Solving For Time When Velocity Equals 30 If V(t) = 5t/4
Have you ever encountered a problem where you're given a velocity function and asked to find the time at which the velocity reaches a certain value? It's a common scenario in physics and calculus, and in this article, we're going to break down one such problem step by step. We'll tackle the question: "If v(t) = 5t/4, what is the value of t when v(t) = 30?" in a way that's easy to understand, even if you're not a math whiz. So, let's dive in and learn how to solve this type of problem!
Understanding the Problem
Before we jump into the solution, let's make sure we understand what the problem is asking. We're given a function, v(t) = 5t/4, which represents the velocity of an object at time t. Velocity, in simple terms, is the rate at which an object is moving. The function tells us how the velocity changes as time (t) changes. Our goal is to find the specific time (t) when the velocity v(t) is equal to 30. In other words, we want to know when the object is moving at a rate of 30 units per time unit. This is a classic problem that combines algebra and the basic concepts of motion, making it a fundamental skill to grasp in both mathematics and physics.
Breaking Down the Velocity Function
To really get a handle on this, let's dissect the velocity function, v(t) = 5t/4. This equation is a linear function, meaning it represents a straight line when graphed. The variable 't' stands for time, which is our independent variable – the one we're trying to solve for. The term '5t/4' tells us how the velocity changes with time. Specifically, it says that the velocity is equal to 5/4 times the time. This fraction, 5/4, is the slope of the line, and it indicates that for every 1 unit increase in time, the velocity increases by 5/4 units. Understanding the components of the function is crucial because it allows us to visualize the relationship between velocity and time. Imagine a car accelerating; its velocity increases over time, and this equation is a simplified way to represent that increase. So, now that we've got a clear picture of what the function means, we're ready to move on to the next step: setting up the equation to solve for 't'.
Setting Up the Equation
The heart of solving this problem lies in setting up the equation correctly. We know that v(t) represents the velocity at time t, and we want to find the time when v(t) is equal to 30. So, we simply replace v(t) in the equation with 30. This gives us the equation: 30 = 5t/4. This equation is the key to unlocking our answer. It states that 30 is the value of the velocity when the time is t. Our next step is to isolate 't' on one side of the equation. This involves using algebraic manipulation, which is a fancy way of saying we're going to do some math to get 't' by itself. Setting up the equation is like laying the foundation for a building; it's the crucial first step that determines the success of the entire solution. Without a solid equation, we'd be wandering in the mathematical wilderness. So, now that we have our equation, 30 = 5t/4, we're ready to roll up our sleeves and start solving for 't'.
Solving for t
Alright, guys, now comes the fun part – solving for 't'! We've got our equation, 30 = 5t/4, and we need to isolate 't'. To do this, we'll use a couple of algebraic tricks that are super useful in math problems like this. The first thing we want to do is get rid of that fraction.
Step 1: Multiplying Both Sides by 4
Fractions can sometimes make equations look scary, but they're really not that bad. To eliminate the fraction in our equation, we'll multiply both sides of the equation by 4. Remember, whatever you do to one side of an equation, you have to do to the other side to keep things balanced. So, we multiply both 30 and 5t/4 by 4. This gives us: 4 * 30 = 4 * (5t/4). On the left side, 4 times 30 is simply 120. On the right side, the 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with 5t. So, our equation now looks like this: 120 = 5t. See how much simpler it looks without the fraction? We're one step closer to finding the value of t. Multiplying by 4 was like using a mathematical vacuum cleaner to suck away the fraction, making our equation much cleaner and easier to handle. Now, let's move on to the next step and finish the job.
Step 2: Dividing Both Sides by 5
We're almost there! We've got the equation 120 = 5t, and we just need to get 't' all by itself. Right now, 't' is being multiplied by 5. To undo that multiplication, we'll do the opposite operation: division. We'll divide both sides of the equation by 5. This gives us: 120 / 5 = (5t) / 5. On the left side, 120 divided by 5 is 24. On the right side, the 5 in the numerator and the 5 in the denominator cancel each other out, leaving us with just 't'. So, our equation now looks like this: 24 = t. That's it! We've solved for t. Dividing by 5 was like using a mathematical sledgehammer to break the bond between 5 and 't', leaving 't' isolated and revealed. We've successfully navigated the algebraic maze and found the value of 't'.
The Solution: t = 24
So, after all that math, we've arrived at our answer: t = 24. But what does this mean in the context of our original problem? Remember, we were trying to find the time (t) when the velocity v(t) is equal to 30. Our solution, t = 24, tells us that the velocity will be 30 units per time unit when the time is 24 units. It's like finding the exact moment in a race when a car hits a certain speed. In this case, the car (or whatever object we're talking about) reaches a velocity of 30 at time 24. Understanding the meaning of the solution is just as important as the math itself. It's what connects the abstract symbols and equations to the real world. So, we've not only found the value of 't', but we've also understood what that value represents. That's a mathematical victory! Now, let's wrap things up and recap what we've learned.
Verifying the Solution
Before we declare victory, it's always a good idea to double-check our work. We can verify our solution by plugging t = 24 back into the original equation, v(t) = 5t/4, and seeing if we get v(t) = 30. So, let's do it! If we substitute t = 24 into the equation, we get: v(24) = 5 * 24 / 4. Now, let's simplify: 5 * 24 is 120, so we have v(24) = 120 / 4. Finally, 120 divided by 4 is 30. So, v(24) = 30. This confirms that our solution, t = 24, is correct. We've successfully verified our answer, which is like getting a gold star on our mathematical homework. This step is crucial because it gives us confidence that we haven't made any mistakes along the way. Verifying the solution is like being a mathematical detective, making sure all the clues add up and the case is closed.
Conclusion
Alright, guys, we've reached the end of our mathematical journey! We successfully solved the problem: "If v(t) = 5t/4, what is the value of t when v(t) = 30?" We found that the value of t is 24. We started by understanding the problem, setting up the equation, solving for 't' using algebraic manipulation, and finally, verifying our solution. This problem is a great example of how math can be used to describe real-world situations, like the motion of an object. The key takeaways here are the importance of understanding the problem, setting up the equation correctly, and using algebraic techniques to isolate the variable you're trying to solve for. Remember, math isn't just about numbers and symbols; it's about problem-solving and critical thinking. So, keep practicing, keep exploring, and keep challenging yourself with new mathematical adventures! You've got this!
By mastering these types of problems, you're not just learning math; you're developing valuable skills that can be applied in many different areas of life. So, congratulations on making it to the end, and keep up the great work!