Let's solve each question step by step.
1. Rectangle Length and Width
Given:
- The length \( L \) of the rectangle is twice its width \( W \): \[ L = 2W \]
- The perimeter \( P \) of the rectangle is 72 meters: \[ P = 2(L + W) \]
Now we can set up the equation using the formula for perimeter: \[ 72 = 2(L + W) \] Dividing both sides by 2: \[ 36 = L + W \]
Now substitute \( L = 2W \) into the perimeter equation: \[ 36 = 2W + W \] \[ 36 = 3W \] Now, solve for \( W \): \[ W = \frac{36}{3} = 12 \text{ meters} \]
Now substitute the width back to find the length: \[ L = 2W = 2 \times 12 = 24 \text{ meters} \]
The dimensions of the rectangle are:
- Width = 12 meters
- Length = 24 meters
2. Slope of the Line
To find the slope \( m \) of the line passing through points \( A(6, 8) \) and \( B(10, 5) \), we use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) = (6, 8) \) and \( (x_2, y_2) = (10, 5) \).
Substituting in the values: \[ m = \frac{5 - 8}{10 - 6} = \frac{-3}{4} \]
The slope of the line is: \[ -\frac{3}{4} \]
3. Finding the Numbers
Let's define the two numbers:
- Let the first number be \( x \).
- The second number, which exceeds the first by 9, is \( x + 9 \).
According to the problem, the sum of the two numbers is 25: \[ x + (x + 9) = 25 \] Combining like terms: \[ 2x + 9 = 25 \] Subtracting 9 from both sides: \[ 2x = 16 \] Now, divide by 2: \[ x = 8 \]
Thus, the first number is \( 8 \) and the second number, being \( x + 9 \), is: \[ 8 + 9 = 17 \]
The two numbers are:
- First Number = 8
- Second Number = 17
Summary of Answers
- Length = 24 meters, Width = 12 meters.
- Slope = \(-\frac{3}{4}\).
- The two numbers are 8 and 17.