]Calculating further, we find:\[t = \frac{-8 + 44.55}{-32} \text{ and } t = \frac{-8 - 44.55}{-32}\]This gives approximately:\[t = -1.1 \, \text{seconds (discarded, as time cannot be negative)}\]\[t = 1.645 \, \text{seconds}\]

Step 6: Interpret the Result

Based on the calculation from the quadratic formula, the valid time solution is approximately \(t \approx 1.645\) seconds. This is the time it takes for the diver to reach the water after diving off the platform.

Key Concepts

Discriminant

In the context of quadratic equations, the discriminant is a valuable tool. It helps us determine the nature of the roots without solving the equation entirely. The discriminant is found within the quadratic formula, specifically in the expression under the square root: \( b^2 - 4ac \). When you calculate the discriminant:

• If it's positive, the quadratic equation has two distinct real roots.
• If it's zero, there is exactly one real root, meaning the parabola touches the x-axis at a single point.
• If it's negative, the equation has no real roots, indicating the parabola does not intersect the x-axis.

In our exercise, the discriminant \( 1984 \) was positive, signaling two real solutions for time. This is crucial for determining when the diver hits the water.

Quadratic Formula

The quadratic formula is a universal method to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). This formula is expressed as:\[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]It's a powerful tool because it allows us to solve the equation for any combination of coefficients \( a \), \( b \), and \( c \). To apply it, you substitute the coefficients into the formula. In this example, with \( a = -16 \), \( b = 8 \), and \( c = 30 \), we substituted these values into the quadratic formula to find the time when the diver reaches the water. This method provided the time values based on the diver's motion described by the quadratic equation.

Roots of Equations

The roots of a quadratic equation are the x-values where the graph of the equation crosses the x-axis. In our scenario, these roots represent the point in time when the diver's height becomes zero (as they hit the water). To find the roots:

• Set the quadratic equation \( -16t^2 + 8t + 30 = 0 \) to zero.
• Use the quadratic formula to calculate the possible values for \( t \).

The calculations showed two potential times, \( t = -1.1 \) seconds and \( t = 1.645 \) seconds. However, a negative time doesn't make sense in this physical situation, so we discard it. Thus, the root \( t = 1.645 \) is when the diver hits the water.

Physics Applications in Math

Quadratic equations often model real-world phenomena, particularly in physics. The example here illustrates a classic physics application: a diver's motion. Such equations describe the relationship between time and height when gravity acts on an object. When a diver jumps from a platform, their height over time can be represented by a quadratic equation, factoring in:

• Initial velocity
• Starting height
• The constant acceleration due to gravity

In this exercise, physics and math converge to determine how long the diver is in the air. By solving the quadratic equation, we calculate the precise moment the diver hits the water, demonstrating how quadratic equations can predict outcomes in dynamic real-world contexts.