Area Of Right Trapezoid Formed By Rods - Calculation Guide

Hey guys! Today, we're diving into a fun geometry problem where we'll calculate the area of a right trapezoid formed by joining four rods of specific lengths. This isn't just about math; it's about understanding how different lengths come together to create shapes and how we can measure their areas. So, let's get started and break down this problem step by step. We'll make sure it's super clear and you'll be a pro at these kinds of calculations in no time!

Understanding the Problem

Before we jump into the calculations, let's make sure we understand the problem completely. We're given four rods with the following lengths:

• 120 * 10^-3 m
• 4 * 10^-2 m
• 3.4 * 10^-1 m

These rods are joined end-to-end to form a right trapezoid. A right trapezoid, for those who might need a quick refresher, is a trapezoid that has at least two right angles. Our mission is to calculate the area of this trapezoid in square meters.

To tackle this, we'll first convert these lengths into more familiar decimal forms. This will make our calculations easier. Then, we'll identify which sides of the trapezoid correspond to which rods. After that, we'll use the formula for the area of a trapezoid, which is: Area = (1/2) * (sum of parallel sides) * height.

Remember, the parallel sides of the trapezoid are the two sides that run parallel to each other, and the height is the perpendicular distance between these parallel sides. Getting these values right is key to finding the correct area. We will explore each length individually and see how they contribute to forming our trapezoid.

Step-by-Step Solution

1. Convert Rod Lengths to Decimal Form

First, let’s convert the lengths of the rods from scientific notation to decimal form. This will help us visualize and use these measurements more easily. Let's break down each one:

• 120 * 10^-3 m = 120 * 0.001 m = 0.12 m. Think of this as moving the decimal point three places to the left. So, 120 becomes 0.120.
• 4 * 10^-2 m = 4 * 0.01 m = 0.04 m. Here, we're moving the decimal point two places to the left. Thus, 4 turns into 0.04.
• 3.4 * 10^-1 m = 3.4 * 0.1 m = 0.34 m. For this one, we move the decimal point one place to the left, changing 3.4 to 0.34.

Now we have our lengths in decimal form: 0.12 m, 0.04 m, and 0.34 m. We know there are four rods, but only three lengths given. This implies that two of the rods have the same length. Looking at the shape we’re forming – a right trapezoid – this makes sense. The two parallel sides of the trapezoid can be different lengths, and the height will be one of the other rods. The remaining rod will form the slanted side of the trapezoid.

2. Identifying the Sides of the Trapezoid

Now, let’s figure out which rods form which sides of the right trapezoid. Remember, a right trapezoid has two parallel sides and two right angles. One of the non-parallel sides is perpendicular to the parallel sides, forming the height of the trapezoid.

Given the lengths 0.12 m, 0.04 m, and 0.34 m, it's reasonable to assume that the two parallel sides are likely to be the 0.12 m and 0.34 m rods, as they are the most distinct lengths. The height of the trapezoid, which is perpendicular to these parallel sides, is likely the 0.04 m rod. Since we're dealing with a right trapezoid, the height corresponds to one of the vertical sides.

The fourth side, which we haven't explicitly mentioned yet, is the slanted side of the trapezoid. Its length is not directly provided, but we don’t actually need it to calculate the area. The area of a trapezoid depends on the lengths of the parallel sides and the height, all of which we've now identified. This is great news because it simplifies our calculations!

3. Applying the Trapezoid Area Formula

Okay, we're in the home stretch! Now that we know the lengths of the parallel sides and the height, we can plug these values into the formula for the area of a trapezoid. Remember, the formula is:

Area = (1/2) * (sum of parallel sides) * height

In our case:

• Parallel side 1 = 0.12 m
• Parallel side 2 = 0.34 m
• Height = 0.04 m

Let's plug these values into the formula:

Area = (1/2) * (0.12 m + 0.34 m) * 0.04 m

First, we add the lengths of the parallel sides:

0. 12 m + 0.34 m = 0.46 m

Now, we plug this sum back into the formula:

Area = (1/2) * (0.46 m) * 0.04 m

Next, we multiply 0.46 m by 0.04 m:

0. 46 m * 0.04 m = 0.0184 m²

Finally, we multiply this result by 1/2 (or 0.5):

Area = 0.5 * 0.0184 m² = 0.0092 m²

So, the area of the right trapezoid formed by the rods is 0.0092 square meters.

Final Answer

Therefore, the area of the right trapezoid formed by the rods with lengths 120 * 10^-3 m, 4 * 10^-2 m, and 3.4 * 10^-1 m is 0.0092 square meters.

We successfully calculated the area by converting the lengths to decimal form, identifying the corresponding sides of the trapezoid, and applying the area formula. Remember, understanding the properties of geometric shapes and knowing the right formulas are key to solving these problems. Great job, guys! You've tackled this geometry challenge like pros. If you found this helpful, stick around for more math adventures!