Soccer Ball Trajectory: Finding The Initial Height

Hey guys! Let's dive into a classic math problem involving a soccer ball's journey through the air. We've got a scenario where a soccer ball gets kicked from a hill, so its movement is described by a specific equation. Our mission? To figure out the ball's initial height. Don't worry, it's not as tricky as it sounds! We'll break it down step by step, using the given function to find the answer. Get ready to put on your thinking caps and let's explore this problem together.

Understanding the Problem: The Ball's Flight

Okay, so the problem sets the scene: a soccer ball is kicked upwards from a hill, and then it eventually comes back down to the ground. The height of the ball at any given time ($t$ seconds) is described by the equation: $h(t) = -16t^2 + 26t + 12$. Now, this equation is a mathematical model that tries to capture the ball's trajectory. It’s a quadratic equation, which means it creates a parabola when graphed. The height, $h(t)$, is measured in feet. The key here is to understand what each part of this equation represents. The $-16t^2$ part relates to the effect of gravity (pulling the ball down), the $26t$ part relates to the initial upward velocity, and the $+12$ is the constant term which is crucial to find the initial height, which is what we are looking for. We will use the formula to find the answer to the question.

Now, what does "initial height" actually mean? Well, it's the height of the ball at the very beginning of its journey – the moment the ball is kicked. In mathematical terms, this is when time ($t$) is equal to zero. So, to find the initial height, all we need to do is figure out what $h(t)$ is when $t = 0$. This is a straightforward concept, but it's essential to grasp it before moving on. By finding the height at the start, we understand where our ball starts its flight.

In essence, we're not trying to find the ball's maximum height or when it hits the ground. We are focused on that instant when the ball leaves the kicker's foot. This is the starting point, and it's represented by the $y$-intercept of the parabola. We can use the information given, the equation $h(t)=-16t^2 + 26t + 12$, to guide our decision and solve the question with the correct answer. The initial height is super important because it tells us where the ball begins its journey. Think of it like the starting line of a race; it sets the stage for everything that follows. Understanding this will make the rest of the problem way easier, trust me!

Solving for the Initial Height: A Simple Calculation

Alright, time to get our hands dirty and actually solve for the initial height. Remember, the initial height is the height of the ball at $t = 0$ seconds. So, let’s plug that value into our equation. Our equation is $h(t) = -16t^2 + 26t + 12$. Substitute $t$ with $0$ and we'll get $h(0) = -16(0)^2 + 26(0) + 12$. This means wherever we see 't' in the equation, we swap it out for a '0'.

Now, let's break down this calculation. First, we have $-16(0)^2$. Zero squared is still zero, and any number multiplied by zero is zero. So, this term becomes 0. Next, we have $26(0)$. Again, any number multiplied by zero is zero, so this term also becomes 0. Finally, we are left with $+12$. Therefore, $h(0) = 0 + 0 + 12$, which simplifies to $h(0) = 12$.

This simple calculation tells us that the initial height of the ball is 12 feet. So, when the soccer ball is first kicked, it starts at a height of 12 feet above the ground. This means the hill is 12 feet above the ground. The constant term in the original equation directly gives us the initial height. By substituting $t = 0$, the other terms disappear, leaving us with the constant. Pretty neat, right? The initial height represents where the ball's journey begins, so it's a vital piece of information. That is why it is very easy to find by making $t=0$, as all the terms with $t$ are canceled out and we just take the constant, which in this case is 12.

Analyzing the Answer Choices: Finding the Right Match

Now that we've calculated the initial height, let's go back to our answer choices and find the one that matches our solution. We found that the initial height is 12 feet. Let's look at the multiple-choice options:

A. $-16$ ft B. $-12$ ft C. $12$ ft D. $22$ ft

The correct answer is C. 12 ft. This option directly reflects the initial height we calculated by substituting $t = 0$ into the equation. It's super important to match the solution we've found to the provided options. The other options, A, B, and D, represent different values, and they aren't the initial height of the ball. The answer C is the one that we found out by substituting $t=0$ in the equation and finding the constant. So, the correct answer is C.

Keep in mind that understanding what each part of the equation means is key. The negative sign in front of the $16t^2$ is important because it tells us that the parabola opens downward, and this is because of gravity pulling the ball back towards the ground. The term $26t$ reflects the initial upward velocity, which determines how high the ball goes. And, of course, the constant of $+12$ tells us the starting height. By breaking down the problem this way, it is easy to pick the correct choice. Great work, guys! You have successfully solved the problem and chosen the right answer. Yay!

Conclusion: Recap and Key Takeaways

Okay, let's wrap things up and recap what we've learned. We started with a word problem about a soccer ball's trajectory described by a quadratic equation. Our goal was to determine the initial height of the ball, which is the height at the moment it’s kicked. We did this by understanding that the initial height occurs when time, $t$, is equal to zero. Plugging $t = 0$ into the equation $h(t) = -16t^2 + 26t + 12$ gave us $h(0) = 12$ feet. Thus, the correct answer is C. 12 ft.

This problem highlights the importance of understanding the concepts behind the math, not just memorizing formulas. By knowing what each part of the equation represents, we can easily solve the problem. The constant term of a quadratic equation represents the initial height (the y-intercept), which is a valuable shortcut. Remember that the initial height is the starting point, the beginning of the ball’s journey. By finding this initial point, we get a solid understanding of the ball's flight path. This knowledge is not only helpful for solving math problems but also for understanding real-world scenarios, like how objects move under the influence of gravity.

So next time you see a problem like this, remember this step-by-step approach. Break down the problem, identify the key parts, use your knowledge of the concepts, and always double-check your work. You've now got the skills to tackle this kind of problem with confidence. Keep practicing and keep up the great work, everyone! And most importantly, keep enjoying the beautiful game of soccer and the mathematics behind it! Have a good one! Keep in mind that math can be fun and exciting, all you have to do is approach it with a positive attitude!