Answer :
To solve these problems effectively, we will use the established scale [tex]\(\frac{1}{2}\)[/tex] inch [tex]\(=10\)[/tex] feet. This means that 1 inch on the blueprints corresponds to [tex]\(20\)[/tex] feet in actual length. We'll apply this conversion factor to each of the questions.
### 1. Actual length of the living room whose distance is 1 inch on the blueprints
Since 1 inch on the blueprint corresponds to 20 feet in actual length:
[tex]\[ \text{Actual length} = 1 \text{ inch} \times 20 = 20 \text{ feet} \][/tex]
Thus, the actual length of the living room is 20 feet.
### 2. Actual width of the living room whose distance is [tex]\(\frac{3}{4}\)[/tex] inch on the blueprints
Apply the same conversion factor:
[tex]\[ \text{Actual width} = \frac{3}{4} \text{ inch} \times 20 = 15 \text{ feet} \][/tex]
Thus, the actual width of the living room is 15 feet.
### 3. Actual height of the living room whose distance is [tex]\(\frac{9}{20}\)[/tex] inch on the blueprints
Similarly, using the conversion factor:
[tex]\[ \text{Actual height} = \frac{9}{20} \text{ inch} \times 20 = 9 \text{ feet} \][/tex]
Thus, the actual height of the living room is 9 feet.
### 4. Length of the kitchen on the blueprints for an actual length of 18 feet
Here we need to reverse the conversion to find out how long the kitchen should be on the blueprints:
[tex]\[ \text{Blueprint length} = \frac{18 \text{ feet}}{20} = 0.9 \text{ inches} \][/tex]
Thus, the length of the kitchen on the blueprints should be 0.9 inches.
### 5. Dimensions on the blueprints of a bedroom that will be 18 feet by 16 feet when the house is built
For the length:
[tex]\[ \text{Blueprint length} = \frac{18 \text{ feet}}{20} = 0.9 \text{ inches} \][/tex]
For the width:
[tex]\[ \text{Blueprint width} = \frac{16 \text{ feet}}{20} = 0.8 \text{ inches} \][/tex]
Thus, the dimensions on the blueprints of the bedroom should be 0.9 inches by 0.8 inches.
### 6. Actual dimensions of the den with blueprint dimensions of 0.5 inch by 0.6 inch
For the length:
[tex]\[ \text{Actual length} = 0.5 \text{ inches} \times 20 = 10 \text{ feet} \][/tex]
For the width:
[tex]\[ \text{Actual width} = 0.6 \text{ inches} \times 20 = 12 \text{ feet} \][/tex]
Thus, the actual dimensions of the den should be 10 feet by 12 feet.
### 7. Diameter of an actual tire on the car given the model dimensions and scale
We need to use the scale of 1:24, where the model dimensions are in the numerator and actual dimensions in the denominator. The diameter of the model tire is [tex]\(\frac{1}{2}\)[/tex] inch. To find the actual tire diameter:
[tex]\[ \text{Actual tire diameter} = \frac{1}{2} \text{ inch} \times 24 = 12 \text{ inches} \][/tex]
Thus, the diameter of an actual tire on the car is 12 inches.
These solutions implement the necessary scale conversions to provide the correct actual or blueprint dimensions as required.
### 1. Actual length of the living room whose distance is 1 inch on the blueprints
Since 1 inch on the blueprint corresponds to 20 feet in actual length:
[tex]\[ \text{Actual length} = 1 \text{ inch} \times 20 = 20 \text{ feet} \][/tex]
Thus, the actual length of the living room is 20 feet.
### 2. Actual width of the living room whose distance is [tex]\(\frac{3}{4}\)[/tex] inch on the blueprints
Apply the same conversion factor:
[tex]\[ \text{Actual width} = \frac{3}{4} \text{ inch} \times 20 = 15 \text{ feet} \][/tex]
Thus, the actual width of the living room is 15 feet.
### 3. Actual height of the living room whose distance is [tex]\(\frac{9}{20}\)[/tex] inch on the blueprints
Similarly, using the conversion factor:
[tex]\[ \text{Actual height} = \frac{9}{20} \text{ inch} \times 20 = 9 \text{ feet} \][/tex]
Thus, the actual height of the living room is 9 feet.
### 4. Length of the kitchen on the blueprints for an actual length of 18 feet
Here we need to reverse the conversion to find out how long the kitchen should be on the blueprints:
[tex]\[ \text{Blueprint length} = \frac{18 \text{ feet}}{20} = 0.9 \text{ inches} \][/tex]
Thus, the length of the kitchen on the blueprints should be 0.9 inches.
### 5. Dimensions on the blueprints of a bedroom that will be 18 feet by 16 feet when the house is built
For the length:
[tex]\[ \text{Blueprint length} = \frac{18 \text{ feet}}{20} = 0.9 \text{ inches} \][/tex]
For the width:
[tex]\[ \text{Blueprint width} = \frac{16 \text{ feet}}{20} = 0.8 \text{ inches} \][/tex]
Thus, the dimensions on the blueprints of the bedroom should be 0.9 inches by 0.8 inches.
### 6. Actual dimensions of the den with blueprint dimensions of 0.5 inch by 0.6 inch
For the length:
[tex]\[ \text{Actual length} = 0.5 \text{ inches} \times 20 = 10 \text{ feet} \][/tex]
For the width:
[tex]\[ \text{Actual width} = 0.6 \text{ inches} \times 20 = 12 \text{ feet} \][/tex]
Thus, the actual dimensions of the den should be 10 feet by 12 feet.
### 7. Diameter of an actual tire on the car given the model dimensions and scale
We need to use the scale of 1:24, where the model dimensions are in the numerator and actual dimensions in the denominator. The diameter of the model tire is [tex]\(\frac{1}{2}\)[/tex] inch. To find the actual tire diameter:
[tex]\[ \text{Actual tire diameter} = \frac{1}{2} \text{ inch} \times 24 = 12 \text{ inches} \][/tex]
Thus, the diameter of an actual tire on the car is 12 inches.
These solutions implement the necessary scale conversions to provide the correct actual or blueprint dimensions as required.
