Answer :
To determine the area of the actual space in Jessica's room where furniture cannot be placed because of the door swing, follow these steps:
1. Understand the scale model dimensions and actual room dimensions:
- The scale model dimensions are 11 inches in length and 8.5 inches in width.
- The door in the scale model is 1.75 inches wide.
- The actual length of the room is 15 feet.
2. Calculate the scaling factor:
The scaling factor is determined by dividing the actual length of the room by the length of the scale model.
[tex]\[ \text{Scaling factor} = \frac{\text{Actual length}}{\text{Scale length}} = \frac{15 \text{ feet}}{11 \text{ inches}} \][/tex]
3. Convert scale model door width to actual door width:
Multiply the door width in the scale model by the scaling factor to get the actual width of the door.
[tex]\[ \text{Actual door width} = \text{Door scale width} \times \text{Scaling factor} = 1.75 \times 1.3636363636363635 \approx 2.39 \text{ feet} \][/tex]
4. Determine the radius of the area impacted by the door swing:
Since the door swings open and forms a circular area, the radius of this area is equivalent to the actual door width. However, we must consider a conservative assumption for the radius, as typical doors may be around 2.5 feet wide or less. Here, the actual calculated door width, 2.39 feet, will be used.
5. Calculate the area where furniture cannot be placed:
The area where furniture cannot be placed due to the door swing is approximately a semi-circle with the radius calculated in the previous step.
[tex]\[ \text{Area without furniture} = \frac{1}{2} \pi \times (\text{Radius})^2 = \frac{1}{2} \pi \times (2.3863636363636362)^2 \][/tex]
[tex]\[ \text{Area without furniture} \approx 8.95 \text{ square feet} \][/tex]
6. Estimate the area:
The most appropriate choice in the given options is:
[tex]\[ \boxed{9 \text{ ft}^2} \][/tex]
Thus, the best estimate of the area in Jessica's room where furniture cannot be placed because of the door swing is approximately 9 square feet.
1. Understand the scale model dimensions and actual room dimensions:
- The scale model dimensions are 11 inches in length and 8.5 inches in width.
- The door in the scale model is 1.75 inches wide.
- The actual length of the room is 15 feet.
2. Calculate the scaling factor:
The scaling factor is determined by dividing the actual length of the room by the length of the scale model.
[tex]\[ \text{Scaling factor} = \frac{\text{Actual length}}{\text{Scale length}} = \frac{15 \text{ feet}}{11 \text{ inches}} \][/tex]
3. Convert scale model door width to actual door width:
Multiply the door width in the scale model by the scaling factor to get the actual width of the door.
[tex]\[ \text{Actual door width} = \text{Door scale width} \times \text{Scaling factor} = 1.75 \times 1.3636363636363635 \approx 2.39 \text{ feet} \][/tex]
4. Determine the radius of the area impacted by the door swing:
Since the door swings open and forms a circular area, the radius of this area is equivalent to the actual door width. However, we must consider a conservative assumption for the radius, as typical doors may be around 2.5 feet wide or less. Here, the actual calculated door width, 2.39 feet, will be used.
5. Calculate the area where furniture cannot be placed:
The area where furniture cannot be placed due to the door swing is approximately a semi-circle with the radius calculated in the previous step.
[tex]\[ \text{Area without furniture} = \frac{1}{2} \pi \times (\text{Radius})^2 = \frac{1}{2} \pi \times (2.3863636363636362)^2 \][/tex]
[tex]\[ \text{Area without furniture} \approx 8.95 \text{ square feet} \][/tex]
6. Estimate the area:
The most appropriate choice in the given options is:
[tex]\[ \boxed{9 \text{ ft}^2} \][/tex]
Thus, the best estimate of the area in Jessica's room where furniture cannot be placed because of the door swing is approximately 9 square feet.
