A point is chosen at random inside a rectangle measuring 6 inches by 5 inches. What is the probability that the randomly selected point is at least one inch from the edge of the rectangle?

Options

A. `(2)/(3)`
B. `(1)/(3)`
C. `(1)/(4)`
D. `(2)/(5)`

Answer

To find the probability that a randomly selected point inside a rectangle measuring 6 inches by 5 inches is at least 1 inch from the edge of the rectangle, we can follow these steps:

### Step 1: Calculate the area of the larger rectangle.
The area of a rectangle is given by the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
For our rectangle:
\[ \text{Length} = 6 \text{ inches}, \quad \text{Breadth} = 5 \text{ inches} \]
Thus, the area of the larger rectangle is:
\[ \text{Area}_{\text{larger}} = 6 \times 5 = 30 \text{ square inches} \]

### Step 2: Determine the dimensions of the smaller rectangle.
To find the dimensions of the smaller rectangle that is at least 1 inch from the edge, we need to subtract 1 inch from each side of the larger rectangle:
- For the length:
\[ \text{New Length} = 6 - 2 \times 1 = 6 - 2 = 4 \text{ inches} \]
- For the breadth:
\[ \text{New Breadth} = 5 - 2 \times 1 = 5 - 2 = 3 \text{ inches} \]

### Step 3: Calculate the area of the smaller rectangle.
Now, we can calculate the area of the smaller rectangle:
\[ \text{Area}_{\text{smaller}} = \text{New Length} \times \text{New Breadth} = 4 \times 3 = 12 \text{ square inches} \]

### Step 4: Calculate the probability.
The probability \( P \) that a randomly selected point is at least 1 inch from the edge of the rectangle is given by the ratio of the area of the smaller rectangle to the area of the larger rectangle:
\[ P = \frac{\text{Area}_{\text{smaller}}}{\text{Area}_{\text{larger}}} = \frac{12}{30} \]

### Step 5: Simplify the probability.
Now we can simplify the fraction:
\[ P = \frac{12 \div 6}{30 \div 6} = \frac{2}{5} \]

Thus, the probability that the randomly selected point is at least 1 inch from the edge of the rectangle is:
\[ \boxed{\frac{2}{5}} \]