Question

what is the area of this figure? 4 mi, 6 mi, 2 mi, 14 mi, 8 mi, 6 mi, square miles

Explanation:

Step1: Divide the figure into two rectangles

We can split the composite figure into a top rectangle and a bottom rectangle. The top rectangle has dimensions \(4\) mi (width) and \(6\) mi (height). The bottom rectangle has dimensions \(6\) mi (width) and \(8\) mi (height), but we need to check if there's an overlap or adjust for the indent. Wait, actually, another way: the total height is \(14\) mi, and the bottom part is \(8\) mi, so the top part's height is \(14 - 8=6\) mi (which matches). The width of the top rectangle is \(4\) mi, and the bottom rectangle's width is \(6\) mi, but there's a protrusion? Wait, no, let's calculate the area by splitting into two rectangles: one with length \(6\) mi (height) and \(4\) mi (width), and another with length \(8\) mi (height) and \(6\) mi (width)? Wait, no, that's not right. Wait, the figure can be seen as a large rectangle minus a smaller rectangle? Wait, no, let's look at the dimensions. The total height is \(14\) mi, total width at the bottom is \(6\) mi, and at the top, the width is \(4\) mi, with a \(2\) mi indent on the left. So another approach: split into two rectangles. The first rectangle: height \(8\) mi, width \(6\) mi. The second rectangle: height \(6\) mi (since \(14 - 8 = 6\)), width \(4\) mi (since the top part is \(4\) mi wide). Then calculate the area of each and add them.

Step2: Calculate area of first rectangle

Area of first rectangle (bottom) = length × width = \(8 \times 6 = 48\) square miles.

Step3: Calculate area of second rectangle

Area of second rectangle (top) = length × width = \(6 \times 4 = 24\) square miles.

Step4: Sum the areas

Total area = \(48 + 24 = 72\)? Wait, no, that can't be. Wait, no, maybe I made a mistake. Wait, the bottom rectangle: height \(8\) mi, width \(6\) mi: area \(8 \times 6 = 48\). The top rectangle: height \(6\) mi (since \(14 - 8 = 6\)), width \(4\) mi: area \(6 \times 4 = 24\). But wait, the indent is \(2\) mi, so maybe another way: the figure is a rectangle of \(14\) mi height and \(6\) mi width, minus a rectangle of \(6\) mi height (since \(14 - 8 = 6\)) and \(2\) mi width (the indent). Let's try that. Area of large rectangle: \(14 \times 6 = 84\). Area of the missing rectangle: \(6 \times 2 = 12\). Then total area = \(84 - 12 = 72\)? Wait, no, that's the same as before. Wait, but let's check again. Wait, the large rectangle would be \(14\) mi (height) × \(6\) mi (width) = \(84\) sq mi. The missing part is a rectangle with height \(6\) mi (since the bottom part is \(8\) mi, so the top part is \(14 - 8 = 6\) mi) and width \(2\) mi (the indent on the left: \(6 - 4 = 2\) mi). So area of missing part: \(6 \times 2 = 12\) sq mi. Then total area = \(84 - 12 = 72\) sq mi. Wait, but let's verify with the two rectangles. First rectangle: bottom part, height \(8\) mi, width \(6\) mi: \(8 \times 6 = 48\). Second rectangle: top part, height \(6\) mi, width \(4\) mi: \(6 \times 4 = 24\). Total: \(48 + 24 = 72\) sq mi. Yes, that matches.

Answer:

\(72\)