Question

1. find the perimeter of the region that is not shaded. 39 ft 50 ft 58 ft 29 ft

Explanation:

Step1: Analyze the unshaded region's perimeter

The unshaded region's perimeter can be found by considering the outer rectangle and the inner "indent" (but we can also use the concept that the perimeter of the unshaded region is equal to the perimeter of the large rectangle, because the horizontal and vertical segments of the indent are accounted for when calculating the perimeter. First, find the length and width of the large rectangle. The total length is 15 ft, and the total height is 14 ft? Wait, no, let's look at the figure. Wait, actually, when we calculate the perimeter of the unshaded region, we can "shift" the horizontal and vertical sides of the shaded part to see that the perimeter is the same as the perimeter of a rectangle with length 15 ft and height 14 ft? Wait, no, let's check the dimensions. Wait, the unshaded region: let's list all the sides. The bottom side is 15 ft. The right side? Wait, no, maybe a better approach: the perimeter of a rectangle is \( P = 2\times (length + width) \). Wait, the large rectangle (if we consider the outer bounds) has length 15 ft and height 14 ft? Wait, no, let's see: the vertical side is 14 ft, and the horizontal side is 15 ft. Wait, but when we have the unshaded region, the perimeter can be calculated by adding all the outer and inner sides. Wait, actually, if we "move" the horizontal segment of the shaded part (the top of the unshaded lower part) and the vertical segment (the left of the unshaded right part), we can see that the perimeter of the unshaded region is equal to the perimeter of the rectangle with length 15 ft and height 14 ft. Wait, let's calculate the perimeter of a rectangle with length 15 and height 14: \( P = 2\times(15 + 14) = 2\times29 = 58 \)? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's look at the figure again. The unshaded region: the left vertical side is 14 ft, the bottom horizontal side is 15 ft, the right vertical side? Wait, no, the shaded region is a rectangle? Wait, the unshaded region has a vertical segment of 4 ft (left part) and a horizontal segment of 6 ft (bottom part). Wait, maybe another approach: the perimeter of the unshaded region is equal to the sum of all its outer and inner sides. Let's list the sides:

- Bottom: 15 ft
- Right: 6 ft (vertical)
- Top of the lower unshaded part: 15 - 4 = 11 ft? Wait, no, maybe I'm overcomplicating. Wait, actually, when you have a rectangular indent, the perimeter of the unshaded region is equal to the perimeter of the original rectangle (before the indent) because the two new sides (the horizontal and vertical from the indent) are equal to the sides they replaced. Wait, the original rectangle (if we consider the outer rectangle) would have length 15 ft and height 14 ft? Wait, no, let's check the given dimensions. The vertical side is 14 ft, the horizontal side is 15 ft. Wait, the shaded region: the horizontal length of the shaded region is 15 - 4 = 11 ft, and the vertical height is 14 - 6 = 8 ft? Wait, no, the unshaded region: the left part is 4 ft wide and 14 ft tall? No, the unshaded region has two parts: the left vertical strip (4 ft wide, 14 ft tall) and the bottom horizontal strip (15 ft long, 6 ft tall), but they overlap at the corner. Wait, actually, the perimeter of the unshaded region can be calculated by adding all the outer edges and the inner edges. Let's do that:

- Left side: 14 ft (vertical)
- Bottom side: 15 ft (horizontal)
- Right side: 6 ft (vertical)
- Top of the bottom strip: 15 ft (horizontal, but wait, no, the top of the bottom strip is adjacent to the shaded region. Wait, maybe a better way: the perimeter of the unshaded region is equal to the perimeter of a rectangle with length 15 and height 14, because when you have the indent, the two new sides (the horizontal one at the top of the bottom unshaded part and the vertical one at the right of the left unshaded part) are equal to the sides that are "indented". Wait, let's calculate the perimeter of the rectangle with length 15 and height 14: \( P = 2\times(15 + 14) = 2\times29 = 58 \) ft. Wait, but let's verify:

The unshaded region's perimeter:

- Left vertical: 14 ft
- Bottom horizontal: 15 ft
- Right vertical: 6 ft
- Top of the bottom unshaded: 15 ft (but wait, no, the top of the bottom unshaded is 15 ft? No, the bottom unshaded is 6 ft tall, so the top of the bottom unshaded is 6 ft from the bottom. Then the right side of the bottom unshaded is 6 ft, and the top of the bottom unshaded is 15 ft. Then the left unshaded is 4 ft wide, 14 ft tall, but the bottom of the left unshaded is 15 ft? No, I think I'm confusing. Wait, another method: the perimeter of the unshaded region is equal to the sum of all the outer sides and the inner sides. The outer sides: the left side (14 ft), the bottom side (15 ft), the right side (14 ft? No, the total height is 14 ft, but the bottom unshaded is 6 ft, so the right side of the bottom unshaded is 6 ft, and the right side of the left unshaded is 4 ft? Wait, no, let's look at the figure again. The figure is a large rectangle with a shaded rectangle cut out from the top right. The large rectangle has length 15 ft and height 14 ft. The shaded rectangle has width (15 - 4) = 11 ft and height (14 - 6) = 8 ft? No, the unshaded region is the left 4 ft (vertical strip) and the bottom 6 ft (horizontal strip), overlapping at the corner (4 ft by 6 ft). So the perimeter of the unshaded region:

- Left vertical strip: height 14 ft, width 4 ft. Its perimeter contribution: left side (14 ft), top (4 ft), right (14 ft - 6 ft = 8 ft), bottom (4 ft). Wait, no, that's not right. Wait, maybe the correct way is to realize that the perimeter of the unshaded region is equal to the perimeter of the large rectangle, because the two sides of the shaded region (the horizontal and vertical) are added to the perimeter. Wait, the large rectangle's perimeter is \( 2\times(15 + 14) = 58 \) ft. Let's check: when you cut out a rectangle from the corner, the perimeter increases by twice the length of the cut? No, wait, no: if you have a rectangle and you cut out a smaller rectangle from the corner, the perimeter remains the same as the original rectangle. Wait, let's take a simple example: a square of side 5. Cut out a 2x2 square from the corner. The original perimeter is 20. The new perimeter: the outer sides are still 5, but the inner sides: the horizontal cut is 2, and the vertical cut is 2, so the new perimeter is 5 + 5 + (5 - 2) + 2 + 2 + (5 - 2) = 5 + 5 + 3 + 2 + 2 + 3 = 20. So yes, the perimeter remains the same. So in this case, the large rectangle has length 15 and height 14, so perimeter is \( 2\times(15 + 14) = 58 \) ft.

Step2: Calculate the perimeter

Using the formula for the perimeter of a rectangle \( P = 2\times (l + w) \), where \( l = 15 \) ft and \( w = 14 \) ft.

\( P = 2\times(15 + 14) = 2\times29 = 58 \) ft.

Answer:

58 ft