Problem 31 - Algebra 2

Question

Rockers Two rockets are launched at the same time, but from different heights. The height \(y\) in feet of one recket after \(t\) seconds is given by \(y=-16 t^{2}+150 t+5 .\) The height of the other rocket is given by \(y=-16 t^{2}+\) 160\(t\) . After how many seconds are the rockets at the same height?

Step-by-Step Solution

Answer

The rockets are at the same height after 0.5 seconds.

Step 1: Set the Equations Equal

To find when both rockets are at the same height, set their height equations equal to each other:\[-16t^2 + 150t + 5 = -16t^2 + 160t\]

Step 2: Simplify the Equation

Cancel the \(-16t^2\) terms on both sides:\[150t + 5 = 160t\]Next, subtract \(150t\) from both sides:\[5 = 10t\]

Step 3: Solve for t

Now, divide both sides of the equation by 10 to find \(t\):\[t = \frac{5}{10} = 0.5\]

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in various mathematical and real-world applications. A typical quadratic equation is of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The equation features a degree of two, making the quadratic equation easily identifiable by its highest exponent.

To solve quadratic equations, we often look for values of \( x \) that satisfy the equation. These solutions can be found through various methods:

• Factoring: Expressing the quadratic as a product of two binomial expressions, if possible.
• Quadratic Formula: Using \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), which works for any quadratic equation.
• Completing the Square: Rewriting the equation in perfect square trinomial form.

Quadratic equations are important for modeling curved trajectories, making them essential for studying the heights of projectiles like rockets.

Heights of Projectiles

When we discuss the heights of projectiles, we often use quadratic equations to determine the vertical position over time. This is due to the influence of gravity on the projectile's motion. In the context of rockets or other objects launched into the air, the equation representing their height frequently includes terms for time squared, linear time, and an initial height.

The height function typically takes the form \( y = -16t^2 + vt + h_0 \), where:

• \( t \) represents time in seconds.
• \( v \) is the initial vertical velocity in feet per second.
• \( h_0 \) is the initial height from which the projectile is launched.
• The term \( -16t^2 \) accounts for the effect of gravity.

The negative coefficient in front of \( t^2 \) indicates that the object will eventually descend due to gravity. By analyzing these equations, we can predict when two projectiles, starting from different heights, will equalize in height. Keeping track of these factors allows us to solve complex problems involving timing and height in projectile motion.

Algebraic Manipulation

Algebraic manipulation is a key skill in solving equations, especially when dealing with systems of equations like those involving multiple projectiles. It involves rearranging and simplifying equations to isolate the variables of interest.

For instance, in the given exercise, both projectile equations were set equal to each other to find when the heights match: \[-16t^2 + 150t + 5 = -16t^2 + 160t\]After equating the equations, similar terms on both sides such as \(-16t^2\) could be cancelled out to simplify the expression, leading to:\[150t + 5 = 160t\]From this step, subtracting \(150t\) from both sides further reduces the equation:\[5 = 10t\]Finally, dividing both sides by 10 isolates \(t\):\[t = 0.5\]
Performing algebraic manipulation allows you to systematically simplify equations and solve for unknowns efficiently. This makes it easier to find when two objects are at the same height during their flight when analyzing projectile motion.