Math Problem: Calculating Inclined Plane Height

Hey guys! Let's dive into a cool geometry problem related to a real-world application. We're looking at a piece designed to help people dealing with acid reflux. It's basically a wedge placed at the head of the bed, so you can rest at an incline. The question is, how high is this inclined part? Let's break it down step-by-step.

Understanding the Problem: Inclined Plane and Trigonometry

Alright, so we have this wedge, right? It's designed to elevate the upper body and ease reflux symptoms. The diagram gives us a 30-degree angle and a hypotenuse length. The key here is understanding that we're dealing with a right-angled triangle. The inclined part of the piece forms the hypotenuse of our triangle, and the height we want to find is one of the legs (specifically, the side opposite the 30-degree angle). This brings trigonometry into play – specifically, the sine function. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, we know the angle (30 degrees) and the hypotenuse length, and we want to find the opposite side (the height). So, we use the sine function to relate these values. This is where the math magic happens, so pay close attention! This type of problem is super common, and understanding the basics of right triangles and trigonometric functions is crucial for many different areas, from construction to physics. That 30-degree angle is our guide. When you see these types of problems, make sure to take the time to draw a good picture, then label the sides, and identify the known and unknown variables. It is really important to be able to visualize the problem, and sometimes a simple sketch will give you a big hand.

Now, let's look at the givens. The piece has a 30-degree angle and a hypotenuse with a certain length (which we will work out). The question is asking us to find the height, which is the side opposite the 30-degree angle. By using the sine function we'll easily determine the height of the inclined plane. This understanding of the problem setup is half the battle. Once you know what you're looking for and which tools to use, the math becomes more straightforward. Being able to break down a complex problem into smaller, manageable steps is a valuable skill, not just in math, but in life in general.

Setting Up the Equation: Using the Sine Function

So, we know we have a right triangle with a 30-degree angle, and we're given a certain length. In a right triangle, the sine of an angle (let's call it θ, pronounced 'theta') is equal to the length of the side opposite the angle divided by the length of the hypotenuse. So, if we let 'h' represent the height we want to find (the side opposite the 30-degree angle) and 'hyp' represent the length of the hypotenuse. We can write the equation like this: sin(θ) = h / hyp. To find the height (h), we need to rearrange this equation to solve for h: h = hyp * sin(θ). Now, we need to calculate the sine of 30 degrees. The sine of 30 degrees is a well-known value: 0.5. This is something you might memorize, or look up in a table of trigonometric values, but you'll encounter it a lot, so it is good to learn! Next, we will multiply the hypotenuse length by the sine of 30 degrees (which is 0.5). Let's assume the hypotenuse length is 30 cm (the picture does not state a number, but we can use one for the purpose of solving the problem). So: h = 30 * 0.5. Calculating this value will give us the height of the inclined plane. Pretty simple, right?

Calculation and Solution: Finding the Height

Now, let's get to the actual calculation. We've established that the formula to find the height (h) is: h = hypotenuse * sin(30°). We will use the information we know from the diagram. Let's assume the hypotenuse is 30 cm. We know that sin(30°) = 0.5. So, plugging the values in: h = 30 cm * 0.5. Multiplying 30 by 0.5, we get 15 cm. So, the height of the inclined plane is 15 cm. That means when someone is resting on this piece, the top of the piece is 15 cm higher than the bottom part. This is the answer we have been looking for! Always remember to include the unit in your answer, which in this case is centimeters (cm). This is how you solve the problem, guys! Always remember to double-check your work to make sure that you have the correct answer! Also, it is important that you take the time to understand the problem, draw the graph, and understand all of the givens. This will help you to solve similar problems in the future.

Conclusion: The Answer and Its Significance

So, we've calculated that the height of the inclined plane is 15 cm. In other words, the correct answer from the options provided is 'a) 15 cm'. Awesome, we did it! But what does this actually mean in the context of the reflux treatment? This 15 cm elevation can make a significant difference in reducing reflux symptoms. By raising the head and upper body, gravity helps prevent stomach acid from flowing back into the esophagus. This can lead to a better night's sleep and less discomfort. In terms of the math, we've successfully applied trigonometry to solve a practical problem. We've seen how understanding basic trigonometric functions can help us find unknown lengths and angles in real-world scenarios. The ability to break down problems and apply the right formulas is a valuable skill in both math and life. So, next time you see a problem like this, you'll know exactly what to do! Keep practicing, and you'll become a pro at these kinds of questions in no time! Finally, remember that understanding the concepts is much more important than just getting the right answer. Take the time to reflect on what you have learned.