The floor plan of a ballroom is shown at right. The scale is 3 centimeters : 4 meters. What are the length and width of the ballroom? Find the ballroom’s area.
Length:
Width:
Area:
The correct answer and explanation is :
To accurately answer this question, I would need to see the floor plan of the ballroom, as it contains the actual measurements in centimeters. Since no image is currently shown or attached, I will explain how to solve this type of problem step-by-step using a hypothetical example, and how to apply the scale conversion and area formula.
Step-by-Step Guide:
1. Understand the Scale:
The scale provided is:
- 3 centimeters : 4 meters
This means for every 3 cm on the drawing, the actual ballroom measures 4 meters in real life.
2. Measure the Dimensions on the Drawing:
Let’s say, for example:
- The length on the drawing is 12 cm
- The width on the drawing is 9 cm
3. Convert Drawing Measurements to Actual Measurements:
Use a proportion:
$\frac{3 \text{ cm (drawing)}}{4 \text{ m (actual)}} = \frac{12 \text{ cm}}{x \text{ m}}$
Solve for $x$:
$$
x = \frac{12 \times 4}{3} = \frac{48}{3} = 16 \text{ meters}
$$
Repeat for width:
$$
x = \frac{9 \times 4}{3} = \frac{36}{3} = 12 \text{ meters}
$$
So:
- Length = 16 meters
- Width = 12 meters
4. Find the Area:
Use the formula for the area of a rectangle:
$$
\text{Area} = \text{Length} \times \text{Width}
= 16 \times 12 = 192 \text{ square meters}
$$
Final Answer:
- Length: 16 meters
- Width: 12 meters
- Area: 192 square meters
Explanation (300+ words):
This problem involves scale drawing and unit conversion, which is a common application in geometry and real-world measurements such as architecture or engineering. The scale provided—3 centimeters : 4 meters—is crucial. It allows us to translate the measurements from the drawing (in centimeters) to actual real-world dimensions (in meters). This ratio tells us that every 3 centimeters on the floor plan represents 4 meters in reality.
To find the real dimensions, we use proportional reasoning. This involves setting up a ratio using the known scale and the measured value from the plan and solving for the unknown actual value. For example, if the drawing shows 12 centimeters as the length, we set up the proportion $\frac{3}{4} = \frac{12}{x}$, where $x$ is the real-life measurement. Cross-multiplying and solving gives us the real-world value.
Once both the length and width are found, we apply the formula for the area of a rectangle:
$$
\text{Area} = \text{Length} \times \text{Width}
$$
This gives us the total space inside the ballroom, which is useful for event planning, capacity estimation, or flooring costs.
If you can upload or describe the actual dimensions shown in centimeters on the drawing, I can give the exact answer. Would you like to do that?