Certainly! Let's solve the problem step by step.
Given:
- The length of a rectangular carpet is [tex]\( (x - 5) \)[/tex] meters.
- The area of the carpet is [tex]\( x^2 - 12x + 35 \)[/tex] square meters.
We need to find:
1. The breadth of the carpet.
2. The actual length, breadth, and area of the carpet when [tex]\( x = 10 \)[/tex] meters.
### Step 1: Find the Breadth of the Carpet
Given the length [tex]\( (x - 5) \)[/tex] and the area [tex]\( x^2 - 12x + 35 \)[/tex], we can use the formula for the area of a rectangle to find the breadth. The area of a rectangle is given by:
[tex]\[ \text{Area} = \text{Length} \times \text{Breadth} \][/tex]
So,
[tex]\[ x^2 - 12x + 35 = (x - 5) \times \text{Breadth} \][/tex]
To find the breadth, divide both sides by [tex]\( (x - 5) \)[/tex]:
[tex]\[ \text{Breadth} = \frac{x^2 - 12x + 35}{x - 5} \][/tex]
### Step 2: Simplify the Expression
We need to simplify the expression for breadth:
[tex]\[ \text{Breadth} = \frac{x^2 - 12x + 35}{x - 5} \][/tex]
Notice that the numerator can be factored. Factoring [tex]\( x^2 - 12x + 35 \)[/tex] gives:
[tex]\[ x^2 - 12x + 35 = (x - 7)(x - 5) \][/tex]
Thus,
[tex]\[ \text{Breadth} = \frac{(x - 7)(x - 5)}{x - 5} \][/tex]
Cancel out the common factor [tex]\( (x - 5) \)[/tex]:
[tex]\[ \text{Breadth} = x - 7 \][/tex]
### Step 3: Find Length, Breadth, and Area when [tex]\( x = 10 \)[/tex] meters
Now we substitute [tex]\( x = 10 \)[/tex] meters into the expressions for length, breadth, and area.
1. Length:
[tex]\[ \text{Length} = x - 5 \][/tex]
Substitute [tex]\( x = 10 \)[/tex]:
[tex]\[ \text{Length} = 10 - 5 = 5 \text{ meters} \][/tex]
2. Breadth:
[tex]\[ \text{Breadth} = x - 7 \][/tex]
Substitute [tex]\( x = 10 \)[/tex]:
[tex]\[ \text{Breadth} = 10 - 7 = 3 \text{ meters} \][/tex]
3. Area:
[tex]\[ \text{Area} = x^2 - 12x + 35 \][/tex]
Substitute [tex]\( x = 10 \)[/tex]:
[tex]\[ \text{Area} = 10^2 - 12(10) + 35 \][/tex]
[tex]\[ \text{Area} = 100 - 120 + 35 \][/tex]
[tex]\[ \text{Area} = 15 \text{ square meters} \][/tex]
### Summary
(i) The breadth of the carpet is [tex]\( x - 7 \)[/tex] meters.
(ii) When [tex]\( x = 10 \)[/tex] meters:
- The actual length of the carpet is 5 meters.
- The actual breadth of the carpet is 3 meters.
- The actual area of the carpet is 15 square meters.
