Question
What is the area of a figure with these side lengths?
Imagine the simple shape of a sailboat. There's only a headsail on top and the boat's base.
The left side of the headsail is labelled "4 m"
Above the right side of the base of the boat is labelled "3 cm"
Inside the base of the boat at the top is labelled "10 m"
Under the base of the boat is labelled "6 m"
Now imagine a line inside the base of the boat a little to the left. Next to that line is labelled "2 m"
Imagine the simple shape of a sailboat. There's only a headsail on top and the boat's base.
The left side of the headsail is labelled "4 m"
Above the right side of the base of the boat is labelled "3 cm"
Inside the base of the boat at the top is labelled "10 m"
Under the base of the boat is labelled "6 m"
Now imagine a line inside the base of the boat a little to the left. Next to that line is labelled "2 m"
Answers
GPT-4o mini
To find the area of the figure you described, which resembles a sailboat with a triangular headsail and a rectangular base, we'll break it down into two parts: the area of the headsail (triangle) and the area of the boat base (rectangle).
### Step 1: Area of the headsail (triangle)
The headsail is represented as a triangle with one side labeled as 4 m, which I assume is the height of the triangle from the base of the headsail to the top point of the headsail.
To compute the area of the triangle, we also need the base. From your description, it appears that the horizontal distance at the base of the triangle is equal to the base of the boat, which is labeled as 10 m.
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Using the values:
- Height = 4 m (the distance from the height of the triangle down to the base)
- Base = 10 m
\[
A_{\text{triangle}} = \frac{1}{2} \times 10 \, \text{m} \times 4 \, \text{m}
\]
\[
A_{\text{triangle}} = \frac{1}{2} \times 40 \, \text{m}^2 = 20 \, \text{m}^2
\]
### Step 2: Area of the base of the boat (rectangle)
The base of the boat is a rectangle. The dimensions of the rectangle are:
- Width (base) = 10 m
- Height = 6 m
The area \( A \) of the rectangle is given by the formula:
\[
A = \text{width} \times \text{height}
\]
Calculating the area of the base:
\[
A_{\text{rectangle}} = 10 \, \text{m} \times 6 \, \text{m} = 60 \, \text{m}^2
\]
### Final Area Calculation
Now, we simply add the area of the triangle and the rectangle:
\[
A_{\text{total}} = A_{\text{triangle}} + A_{\text{rectangle}}
\]
\[
A_{\text{total}} = 20 \, \text{m}^2 + 60 \, \text{m}^2 = 80 \, \text{m}^2
\]
Thus, the total area of the figure (sailboat) is:
\[
\boxed{80 \, \text{m}^2}
\]
### Step 1: Area of the headsail (triangle)
The headsail is represented as a triangle with one side labeled as 4 m, which I assume is the height of the triangle from the base of the headsail to the top point of the headsail.
To compute the area of the triangle, we also need the base. From your description, it appears that the horizontal distance at the base of the triangle is equal to the base of the boat, which is labeled as 10 m.
The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
Using the values:
- Height = 4 m (the distance from the height of the triangle down to the base)
- Base = 10 m
\[
A_{\text{triangle}} = \frac{1}{2} \times 10 \, \text{m} \times 4 \, \text{m}
\]
\[
A_{\text{triangle}} = \frac{1}{2} \times 40 \, \text{m}^2 = 20 \, \text{m}^2
\]
### Step 2: Area of the base of the boat (rectangle)
The base of the boat is a rectangle. The dimensions of the rectangle are:
- Width (base) = 10 m
- Height = 6 m
The area \( A \) of the rectangle is given by the formula:
\[
A = \text{width} \times \text{height}
\]
Calculating the area of the base:
\[
A_{\text{rectangle}} = 10 \, \text{m} \times 6 \, \text{m} = 60 \, \text{m}^2
\]
### Final Area Calculation
Now, we simply add the area of the triangle and the rectangle:
\[
A_{\text{total}} = A_{\text{triangle}} + A_{\text{rectangle}}
\]
\[
A_{\text{total}} = 20 \, \text{m}^2 + 60 \, \text{m}^2 = 80 \, \text{m}^2
\]
Thus, the total area of the figure (sailboat) is:
\[
\boxed{80 \, \text{m}^2}
\]