If 5, 5r and `5r^(2)` are the lengths of the sides of a triangle, then r cannot be equal to

Options

A. `(3)/(2)`
B. `(3)/(4)`
C. `(5)/(4)`
D. `(7)/(4)`

Answer

To determine the value of \( r \) for which the lengths \( 5 \), \( 5r \), and \( 5r^2 \) cannot form a triangle, we will use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides must be greater than the length of the third side.

### Step 1: Set up the inequalities
We will consider the three inequalities that must hold for the sides of the triangle:
1. \( 5 + 5r > 5r^2 \)
2. \( 5 + 5r^2 > 5r \)
3. \( 5r + 5r^2 > 5 \)

### Step 2: Solve the first inequality
Starting with the first inequality:
\[ 5 + 5r > 5r^2 \]
Dividing through by 5:
\[ 1 + r > r^2 \]
Rearranging gives:
\[ r^2 - r - 1 < 0 \]
Now, we will find the roots of the quadratic equation \( r^2 - r - 1 = 0 \) using the quadratic formula:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2} \]
The roots are:
\[ r_1 = \frac{1 + \sqrt{5}}{2}, \quad r_2 = \frac{1 - \sqrt{5}}{2} \]
The inequality \( r^2 - r - 1 < 0 \) holds between the roots:
\[ \frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \]

### Step 3: Solve the second inequality
Now, we solve the second inequality:
\[ 5 + 5r^2 > 5r \]
Dividing through by 5:
\[ 1 + r^2 > r \]
Rearranging gives:
\[ r^2 - r + 1 > 0 \]
This quadratic has a discriminant of:
\[ (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
Since the discriminant is negative, \( r^2 - r + 1 \) is always positive for all \( r \).

### Step 4: Solve the third inequality
Now, we solve the third inequality:
\[ 5r + 5r^2 > 5 \]
Dividing through by 5:
\[ r + r^2 > 1 \]
Rearranging gives:
\[ r^2 + r - 1 > 0 \]
The roots of this quadratic are the same as before:
\[ r = \frac{-1 \pm \sqrt{5}}{2} \]
The inequality \( r^2 + r - 1 > 0 \) holds outside the roots:
\[ r < \frac{-1 - \sqrt{5}}{2} \quad \text{or} \quad r > \frac{-1 + \sqrt{5}}{2} \]

### Step 5: Combine the results
Now we combine the results from the three inequalities:
1. From the first inequality: \( \frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \)
2. From the second inequality: Always true.
3. From the third inequality: \( r < \frac{-1 - \sqrt{5}}{2} \) or \( r > \frac{-1 + \sqrt{5}}{2} \)

### Step 6: Determine the value of \( r \) that cannot be
The critical point is \( \frac{-1 + \sqrt{5}}{2} \) which is approximately \( 0.618 \). Thus, \( r \) cannot be equal to \( \frac{7}{4} \) (or \( 1.75 \)), as it falls outside the valid range determined by the first inequality.

### Conclusion
Thus, \( r \) cannot be equal to \( \frac{7}{4} \).