Question

1. what is the area of the trapezoid represented by the scale drawing? lesson 7-3 7.gr.1.5 5. the patio in 1 in. = 5 ft

Explanation:

Step1: Find the lengths of the two bases of the trapezoid in the scale drawing.

The top base length: Let's calculate the top base. From the diagram, the bottom base has segments 2 in, 13 in, and 2 in. Wait, actually, for a trapezoid, we need the two parallel sides (bases). Let's see, the height in the scale drawing is 4 in? Wait, no, the scale is 1 in = 5 ft. Wait, first, let's find the lengths of the two bases in the scale drawing.

Looking at the diagram, the bottom base (let's say base \( b_1 \)): 2 in + 13 in + 2 in? Wait, no, maybe the top base and bottom base. Wait, the trapezoid has two parallel sides. Let's re-examine. The diagram shows: on the bottom, there are segments 2 in, 13 in, 2 in. Wait, maybe the top base is \( 13 \) in? No, wait, the left side has a height of 4 in (scale drawing). Wait, maybe the two bases are: let's find the lengths of the two parallel sides (bases) in the scale drawing.

Wait, the bottom base (let's call it \( b_1 \)): 2 in + 13 in + 2 in? No, that can't be. Wait, maybe the top base is \( 13 \) in, and the bottom base is \( 2 + 13 + 2 = 17 \) in? Wait, no, maybe I misread. Wait, the scale is 1 in = 5 ft. Wait, first, let's find the lengths of the two bases in the scale drawing.

Wait, the trapezoid in the scale drawing: let's assume that the two parallel sides (bases) are:

Top base (\( b_2 \)): 13 in? No, wait, the bottom has 2 in, 13 in, 2 in. Wait, maybe the top base is \( 13 \) in, and the bottom base is \( 2 + 13 + 2 = 17 \) in? Wait, no, maybe the left and right sides are the non-parallel sides. Wait, the height in the scale drawing is 4 in (as per the diagram, the vertical segment is 4 in).

Wait, first, let's recall the formula for the area of a trapezoid: \( A = \frac{1}{2}(b_1 + b_2)h \), where \( b_1 \) and \( b_2 \) are the lengths of the two bases (parallel sides), and \( h \) is the height (the perpendicular distance between the bases).

Now, in the scale drawing:

First, find the lengths of \( b_1 \) and \( b_2 \) (the two bases) in the scale drawing.

Looking at the diagram:

- The bottom base (let's say \( b_1 \)): 2 in + 13 in + 2 in = 17 in? Wait, no, maybe the top base is \( 13 \) in, and the bottom base is \( 2 + 13 + 2 = 17 \) in? Wait, that might be. Wait, the left side has a height of 4 in (scale drawing).

Wait, no, maybe the two bases are:

Top base (\( b_2 \)): 13 in?

Bottom base (\( b_1 \)): 2 in + 13 in + 2 in = 17 in?

Height (\( h \)) in scale drawing: 4 in?

Wait, but the scale is 1 in = 5 ft. So we need to convert the scale drawing lengths to actual lengths (in feet) first, or calculate the area in the scale drawing and then convert? Wait, no, better to convert the lengths to actual feet using the scale, then apply the trapezoid area formula.

Scale: 1 in = 5 ft. So 1 in (scale) = 5 ft (actual).

First, find the lengths of the two bases ( \( b_1 \) and \( b_2 \) ) and the height ( \( h \) ) in actual feet.

In scale drawing:

- Let's find \( b_1 \) (bottom base): 2 in + 13 in + 2 in = 17 in (scale). So actual length: \( 17 \times 5 = 85 \) ft.

- \( b_2 \) (top base): Wait, maybe the top base is 13 in (scale)? Wait, no, maybe I made a mistake. Wait, the diagram: the top side is parallel to the bottom side. Let's look again. The bottom has 2 in, 13 in, 2 in. So the top base is 13 in (scale), and the bottom base is 2 + 13 + 2 = 17 in (scale). Yes, that makes sense. So top base ( \( b_2 \) ) scale: 13 in, bottom base ( \( b_1 \) ) scale: 17 in.

Height ( \( h \) ) in scale drawing: 4 in (scale). So actual height: \( 4 \times 5 = 20 \) ft.

Now, apply the trapezoid area formula: \( A = \frac{1}{2}(b_1 + b_2)h \)

Substitute \( b_1 = 85 \) ft, \( b_2 = 13 \times 5 = 65 \) ft? Wait, wait, no! Wait, I messed up. Wait, the top base: if the bottom base is 2 + 13 + 2 = 17 in (scale), then what is the top base? Wait, maybe the top base is 13 in (scale), and the bottom base is 17 in (scale). Wait, let's check again.

Wait, the diagram: the left side has a height of 4 in (scale). The bottom has segments 2 in, 13 in, 2 in. So the bottom base (the longer base) is 2 + 13 + 2 = 17 in (scale), and the top base (the shorter base) is 13 in (scale). The height (distance between the two bases) in scale is 4 in.

So converting to actual feet:

- \( b_1 \) (bottom base, scale) = 17 in. Actual: \( 17 \times 5 = 85 \) ft.

- \( b_2 \) (top base, scale) = 13 in. Actual: \( 13 \times 5 = 65 \) ft.

- \( h \) (height, scale) = 4 in. Actual: \( 4 \times 5 = 20 \) ft.

Now, area of trapezoid: \( A = \frac{1}{2}(b_1 + b_2)h \)

Substitute \( b_1 = 85 \) ft, \( b_2 = 65 \) ft, \( h = 20 \) ft.

Step2: Calculate the area using the trapezoid formula.

\( A = \frac{1}{2}(85 + 65) \times 20 \)

First, calculate \( 85 + 65 = 150 \)

Then, \( \frac{1}{2} \times 150 = 75 \)

Then, \( 75 \times 20 = 1500 \) square feet.

Wait, but let's check again. Maybe the height in the scale drawing is 4 in, but is that the height? Wait, the left side has a vertical segment of 4 in (scale), which is the height (perpendicular distance between the two bases). So that's correct.

Alternatively, maybe I made a mistake in the base lengths. Let's re-examine the scale drawing.

Wait, the problem says "the trapezoid represented by the scale drawing". The scale is 1 in = 5 ft. The scale drawing has:

- The two bases: let's see, the bottom base (length in scale): 2 in + 13 in + 2 in = 17 in.

- The top base (length in scale): 13 in (since the middle segment is 13 in, and the top is parallel to the bottom, so the top base is 13 in? Wait, no, that doesn't make sense. Wait, maybe the bottom base is 13 in, and the top base is 13 - 2 - 2 = 9 in? No, that can't be. Wait, maybe the diagram is: the trapezoid has a bottom base of (2 + 13 + 2) in = 17 in, top base of 13 in, height of 4 in (all in scale). Then converting to actual:

Bottom base: 17 in * 5 ft/in = 85 ft

Top base: 13 in * 5 ft/in = 65 ft

Height: 4 in * 5 ft/in = 20 ft

Then area: (85 + 65)/2 * 20 = (150)/2 * 20 = 75 * 20 = 1500 sq ft.

Yes, that seems right.

Answer:

The area of the trapezoid is \(\boxed{1500}\) square feet.