Problem 55 - Beginning and Intermediate Algebra

Question

A model rocket is projected vertically upward from the ground. Its distance \(s\) in feet above the ground after t seconds is given by the quadratic function $$ s(t)=-16 t^{2}+256 t $$ Work Exercise in order, to see how quadratic equations and inequalities are related. At what times will the rocket be 624 ft above the ground? (Hint: Let \(s(t)=624\) and solve the quadratic equation.)

Step-by-Step Solution

Answer

The rocket will be 624 feet above the ground at 3 seconds and 13 seconds.

Step 1: Identify the Given Information

The distance function is given as: \[s(t) = -16t^2 + 256t\]. We need to find the times when the rocket is 624 feet above the ground. Set \(s(t) = 624\).

Step 2: Set Up the Equation

Substitute 624 into the distance function: \[624 = -16t^2 + 256t\].

Step 3: Rewrite the Equation

Rewrite the equation in standard quadratic form: \[-16t^2 + 256t - 624 = 0\].

Step 4: Simplify the Equation

Divide the entire equation by -16 to simplify: \[t^2 - 16t + 39 = 0\].

Step 5: Solve the Quadratic Equation

To solve \(t^2 - 16t + 39 = 0\), use the quadratic formula: \[t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here, \(a = 1\), \(b = -16\), and \(c = 39\).

Step 6: Calculate the Discriminant

Calculate the discriminant: \[b^2 - 4ac = (-16)^2 - 4(1)(39) \] \[= 256 - 156 = 100\].

Step 7: Find the Solutions

Substitute the discriminant into the quadratic formula: \[t = \frac{16 \pm \sqrt{100}}{2} \] \[= \frac{16 \pm 10}{2}\]. This gives two solutions: \[t = \frac{16 + 10}{2} = 13\] and \[t = \frac{16 - 10}{2} = 3\].

Key Concepts

quadratic functions

Quadratic functions are a type of polynomial function that are characterized by their degree, which is two. This means the highest power of the variable is squared. The general form of a quadratic function is: For example, let's look at the function given in our exercise: This is a quadratic function where: Quadratic functions graph as parabolas. Their U-shaped curves can open upwards or downwards, depending on the sign of the coefficient in front of the squared term. In our function, the parabola opens downwards because the coefficient in front of the squared term is negative.

projectile motion

Projectile motion is a type of motion experienced by an object that is launched into the air and affected only by gravity and its initial thrust. The path of this motion is typically linear in horizontal direction but parabolic vertically. For the model rocket in our problem, its height above the ground over time is described by the quadratic function: This means at any time, the height can be calculated using the quadratic function.

quadratic formula

The quadratic formula is a tool used to find the solutions of a quadratic equation. The standard form of a quadratic equation is , and it can be solved using the quadratic formula: By substituting the values of , and into the formula, you can find the solutions of the quadratic equation. In our exercise, the equation to solve was . By applying the quadratic formula: Therefore, the solutions are .

discriminant

The discriminant is a crucial part of the quadratic formula that helps determine the nature of the roots (solutions) of a quadratic equation. The discriminant is the expression under the square root in the quadratic formula: The value of the discriminant tells us: For our equation , the discriminant is computed as: Since the discriminant is positive, we have two real and distinct solutions: .