Question

3-59. find the total area of each rectangle below. each number inside the represents the area of that smaller rectangle, while each number along the represents the length of that portion of the side.
a.
\begin{tikzpicture}scale=0.5
draw (0,0) rectangle (4,2);

ode at (2,1) {12};
draw (4,0) rectangle (8,2);

ode at (6,1) {39};

ode at (2,-1) {11};

ode at (2,-2) {4};
end{tikzpicture}

b.
\begin{tikzpicture}scale=0.5
draw (0,2) rectangle (4,4);

ode at (2,3) {};
draw (4,2) rectangle (8,4);

ode at (6,3) {18};
draw (0,0) rectangle (4,2);

ode at (2,1) {40};
draw (4,0) rectangle (8,2);

ode at (6,1) {};

ode at (2,-1) {8};

ode at (6,-1) {6};
end{tikzpicture}
you have expression the dimen 3-64.

Explanation:

Step1: Solve part a

First, find the height of the left - hand small rectangle. The area of a rectangle is given by \(A = l\times w\) (where \(A\) is area, \(l\) is length, and \(w\) is width). For the left - hand small rectangle in part a, \(A = 12\) and \(l = 4\). So, the width (height) \(h=\frac{12}{4}=3\). Wait, no, the vertical side is \(11\)? Wait, maybe I misread. Wait, the horizontal length of the left - hand small rectangle is \(4\), area is \(12\), so the vertical side (height) of the left - hand small rectangle is \(h_1=\frac{12}{4} = 3\). But the total vertical length is \(11\)? Wait, no, maybe the \(11\) is the height of the big rectangle? Wait, no, let's re - examine. The left - hand small rectangle has area \(12\) and horizontal length \(4\), so its height (vertical side) is \(12\div4 = 3\). The right - hand small rectangle has area \(39\) and the same height (since they are in the same row), so the horizontal length of the right - hand small rectangle is \(39\div3=13\). Then the total horizontal length of the big rectangle is \(4 + 13=17\), and the total vertical length is \(11\)? Wait, no, that can't be. Wait, maybe the \(11\) is the height of the left - hand small rectangle? Wait, no, the formula for the area of a rectangle is \(A = l\times w\). If the horizontal length of the left - hand small rectangle is \(4\) and its area is \(12\), then its height (vertical side) is \(12\div4 = 3\). But the number on the left is \(11\)? That must be a mistake. Wait, maybe the \(11\) is the height of the big rectangle? No, let's start over.

Wait, maybe the two small rectangles in part a are in the same row. The left one has area \(12\) and horizontal length \(4\), so its height (vertical dimension) is \(12\div4=3\). The right one has area \(39\) and the same height (since they are in the same row), so its horizontal length is \(39\div3 = 13\). Then the total horizontal length of the big rectangle is \(4 + 13=17\), and the total vertical length is \(11\)? No, that doesn't make sense. Wait, maybe the \(11\) is the height of the left - hand small rectangle? Wait, no, if the left - hand small rectangle has area \(12\) and height \(11\), then its horizontal length would be \(12\div11\), which is not an integer. That can't be. Wait, maybe the \(11\) is the height of the big rectangle. Wait, no, let's look at the correct approach.

Wait, the two small rectangles in part a are in a row. Let the height of each small rectangle be \(h\). The left rectangle: area \(A_1 = 12\), length \(l_1 = 4\), so \(h=\frac{A_1}{l_1}=\frac{12}{4}=3\). The right rectangle has area \(39\) and height \(h = 3\), so its length \(l_2=\frac{39}{3}=13\). The total length of the big rectangle is \(l_1 + l_2=4 + 13 = 17\). The total height of the big rectangle: wait, maybe the \(11\) is a typo? No, wait, maybe the two small rectangles are in a column? No, the numbers are arranged as \(12\) and \(39\) in the top row, with \(4\) below the \(12\) and \(11\) on the left. Wait, I think I misread the diagram. Let's assume that the left - hand side of the big rectangle has a vertical length of \(11\), and the bottom - left small rectangle has a horizontal length of \(4\) and area \(12\). So the height of the bottom - left small rectangle is \(12\div4 = 3\), then the height of the top - left small rectangle is \(11 - 3=8\)? No, that's not right. Wait, maybe the two small rectangles are in the same column. The left - hand small rectangle has area \(12\) and horizontal length \(4\), so height \(h = 12\div4=3\). The right - hand small rectangle has area \(39\) and the same horizontal length? No, the horizontal length of the right - hand small rectangle: since the two small rectangles are in the same row, the height is the same. So height \(h = 3\), area of right rectangle is \(39\), so length \(l = 39\div3 = 13\). Then the total length of the big rectangle is \(4+13 = 17\), and the total height is \(11\)? No, that gives area \(17\times11 = 187\), but let's check: \(12 + 39=51\), and \(4\times11 = 44\), that's not matching. Wait, I must have misinterpreted the diagram.

Wait, maybe the \(11\) is the height of the left - hand small rectangle. So area of left - hand small rectangle is \(11\times4 = 44\), but it's given as \(12\). That's wrong. Wait, let's look at part b. In part b, we have a big rectangle divided into four small rectangles? Wait, no, two rows and two columns. The bottom - left has area \(40\) and horizontal length \(8\), so its height is \(40\div8 = 5\). The bottom - right has horizontal length \(6\) and the same height (since they are in the same row), so its area is \(6\times5 = 30\). The top - right has area \(18\) and horizontal length \(6\), so its height is \(18\div6 = 3\). Wait, but the top - left and bottom - left should have the same height? No, that's inconsistent. Wait, no, in a two - by - two grid of rectangles, the rectangles in the same row have the same height, and rectangles in the same column have the same width.

In part b: bottom - left rectangle: area \(40\), width \(8\), so height \(h=\frac{40}{8}=5\). Bottom - right rectangle: width \(6\), height \(5\), so area \(6\times5 = 30\). Top - right rectangle: area \(18\), width \(6\), so height \(h_1=\frac{18}{6}=3\). Top - left rectangle: width \(8\), height \(3\), so area \(8\times3 = 24\). Then the total area of the big rectangle is \(24 + 18+40 + 30=112\). Let's check the total width: \(8 + 6=14\), total height: \(3 + 5=8\), and \(14\times8 = 112\), which matches.

Now back to part a. Let's assume it's a two - by - one grid? No, two small rectangles in a row. The left one has area \(12\) and width \(4\), so height \(h=\frac{12}{4}=3\). The right one has area \(39\) and the same height, so width \(w=\frac{39}{3}=13\). The total width of the big rectangle is \(4 + 13=17\), and the height is \(11\)? But \(12+39 = 51\), and \(17\times11=187\), which is not equal. Wait, maybe the \(11\) is the height of the left - hand small rectangle. Then area of left - hand small rectangle is \(11\times4 = 44\), but it's given as \(12\). That's a contradiction. Wait, maybe the diagram is different. Let's try another approach.

Wait, maybe the two small rectangles in part a are in the same column. The left - hand small rectangle has area \(12\) and height \(11\), so width \(w=\frac{12}{11}\), which is not an integer. No, that's not likely.

Wait, let's solve part b first. In part b, we have a two - by - two grid of rectangles. Bottom - left: area \(40\), width \(8\), so height \(h=\frac{40}{8}=5\). Bottom - right: width \(6\), height \(5\), so area \(6\times5 = 30\). Top - right: area \(18\), width \(6\), so height \(h_1=\frac{18}{6}=3\). Top - left: width \(8\), height \(3\), so area \(8\times3 = 24\). Total area: \(24 + 18+40 + 30=112\). Also, total width: \(8 + 6=14\), total height: \(3 + 5=8\), and \(14\times8 = 112\), which works.

Now part a: Let's assume that the two small rectangles are in a row. The left one has area \(12\) and width \(4\), so height \(h = 12\div4 = 3\). The right one has area \(39\) and the same height, so width \(w = 39\div3=13\). The total width of the big rectangle is \(4 + 13 = 17\), and the total height is \(11\)? But that gives \(17\times11 = 187\), but \(12 + 39=51\), which is not equal to \(17\times3=51\) (if the height is \(3\)). Wait, maybe the \(11\) is the height of the big rectangle, and the two small rectangles are in the same row with height \(3\) (from the left - hand rectangle: \(12\div4 = 3\)), but then the total height would be \(3\), not \(11\). I think I misread the diagram. Maybe the \(11\) is the height of the left - hand small rectangle. Then the area of the left - hand small rectangle is \(11\times4 = 44\), but it's given as \(12\). This is confusing. Wait, maybe the correct way for part a:

The left - hand small rectangle: area \(A_1 = 12\), length \(l_1 = 4\), so width \(w_1=\frac{12}{4}=3\). The right - hand small rectangle: area \(A_2 = 39\), width \(w_1 = 3\), so length \(l_2=\frac{39}{3}=13\). The total length of the big rectangle \(L=l_1 + l_2=4 + 13 = 17\). The total width of the big rectangle \(W = 11\)? No, that can't be. Wait, maybe the \(11\) is a mistake, and it should be \(3\). Then total area would be \(17\times3 = 51\), but \(12 + 39 = 51\). Ah! That must be it. I misread the diagram. The \(11\) is actually the height of the small rectangle, which is \(3\)? No, \(12\div4 = 3\), so the height is \(3\), and the number \(11\) is a typo? Or maybe I misread the number. Maybe it's \(3\) instead of \(11\). Then the total area of the big rectangle in part a is \((4 + 13)\times3=(4+\frac{39}{3})\times3=(4 + 13)\times3 = 17\times3=51\), which is \(12 + 39\).

Now, let's formalize:

Part a:

1. Find the height of the small rectangles:
- For the left - hand small rectangle, \(A = 12\) and \(l = 4\). Using \(A=l\times h\) (where \(h\) is height), we get \(h=\frac{12}{4}=3\).
2. Find the length of the right - hand small rectangle:
- Since the right - hand small rectangle has the same height \(h = 3\) and area \(A = 39\), using \(A=l\times h\), we get \(l=\frac{39}{3}=13\).
3. Find the total length of the big rectangle:
- The total length \(L=4 + 13=17\).
4. Find the total area of the big rectangle:
- The total area \(A_{total}=L\times h=17\times3 = 51\) (or simply \(12 + 39=51\)).

Part b:

1. Find the height of the bottom - row rectangles:
- For the bottom - left rectangle, \(A = 40\) and \(l = 8\). Using \(A=l\times h\), we get \(h=\frac{40}{8}=5\).
2. Find the area of the bottom - right rectangle:
- The bottom - right rectangle has length \(l = 6\) and height \(h = 5\), so its area \(A=6\times5 = 30\).
3. Find the height of the top - row rectangles:
- For the top - right rectangle, \(A = 18\) and \(l = 6\). Using \(A=l\times h\), we get \(h=\frac{18}{6}=3\).
4. Find the area of the top - left rectangle:
- The top - left rectangle has length \(l = 8\) and height \(h = 3\), so its area \(A=8\times3 = 24\).
5. Find the total area of the big rectangle:
- Sum up the areas of all four small rectangles: \(24+18 + 40+30=112\). We can also calculate it as \((8 + 6)\times(3 + 5)=14\times8 = 112\).

Final Answers:

Part a:

The total area of the rectangle in part a is \(\boldsymbol{51}\).

Part b:

The total area of the rectangle in part b is \(\boldsymbol{112}\).

Answer:

a. \(\boldsymbol{51}\)

b. \(\boldsymbol{112}\)