Problem 71 - Algebra 1

Question

The path of a diver diving from a 10 -foot high diving board is $$h=-0.44 x^{2}+2.61 x+10$$ where \(h\) is the height of the diver above water (in feet) and \(x\) is the horizontal distance (in feet) from the end of the board. How far from the end of the board will the diver enter the water?

Step-by-Step Solution

Answer

The diver's distance from the end of the board when she enters the water using the given condition is the positive solution of the quadratic equation.

Step 1: Formulate the equation

The equation according to the problem is \(h=-0.44x^{2}+2.61x+10\). We are asked to solve for when the diver hits the water i.e. when the height \(h\) is 0. So, the equation becomes \(0=-0.44x^{2}+2.61x+10\).

Step 2: Solve the quadratic equation

Now, we need to solve for \(x\). The quadratic formula is the appropriate tool here which is \(x=[-b ± sqrt(b^{2}-4ac)] / 2a\). In the given equation \(a = -0.44, b = 2.61, c = 10\). We plug these values into the quadratic formula giving us \(x=[-2.61 ± sqrt((2.61)^{2}-4*(-0.44)*10)] / 2*(-0.44)\).

Step 3: Compute the solutions

Calculating these values, we get two solutions for \(x\). The quadratic formula will always give two solutions, one with the positive square root and one with the negative square root. However, because \(x\) represents a distance, only the positive solution is relevant to this problem.

Key Concepts

Parabolic Motion

Parabolic motion is a type of trajectory that objects follow when they are subject to gravity, like the path of a diver jumping off a diving board. This kind of motion can be described mathematically using quadratic equations. A diver, for instance, creates a parabola-shaped path when diving off a high board.
The general form of the equation representing this motion is given by \( h = ax^2 + bx + c \), where \( h \) is the height of the object. In our example, \( x \) is the horizontal distance from the edge of the diving board.

• The initial height of the parabola (where \( x = 0 \)) is marked by the constant \( c \). For the diver, it starts at 10 feet above the water.
• The coefficient \( a \) determines the curvature of the parabola, while \( b \) influences the direction and steepness of the trajectory.

Understanding parabolic motion is crucial when analyzing real-world situations, as it allows us to calculate where and when a falling object—in this case, our diver—will hit the ground or, more aptly, enter the water.

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the roots of quadratic equations, which are of the form \( ax^2 + bx + c = 0 \). It provides a straightforward way to solve for the unknowns in these equations.
The formula is expressed as:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here's how it applies to our diver's equation:

• \( a = -0.44 \), \( b = 2.61 \), and \( c = 10 \).
• We substitute these into the formula to find the values of \( x \), representing potential distances from the board when the diver enters the water.

We solve the equation for \( x \) by plugging in known values, which reveals the potential points where the diver might enter the water. The quadratic formula helps ensure accuracy while factoring in both positive and negative solutions that arise due to the nature of the square root component. This dual solution characteristic is because quadratic equations typically intersect the x-axis at two distinct points.

Solving Equations

When approaching problems like the diver's trajectory, solving quadratic equations becomes a vital step. The goal is to determine when \( h \), the height above water, equals zero, meaning the diver has reached the water surface.
To solve the quadratic equation:

• First, write the equation in its standard form: \( 0 = -0.44x^2 + 2.61x + 10 \).
• Apply the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), to find the values of \( x \).
• After calculations, choose the positive solution, since \( x \) is a distance and must be non-negative.

This process allows you to pinpoint how far from the diving board the diver will enter the water. Solving equations like these is essential not just in academic contexts but is also applicable in various real-life situations involving any type of projectile motion. With practice, the steps become second nature, allowing students to confidently tackle a wide range of mathematical problems and understand the principles behind physical phenomena.