Triangle Area Problem: Find Width & Height

Hey guys! Ever get those math problems that seem like they're speaking another language? Today, we're diving into one about a triangle, its area, width, and height. Don't worry; we'll break it down step by step so it's super easy to follow. Let's get started!

Understanding the Problem

Okay, so the problem states that we have a triangle. We know its area is a comfy 40 square inches. Now, here's the quirky part: the height of this triangle has a special relationship with its width. Specifically, the height is four less than six times the width. Sounds like a mouthful, right? Basically, if you take the width, multiply it by six, and then subtract four, you'll get the height. Our mission, should we choose to accept it (and we do!), is to find out exactly what the width and height of this triangle are. We need to figure out the dimensions that fit both the area and the described relationship between height and width. Think of it as a geometrical puzzle – we have the pieces; we just need to arrange them correctly. To solve this kind of problem effectively, it's vital to have a good grasp of the formula for the area of a triangle. Remember, Area = 1/2 * base * height. In our case, the 'base' is what we're calling the width. Also, translating the word problem into algebraic expressions is key. When they say the height is "four less than six times the width," we can directly translate this into an equation that we can use to substitute into the area formula. Breaking down the problem into smaller, manageable parts makes it way less intimidating and much easier to solve. We’re not just solving for the sake of solving; we’re honing our problem-solving skills, which are super useful in all aspects of life! So, let's put on our detective hats and get ready to crack this triangle code!

Setting Up the Equations

Alright, let's translate this word problem into math we can actually work with. The most crucial step here is to convert the given information into algebraic equations. This is where the magic happens! We know the area of a triangle is calculated by the formula: Area = (1/2) * base * height. In our case, we're calling the base the width (w), so we can rewrite this as Area = (1/2) * w * h. We're given that the area is 40 square inches, so we can substitute that in: 40 = (1/2) * w * h. Now for the tricky part: the relationship between the height and width. The problem tells us that the height (h) is four less than six times the width (w). This translates directly into the equation: h = 6w - 4. See how we turned words into a neat little algebraic expression? Now we have two equations: 40 = (1/2) * w * h and h = 6w - 4. We can use these two equations to solve for our two unknowns, w and h. This is a classic system of equations, and we can use substitution to solve it. We'll substitute the expression for h from the second equation into the first equation. This will give us an equation with only one variable, w, which we can then solve. Setting up these equations correctly is absolutely essential. If we mess up the initial translation, the whole problem goes south. So, double-check that you've accurately represented the problem's conditions in your equations. Remember, precision and attention to detail are key in mathematics. With our equations in place, we're now ready to move on to the next step: solving for the width. This is where the algebra skills really come into play. Let's do it!

Solving for the Width

Okay, now for the fun part: solving for the width! We've got our two equations: 40 = (1/2) * w * h and h = 6w - 4. We're going to use substitution to eliminate one of the variables. Since we already have h expressed in terms of w in the second equation, we'll substitute that into the first equation. So, wherever we see 'h' in the first equation, we'll replace it with '6w - 4'. This gives us: 40 = (1/2) * w * (6w - 4). Now, let's simplify this equation. First, multiply both sides by 2 to get rid of the fraction: 80 = w * (6w - 4). Next, distribute the w: 80 = 6w^2 - 4w. Now we have a quadratic equation! To solve it, we need to set it equal to zero: 6w^2 - 4w - 80 = 0. This looks a bit intimidating, but we can simplify it by dividing the entire equation by 2: 3w^2 - 2w - 40 = 0. Now we can use the quadratic formula to solve for w. Remember the quadratic formula? It's: w = (-b ± √(b^2 - 4ac)) / (2a). In our equation, a = 3, b = -2, and c = -40. Plugging these values into the quadratic formula, we get: w = (2 ± √((-2)^2 - 4 * 3 * -40)) / (2 * 3). Simplifying further: w = (2 ± √(4 + 480)) / 6. w = (2 ± √484) / 6. w = (2 ± 22) / 6. This gives us two possible solutions for w: w = (2 + 22) / 6 = 24 / 6 = 4 and w = (2 - 22) / 6 = -20 / 6 = -10/3. Since the width of a triangle can't be negative, we discard the negative solution. Therefore, the width of the triangle is 4 inches. Woohoo! We found the width. Now that we have the width, we can easily find the height. Let's move on to the next step.

Calculating the Height

Alright, now that we've nailed down the width, finding the height is a piece of cake! We know that the height (h) is related to the width (w) by the equation: h = 6w - 4. We found that the width (w) is 4 inches. So, all we need to do is plug that value into our equation: h = 6 * 4 - 4. Let's do the math: h = 24 - 4. h = 20. So, the height of the triangle is 20 inches. See? That wasn't so bad. Once you have one variable, finding the other one is usually pretty straightforward. Now we have both the width and the height. But before we declare victory, let's double-check our work to make sure our answers make sense. We found that the width is 4 inches and the height is 20 inches. Let's plug these values back into the area formula to see if we get 40 square inches: Area = (1/2) * w * h. Area = (1/2) * 4 * 20. Area = (1/2) * 80. Area = 40. Yep, it checks out! Our answers are consistent with the given information. So, we can confidently say that the width of the triangle is 4 inches and the height is 20 inches. Great job, guys! We successfully solved the problem. This whole process underscores the importance of careful setup, accurate algebra, and thorough checking. You got this!

Final Answer

Okay, drumroll please... After all that calculating and problem-solving, here's the final answer: The width of the triangle is 4 inches, and the height of the triangle is 20 inches. We started with a word problem that seemed a bit complicated, but we broke it down into smaller, manageable steps. We translated the words into algebraic equations, used substitution to solve for the width, and then used the width to find the height. We even double-checked our work to make sure our answers were correct. This problem demonstrates how algebra can be used to solve real-world problems. Triangles are everywhere, from architecture to engineering to design. Understanding how to calculate their area and dimensions is a valuable skill. So, next time you see a triangle, you'll know exactly how to figure out its width and height! Remember, the key to solving math problems is to take your time, read the problem carefully, and break it down into smaller steps. Don't be afraid to ask for help if you get stuck. And most importantly, practice, practice, practice! The more you practice, the better you'll become at problem-solving. And who knows, maybe one day you'll be solving even more complex geometrical problems. But for now, let's celebrate our success in solving this triangle problem. You did it!