top
3/4\ 1-1/4\
3/4\ 3/4\
front
1/4\ 1/2\ 1-1/4\
1/2\
side
orthographic problem set #1

Question

Accepted Answer

Since the problem is an orthographic projection problem (related to interpreting 2D views of a 3D object, likely for calculating volume, surface area, or verifying dimensions), we can assume a common task like finding the volume. Let's assume we need to find the volume of the 3D object. First, we analyze the dimensions from the top, front, and side views.

### Step 1: Determine the length, width, and height components
From the top view:
- Total length (along the horizontal) is $ 2'' $, with segments $ \frac{3}{4}'' $ and $ 1\frac{1}{4}'' $ (which adds up: $ \frac{3}{4} + 1\frac{1}{4} = 2 $, correct).
- The vertical segments on the top view are $ \frac{3}{4}'' $ and $ \frac{3}{4}'' $, so total height (vertical in top view, which is depth in 3D) is $ \frac{3}{4} + \frac{3}{4} = \frac{3}{2}'' $? Wait, no, let's check the front view.

From the front view:
- The total height (vertical) is $ 1\frac{1}{4}'' $, with segments $ \frac{1}{4}'' $, $ \frac{1}{2}'' $, and $ \frac{1}{2}'' $ ( $ \frac{1}{4} + \frac{1}{2} + \frac{1}{2} = 1\frac{1}{4} $, correct).
- The length (horizontal) in front view: the leftmost rectangle has width $ \frac{3}{4}'' $ (from top view), and the right part has width $ 1\frac{1}{4}'' $.

Let's break the object into parts. Let's consider the 3D object as composed of rectangular prisms.

Part 1: Leftmost vertical prism (from top view: width $ \frac{3}{4}'' $, depth (from top view's vertical) let's see from side view. Wait, maybe better to use the three views:

- Length (x - axis): from top view, total length is $ 2'' $, with left segment $ \frac{3}{4}'' $ and right $ 1\frac{1}{4}'' $.
- Width (y - axis): from top view, the vertical segments (depth) are $ \frac{3}{4}'' $ and $ \frac{3}{4}'' $, so total depth? Wait, no, top view: the horizontal is length, vertical is depth. So depth (y) from top view: the left rectangle has height (depth) $ \frac{3}{4} + \frac{3}{4} = \frac{3}{2}'' $? Wait, no, the top view's vertical dimension is the depth (into the page). Wait, maybe the front view's vertical is height (z - axis), front view's horizontal is length (x - axis), and side view's horizontal is depth (y - axis), side view's vertical is height (z - axis).

Let's define:
- Length (x): from top view, left part $ \frac{3}{4}'' $, right part $ 1\frac{1}{4}'' $.
- Depth (y): from side view, let's see. The side view has a notch, but maybe the total depth is the same as the top view's vertical? Wait, maybe the object can be divided into three rectangular prisms:

1. Left - most prism: length $ \frac{3}{4}'' $, depth (let's say) $ d $, height $ 1\frac{1}{4}'' $ (from front view's total height). But no, front view shows a notch. Wait, front view: the left rectangle is tall, then a middle and bottom. Wait, maybe the correct approach is to calculate the volume by subtracting the notched part, but maybe it's easier to add the volumes of the three visible prisms.

Wait, maybe the problem is to verify the dimensions or calculate volume. Let's assume we need to find the volume. Let's list the dimensions:

From top view:
- Left rectangle: width (x) $ \frac{3}{4}'' $, depth (y) $ \frac{3}{4} + \frac{3}{4} = \frac{3}{2}'' $ (since the right part has two $ \frac{3}{4}'' $ segments? No, top view: the left rectangle is vertical (height in top view) $ \frac{3}{4} + \frac{3}{4} = \frac{3}{2}'' $, and width (x) $ \frac{3}{4}'' $. The right part is two rectangles stacked, each with width (x) $ 1\frac{1}{4}'' $, height (in top view) $ \frac{3}{4}'' $ each.

From front view:
- The left rectangle has height (z) $ 1\frac{1}{4}'' $, width (x) $ \frac{3}{4}'' $.
- The middle rectangle (right part, middle) has height (z) $ \frac{1}{2}'' $, width (x) $ 1\frac{1}{4}'' $, and depth (y) $ \frac{3}{4}'' $ (from top view's right - top segment).
- The bottom rectangle (right part, bottom) has height (z) $ \frac{1}{2}'' $, width (x) $ 1\frac{1}{4}'' $, and depth (y) $ \frac{3}{4}'' $ (from top view's right - bottom segment). Wait, but front view has a notch: the middle right rectangle is indented by $ \frac{1}{4}'' $ in height? Wait, front view: the total height is $ 1\frac{1}{4}'' $, with the top right part having height $ \frac{1}{4}'' $, middle $ \frac{1}{2}'' $, bottom $ \frac{1}{2}'' $? No, the front view's dimensions: the vertical segments are $ \frac{1}{4}'' $, $ \frac{1}{2}'' $, $ \frac{1}{2}'' $, summing to $ 1\frac{1}{4}'' $. The horizontal: left part is $ \frac{3}{4}'' $, right part is $ 1\frac{1}{4}'' $, but the middle right rectangle is indented by $ \frac{1}{4}'' $ in height? Wait, maybe I'm overcomplicating. Let's use the formula for the volume of a rectangular prism: $ V = l \times w \times h $.

Alternative approach: The object can be considered as a large rectangular prism minus the notched part.

Large prism dimensions:
- Length (x): $ 2'' $ (from top view)
- Depth (y): Let's see from side view. The side view has a similar notch, but maybe the total depth is $ \frac{3}{4} + \frac{3}{4} = \frac{3}{2}'' $ (from top view's vertical segments)
- Height (z): $ 1\frac{1}{4}'' $ (from front view's total height)

Volume of large prism: $ V_{large} = 2 \times \frac{3}{2} \times 1\frac{1}{4} = 2 \times \frac{3}{2} \times \frac{5}{4} = \frac{15}{4} = 3\frac{3}{4} $ cubic inches.

Now, the notched part: From front view, the notch is a rectangular prism with:
- Length (x): $ 1\frac{1}{4}'' $ (right part)
- Depth (y): Let's see, from side view, the notch's depth. Wait, side view shows a notch, so maybe the notch has depth $ \frac{3}{4}'' $ (from top view's right - top segment) and height $ \frac{1}{4}'' $ (from front view's top notch) and length $ 1\frac{1}{4}'' $? No, this is getting too confusing. Maybe the problem is to verify that the sum of the segments equals the total. Let's check the top view horizontal: $ \frac{3}{4} + 1\frac{1}{4} = \frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2'' $, which matches the total. Top view vertical: $ \frac{3}{4} + \frac{3}{4} = \frac{6}{4} = \frac{3}{2}'' $. Front view vertical: $ \frac{1}{4} + \frac{1}{2} + \frac{1}{2} = \frac{1}{4} + 1 = 1\frac{1}{4}'' $. Front view horizontal: $ \frac{3}{4} + 1\frac{1}{4} = 2'' $, which matches top view.

If the problem is to find the volume, let's correctly decompose the object:

1. Left rectangular prism:
   - Length (x): $ \frac{3}{4}'' $
   - Depth (y): $ \frac{3}{4} + \frac{3}{4} = \frac{3}{2}'' $ (from top view's vertical)
   - Height (z): $ 1\frac{1}{4}'' $ (from front view's total height)
   - Volume: $ V_1 = \frac{3}{4} \times \frac{3}{2} \times \frac{5}{4} = \frac{45}{32} $

2. Right - top rectangular prism:
   - Length (x): $ 1\frac{1}{4}'' = \frac{5}{4}'' $
   - Depth (y): $ \frac{3}{4}'' $ (from top view's right - top segment)
   - Height (z): $ \frac{1}{4}'' $ (from front view's top segment)
   - Volume: $ V_2 = \frac{5}{4} \times \frac{3}{4} \times \frac{1}{4} = \frac{15}{64} $

3. Right - middle rectangular prism:
   - Length (x): $ 1\frac{1}{4}'' = \frac{5}{4}'' $
   - Depth (y): $ \frac{3}{4}'' $ (from top view's right - middle segment? Wait, top view's right part is two rectangles stacked, each with depth $ \frac{3}{4}'' $)
   - Height (z): $ \frac{1}{2}'' $ (from front view's middle segment)
   - Volume: $ V_3 = \frac{5}{4} \times \frac{3}{4} \times \frac{1}{2} = \frac{15}{32} $

4. Right - bottom rectangular prism:
   - Length (x): $ 1\frac{1}{4}'' = \frac{5}{4}'' $
   - Depth (y): $ \frac{3}{4}'' $ (from top view's right - bottom segment)
   - Height (z): $ \frac{1}{2}'' $ (from front view's bottom segment)
   - Volume: $ V_4 = \frac{5}{4} \times \frac{3}{4} \times \frac{1}{2} = \frac{15}{32} $

Now, sum all volumes:

$ V = V_1 + V_2 + V_3 + V_4 $

$ V = \frac{45}{32} + \frac{15}{64} + \frac{15}{32} + \frac{15}{32} $

Convert to 64 denominator:

$ \frac{45}{32} = \frac{90}{64} $, $ \frac{15}{32} = \frac{30}{64} $

So $ V = \frac{90}{64} + \frac{15}{64} + \frac{30}{64} + \frac{30}{64} = \frac{90 + 15 + 30 + 30}{64} = \frac{165}{64} \approx 2.578 $ cubic inches.

But maybe the problem is simpler, like verifying the sum of the length segments: $ \frac{3}{4} + 1\frac{1}{4} = 2 $, which is correct. Or the sum of the height segments in top view: $ \frac{3}{4} + \frac{3}{4} = \frac{3}{2} $, and in front view: $ \frac{1}{4} + \frac{1}{2} + \frac{1}{2} = 1\frac{1}{4} $.

If the question is to find the total length (horizontal) from the top view, the answer is $ 2'' $, since $ \frac{3}{4} + 1\frac{1}{4} = 2 $.

### Step 1: Add the two length segments
The two segments are $ \frac{3}{4}'' $ and $ 1\frac{1}{4}'' $. Convert $ 1\frac{1}{4}'' $ to improper fraction: $ 1\frac{1}{4} = \frac{5}{4} $.

Sum: $ \frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2 $.

# Answer:
The total length (horizontal) from the top view is $ \boldsymbol{2''} $.

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QUESTION IMAGE

Question

Explanation:

Step 1: Determine the length, width, and height components

Answer: