Problem 157 - Elementary Algebra

Question

A firework rocket is shot upward at a rate of \(640 \mathrm{ft} / \mathrm{sec}\). Use the \(\quad\) projectile formula \(h=-16 t^{2}+v_{0} t\) to determine when the height of the firework rocket will be 1200 feet.

Step-by-Step Solution

Answer

1.97 seconds and 38.03 seconds

Step 1: Understand the Given Formula and Variables

The projectile formula is given by: \[ h = -16t^2 + v_0 t \]where:- \(h\) is the height of the object (in feet)- \(t\) is the time (in seconds)- \(v_0\) is the initial velocity (in feet per second)Given: \( h = 1200 \) feet and \( v_0 = 640 \) feet/second.

Step 2: Set Up the Equation

Substitute the given values into the formula:\[ 1200 = -16t^2 + 640t \]

Step 3: Move All Terms to One Side

Rearrange the equation to set it to zero:\[ -16t^2 + 640t - 1200 = 0 \]

Step 4: Simplify the Equation

Divide all terms by -16 to make the equation simpler:\[ t^2 - 40t + 75 = 0 \]

Step 5: Solve the Quadratic Equation

Solve the quadratic equation using the quadratic formula:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this equation, \(a = 1\), \(b = -40\), and \(c = 75\).

Step 6: Calculate the Discriminant

Calculate the discriminant \[ b^2 - 4ac = (-40)^2 - 4(1)(75) = 1600 - 300 = 1300 \]

Step 7: Find the Roots

Substitute the values back into the quadratic formula to find the roots:\[ t = \frac{40 \pm \sqrt{1300}}{2} \approx \frac{40 \pm 36.06}{2} \]This gives two solutions:\[ t_1 = \frac{40 + 36.06}{2} = 38.03 \]\[ t_2 = \frac{40 - 36.06}{2} = 1.97 \]

Step 8: Interpret the Results

The solutions for \(t\) represent the times at which the firework rocket is at 1200 feet. These times are approximately 1.97 seconds and 38.03 seconds after launch.

Key Concepts

quadratic equation

In solving projectile motion problems, quadratic equations appear often. A quadratic equation is of the form discriminant The quadratic formula, helps solve for the variable, typically time (t). Be familiar with the form Quadratic equations usually yield two solutions for given real-world problems.

initial velocity

Initial velocity (Planned projectile problems like the firework exercise must know this value. Knowing this leads to correct mathematical models.Initial velocity (v_0) = 640 feet/second as per the problem.

height calculation

In projectile motion problems, knowing height (h) calculations is crucial. Formulas like Use the polynomial form (examples differ), find heights at given times. The process often involves substitution and algebraic manipulation.For instance, solving when We set up: Solving gives us heights at times.

discriminant

A discriminant in quadratic equations reveals solution count and nature. Calculate using: discriminant Three cases arise:positive value = two distinct real rootszero = one real rootnegative = no real roots, irrational solutionsdiscriminant = (-40)^2 - 4(1)(75),=1600-300,=1300.Given value=2 real roots confirming times calculated (1.97 sec,38.03 sec).