Question

problem 11: spiral review
this shape is made up of four rectangles.
determine the total perimeter of the shape.
(the image shows a composite shape with rectangles labeled 30 sq. m, 40 sq. m, 50 sq. m, 60 sq. m, with dimensions 15 m, 8 m, 10 m.)

Explanation:

Step1: Analyze the horizontal and vertical lengths

The total horizontal length is 15 m. The total vertical length: we can see one side is 10 m and another is 8 m, but we need to check the combined vertical extent. Wait, actually, for the perimeter, we can use the concept of the "outer" dimensions. Let's first find the total height (vertical) and total width (horizontal). The horizontal length is 15 m. For the vertical, let's see the maximum vertical length: from the bottom to the top, the vertical segments: 10 m and 8 m? Wait, no, let's look at the shape. Alternatively, we can use the formula for the perimeter of a composite shape by considering the outer rectangle (if we can "unfold" the indentations, but actually, in such composite shapes, the perimeter can be calculated by adding the total horizontal and vertical lengths and doubling them, considering that the indentations' inner sides are balanced by outer sides? Wait, no, let's do it step by step.

First, let's find the total horizontal length: 15 m (given at the top). Now, the vertical length: let's check the vertical sides. The left side has a vertical length of 10 m, and the right side? Wait, the top part has a height of 8 m, but the bottom part? Wait, maybe a better approach: the perimeter of a composite shape made of rectangles can be calculated by adding all the outer sides. Let's list all the outer sides:

Horizontal sides (top and bottom):
- Top: 15 m
- Bottom: 15 m (since the bottom should be equal to the top in a symmetric way, but wait, no, let's check the vertical sides.

Vertical sides (left and right, and the middle vertical sides? Wait, no, let's look at the shape. Let's first find the total height (vertical) and total width (horizontal). Wait, the horizontal length is 15 m. The vertical length: let's see the two vertical segments: 10 m and 8 m? Wait, no, 10 m and 8 m: but maybe the total vertical length is 10 + 8 - overlap? Wait, no, maybe not. Wait, let's calculate the perimeter by adding all the outer edges.

Alternatively, let's consider that the perimeter is equal to 2*(total horizontal length + total vertical length). Let's find the total horizontal length: 15 m (as given). Now, the total vertical length: let's see the vertical sides. The leftmost vertical side is 10 m, and the rightmost vertical side? Wait, the top right has a height of 8 m, but the bottom right? Wait, maybe the total vertical length is 10 + 8 = 18 m? No, that can't be. Wait, no, let's look at the areas to find the missing dimensions. Wait, maybe we can find the lengths of the rectangles to confirm.

Wait, the first rectangle (30 sq. m): area = length * width. Let's say its length is L1 and width W1. 30 = L1 * W1. The second rectangle (40 sq. m): 40 = L2 * W2. The third (50 sq. m): 50 = L3 * W3. The fourth (60 sq. m): 60 = L4 * W4.

But maybe a simpler way: the perimeter of this composite shape is equal to 2*(15 + (10 + 8))? Wait, 15 is the horizontal length, and the vertical length is 10 + 8? Wait, 10 + 8 = 18? No, 10 + 8 = 18? Wait, 10 m and 8 m: if we add them, 18 m. Then perimeter would be 2*(15 + 18) = 2*33 = 66? Wait, no, that can't be. Wait, maybe I made a mistake. Wait, let's check the vertical sides again.

Wait, the left vertical side: 10 m. The right vertical side: let's see, the top part has a height of 8 m, and the bottom part? Wait, the bottom rectangle (60 sq. m) and the middle (50 sq. m) and the right top (40 sq. m). Wait, maybe the total vertical length is 10 + 8 = 18? No, 10 and 8: are they in the same vertical line? Wait, the left side is 10 m (from bottom to the middle), and the top right is 8 m (from top to the middle). So the total vertical height is 10 + 8 = 18 m? Wait, no, 10 m is from bottom to the middle of the left, and 8 m is from top to the middle of the right. So the total vertical length is 10 + 8 = 18 m? Then the horizontal length is 15 m. Then perimeter is 2*(15 + 18) = 2*33 = 66 m? Wait, but let's verify with the areas.

Wait, maybe another approach: the perimeter of a composite shape can be calculated by adding all the outer edges. Let's list all the outer edges:

Top: 15 m

Right side: let's see, the vertical segments on the right: from bottom to the top of the 60 sq. m rectangle, then up to the 40 sq. m, then up to the top? Wait, no, the 60 sq. m is at the bottom right, 40 sq. m is above it, 30 sq. m is at the top left, 50 sq. m is in the middle left.

Wait, maybe it's easier to use the formula for the perimeter of a rectangle that would enclose the shape, but with some adjustments. Wait, actually, in such "L" or "T" shaped composite figures, the perimeter is equal to the perimeter of the larger rectangle that would contain the shape, because the inner indentations' horizontal and vertical sides are canceled out by the outer sides. Wait, let's check the horizontal and vertical extents.

Horizontal extent: 15 m (given at the top).

Vertical extent: the maximum vertical length is 10 m (left side) + 8 m (top right) - overlap? Wait, no, 10 m and 8 m: are they overlapping? Let's see, the 50 sq. m rectangle is between the 30 sq. m (top left) and 60 sq. m (bottom right), and the 40 sq. m is to the right of 30 sq. m and above 60 sq. m. So the vertical length from bottom to top is 10 m (from bottom to the top of 50 sq. m) plus 8 m (from the bottom of 40 sq. m to the top of 40 sq. m)? Wait, no, 10 m is the height of the 50 sq. m rectangle, and 8 m is the height of the 30 sq. m and 40 sq. m rectangles? Wait, 30 sq. m: area = 30, if its width is, say, let's find the length of the 30 sq. m rectangle. The top horizontal length is 15 m, so the length of 30 sq. m + length of 40 sq. m = 15 m? Wait, 30 sq. m: let's say its height is h1, and 40 sq. m's height is h2. Wait, the top part has a total height of 8 m (given on the right). So 30 sq. m and 40 sq. m are in the top row, with height 8 m? Wait, 30 = length1 * 8? No, 30/8 is not integer. Wait, maybe 30 sq. m has a height of 2 m? Wait, 30 = 15 * 2? No, 15 is the total top length. Wait, maybe I'm overcomplicating. Let's go back to the perimeter.

Alternative method: the perimeter of a composite shape made of rectangles is equal to 2*(sum of the maximum horizontal length + sum of the maximum vertical length). Let's find the maximum horizontal length: 15 m (given). Maximum vertical length: let's see the vertical sides. The left side has a vertical length of 10 m, and the right side has a vertical length of 8 m? No, that can't be. Wait, the 50 sq. m rectangle has a height of 10 m (given on the left), and the 40 sq. m rectangle has a height of 8 m (given on the right). So the total vertical length is 10 + 8 = 18 m? Wait, no, 10 m and 8 m: are they in the same vertical line? No, the 10 m is on the left, 8 m on the right. So the total vertical extent is 10 + 8? No, that would be overlapping. Wait, maybe the total vertical length is the maximum of 10 m and 8 m? No, that doesn't make sense. Wait, let's draw the shape mentally:

- Top row: two rectangles, 30 sq. m (left) and 40 sq. m (right), with total width 15 m, height 8 m (as per the right side label).

- Middle row: one rectangle, 50 sq. m (left), adjacent to the 30 sq. m and 40 sq. m? Wait, no, the 50 sq. m is below the 30 sq. m, and to the left of the 60 sq. m.

- Bottom row: one rectangle, 60 sq. m (right), adjacent to the 50 sq. m.

So the left side of the shape: from bottom to top, the vertical length is 10 m (height of 50 sq. m) plus the height of 30 sq. m? Wait, no, 30 sq. m is on top of 50 sq. m? No, the 30 sq. m is to the left of 40 sq. m, and 50 sq. m is below 30 sq. m, and 60 sq. m is to the right of 50 sq. m and below 40 sq. m.

So the vertical sides:

- Leftmost vertical side: height = 10 m (from bottom to the top of 50 sq. m) + height of 30 sq. m. Wait, but 30 sq. m's height: if 30 sq. m is in the top row with height 8 m, then the leftmost vertical side would be 10 m (from bottom to 50 sq. m top) + 8 m (from 50 sq. m top to 30 sq. m top) = 18 m? Wait, that makes sense. So total vertical height is 18 m, total horizontal width is 15 m. Then perimeter is 2*(15 + 18) = 2*33 = 66 m.

Wait, let's verify with another approach. Let's calculate all the outer sides:

Top: 15 m

Right side: from top of 40 sq. m down to bottom of 60 sq. m: height = 8 m (top part) + height of 60 sq. m. Wait, 60 sq. m: area = 60, if its width is, say, let's find the length of 60 sq. m. The total horizontal length is 15 m, and the 50 sq. m's width: 50 sq. m, height 10 m, so width is 50/10 = 5 m. Then 60 sq. m's width is 15 - 5 = 10 m (since 50 sq. m is 5 m wide, 60 sq. m is 10 m wide). Then 60 sq. m's height is 60/10 = 6 m. So the right side: from top of 40 sq. m (height 8 m) down to bottom of 60 sq. m: 8 m + 6 m = 14 m? Wait, that contradicts the earlier thought. Wait, now I'm confused. Let's recalculate:

50 sq. m: area = 50, height = 10 m (given on the left), so width = 50 / 10 = 5 m.

60 sq. m: width = 15 - 5 = 10 m (since total horizontal is 15 m, and 50 sq. m is 5 m wide), so height = 60 / 10 = 6 m.

40 sq. m: height = 8 m (given on the right), so width = 40 / 8 = 5 m.

30 sq. m: width = 15 - 5 = 10 m (since 40 sq. m is 5 m wide), so height = 30 / 10 = 3 m. Wait, but the top height is given as 8 m, but 3 m + 5 m? No, that doesn't match. Wait, no, 30 sq. m and 40 sq. m are in the top row, so their total height should be the same. So 30 sq. m: height h, width w1; 40 sq. m: height h, width w2; w1 + w2 = 15 m. So 30 = w1 * h, 40 = w2 * h. Then (30 + 40) = h*(w1 + w2) => 70 = h*15 => h = 70/15 = 14/3 ≈ 4.666 m. But the right side is labeled 8 m, so that's a contradiction. So my initial assumption is wrong.

Wait, the right side is labeled 8 m, so the height of the top row (30 and 40 sq. m) is 8 m. So 30 sq. m: width = 30 / 8 = 3.75 m, 40 sq. m: width = 40 / 8 = 5 m. Then total top width: 3.75 + 5 = 8.75 m, but the top is labeled 15 m. So that's not possible. So clearly, my approach is wrong.

Alternative approach: look at the vertical and horizontal sides. The left side has a vertical length of 10 m, and the right side has a vertical length of 8 m, but there's a horizontal segment between them. Wait, the perimeter of a composite shape is the sum of all outer sides. Let's list all the outer sides:

Horizontal sides (top and bottom, and the middle horizontal sides? No, only the outer ones):

- Top: 15 m (given)

- Bottom: 15 m (since the bottom should be equal to the top, as there's no horizontal indentation that's not balanced)

Vertical sides:

- Left: 10 m (given)

- Right: 8 m (given)

- Then, the other vertical sides: wait, no, there are more vertical sides. Wait, the shape has indentations, but in reality, when you have a composite shape made of rectangles, the perimeter can be calculated by adding 2*(total horizontal length + total vertical length), where total horizontal length is the maximum horizontal extent, and total vertical length is the maximum vertical extent. Let's find the maximum horizontal extent: 15 m (given). Maximum vertical extent: the vertical length from the bottom to the top. The bottom is at the bottom of the 50 and 60 sq. m rectangles, and the top is at the top of the 30 and 40 sq. m rectangles. So the vertical length is 10 m (height of 50 sq. m) + 8 m (height of 30 and 40 sq. m) - overlap? Wait, is there an overlap? The 50 sq. m is below the 30 sq. m, so the vertical distance between the bottom of 50 and the top of 30 is zero? No, the 30 sq. m is on top of the 50 sq. m? No, the diagram shows 30 sq. m to the left, 50 sq. m below it, 40 sq. m to the right of 30, and 60 sq. m to the right of 50 and below 40. So the vertical extent is from the bottom of 50/60 (which is the same level) to the top of 30/40 (same level). So the vertical length is 10 m (height of 50) + 8 m (height of 30/40) = 18 m. The horizontal extent is 15 m. So perimeter is 2*(15 + 18) = 66 m. That must be it, because the inner indentations' vertical and horizontal sides are internal and thus not part of the perimeter (they are canceled by the outer sides). So even though there are inner rectangles, the outer perimeter is like a rectangle with length 15 m and height 18 m (10 + 8), so perimeter is 2*(15 + 18) = 66 m.

Step2: Calculate the perimeter

Perimeter = 2*(length + width) = 2*(15 + (10 + 8)) = 2*(15 + 18) = 2*33 = 66 meters.

Answer:

66