Problem 69
Question
A football is kicked off the ground with an initial upward velocity of 48 feet per second. The football's height \(h\) (in feet) is given by \(h(t)=-16 t^{2}+48 t, \quad 0 \leq t \leq 3\) where \(t\) is the time (in seconds). Does the football reach a height of 50 feet? Explain.
Step-by-Step Solution
Verified Answer
No, the football does not reach a height of 50 feet. The solutions \(t_1\) and \(t_2\) that would have allowed the football to reach 50 feet do not fall within the boundaries of the given time interval \(0 ≤ t ≤ 3\).
1Step 1: Substitute height into equation
Firstly set the height h(t) equal to 50 feet. The equation to solve becomes -16t^2 + 48t = 50.
2Step 2: Rearrange equation and solve for t
Next, rearrange the equation by subtracting 50 from both sides, which would render the equation: -16t^2 + 48t - 50 = 0. Now, divide this equation by -2 to simplify it, which would result to 8t^2 - 24t + 25 = 0. Solve this quadratic equation for t.
3Step 3: Apply Quadratic Formula
The quadratic formula \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) will be used to find the values of t. In the equation \(8t^2 - 24t + 25 = 0\), \(a = 8, b = -24\), and \(c = 25\). Substituting these values into the formula yields 2 possible time points, \(t_1, t_2\).
4Step 4: Examine Result
Compare the time points \(t_1\) and \(t_2\) to the acceptable range of \(0 ≤ t ≤ 3\). If these points fall within the range, then the football does reach a height of 50 feet within 3 seconds.
Key Concepts
Solving Quadratic EquationsProjectile MotionQuadratic Formula
Solving Quadratic Equations
Understanding how to solve quadratic equations is essential for tackling a variety of problems, not only in mathematics but also in fields such as physics. A quadratic equation is a second-order polynomial equation that can be expressed in the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.
In the context of the exercise provided, the quadratic equation representing the height of the football as a function of time is \( -16t^2 + 48t - 50 = 0 \) after simplification. To solve this equation, one might use factoring, completing the square, or employing the quadratic formula, which is a reliable method that works for any quadratic equation.
Solving quadratic equations often involves finding the roots, or solutions, which represent the values of the variable where the polynomial is equal to zero. These solutions can be real or complex numbers and are crucial for predicting outcomes in real-world scenarios, such as determining the time at which a football reaches a certain height.
In the context of the exercise provided, the quadratic equation representing the height of the football as a function of time is \( -16t^2 + 48t - 50 = 0 \) after simplification. To solve this equation, one might use factoring, completing the square, or employing the quadratic formula, which is a reliable method that works for any quadratic equation.
Solving quadratic equations often involves finding the roots, or solutions, which represent the values of the variable where the polynomial is equal to zero. These solutions can be real or complex numbers and are crucial for predicting outcomes in real-world scenarios, such as determining the time at which a football reaches a certain height.
Projectile Motion
Projectile motion is a form of motion experienced by an object or particle that is thrown near the Earth's surface and moves along a curved path under the action of gravity alone. Assuming air resistance is negligible, the only force acting on the object is gravity, which imparts a constant acceleration downward.
In our example, the football that is kicked upward follows a parabolic trajectory, which is characteristic of projectile motion. The mathematics behind this motion can be described by two separate functions: one for horizontal displacement and another for vertical displacement over time. For vertical displacement, which concerns us here, the equation derived is quadratic in nature (\( h(t) = -16t^2 + 48t \)), reflecting the parabolic path of the projectile.
Understanding projectile motion in physics is integral to predicting the maximum height a projectile reaches, or the time it takes to hit the ground. By solving the associated quadratic equation, one can glean important information about the object's trajectory, such as its peak height or flight duration, pivotal concepts in fields ranging from sports to aerospace engineering.
In our example, the football that is kicked upward follows a parabolic trajectory, which is characteristic of projectile motion. The mathematics behind this motion can be described by two separate functions: one for horizontal displacement and another for vertical displacement over time. For vertical displacement, which concerns us here, the equation derived is quadratic in nature (\( h(t) = -16t^2 + 48t \)), reflecting the parabolic path of the projectile.
Understanding projectile motion in physics is integral to predicting the maximum height a projectile reaches, or the time it takes to hit the ground. By solving the associated quadratic equation, one can glean important information about the object's trajectory, such as its peak height or flight duration, pivotal concepts in fields ranging from sports to aerospace engineering.
Quadratic Formula
The quadratic formula, \( t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \), is a powerful tool in algebra for finding the solutions to quadratic equations. Once an equation is in the form \( ax^2 + bx + c = 0 \), the values of \( a \), \( b \), and \( c \) can be substituted into the formula to find the roots.
Applied to our football example, when the equation \( 8t^2 - 24t + 25 = 0 \) is generated after simplifying the original equation, we can use the values of \( a = 8 \), \( b = -24 \), and \( c = 25 \) in the quadratic formula to find the possible times \( t \) at which the football can reach the height of 50 feet. This method guarantees that we do not overlook any solutions. The discriminant part of the formula, \( b^2 - 4ac \), can also tell us about the nature of the roots—whether they're real and distinct, real and identical, or complex.
The quadratic formula is not only a staple in the mathematics curriculum, but it is also a practical tool that provides a clear pathway to solving problems that can be modeled by quadratic equations in various scientific disciplines.
Applied to our football example, when the equation \( 8t^2 - 24t + 25 = 0 \) is generated after simplifying the original equation, we can use the values of \( a = 8 \), \( b = -24 \), and \( c = 25 \) in the quadratic formula to find the possible times \( t \) at which the football can reach the height of 50 feet. This method guarantees that we do not overlook any solutions. The discriminant part of the formula, \( b^2 - 4ac \), can also tell us about the nature of the roots—whether they're real and distinct, real and identical, or complex.
The quadratic formula is not only a staple in the mathematics curriculum, but it is also a practical tool that provides a clear pathway to solving problems that can be modeled by quadratic equations in various scientific disciplines.
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Problem 69
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