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 \( r \) that cannot be equal to when \( 5, 5r, \) and \( 5r^2 \) are the lengths of the sides of a triangle, we will use the triangle inequality theorem.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
The sides are given as \( a = 5 \), \( b = 5r \), and \( c = 5r^2 \).

2. **Apply the triangle inequality**:
The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side. We will set up the inequalities:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

3. **Set up the inequalities**:
- From \( 5 + 5r > 5r^2 \):
\[ 5 + 5r > 5r^2 \implies 5r^2 - 5r - 5 < 0 \implies r^2 - r - 1 < 0 \]
- From \( 5 + 5r^2 > 5r \):
\[ 5 + 5r^2 > 5r \implies 5r^2 - 5r + 5 > 0 \]
- From \( 5r + 5r^2 > 5 \):
\[ 5r + 5r^2 > 5 \implies 5r^2 + 5r - 5 > 0 \]

4. **Solve the first inequality**:
We focus on the first inequality \( r^2 - r - 1 < 0 \). To find the roots, we use 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:
\[ r_1 = \frac{1 - \sqrt{5}}{2}, \quad r_2 = \frac{1 + \sqrt{5}}{2} \]

5. **Determine the intervals**:
The inequality \( r^2 - r - 1 < 0 \) holds between the roots:
\[ \frac{1 - \sqrt{5}}{2} < r < \frac{1 + \sqrt{5}}{2} \]

6. **Calculate the numerical values**:
- \( \frac{1 - \sqrt{5}}{2} \approx -0.618 \)
- \( \frac{1 + \sqrt{5}}{2} \approx 1.618 \)

7. **Analyze the second inequality**:
The second inequality \( 5r^2 - 5r + 5 > 0 \) is always true since the discriminant \( (-5)^2 - 4 \cdot 5 \cdot 5 < 0 \).

8. **Analyze the third inequality**:
The third inequality \( 5r^2 + 5r - 5 > 0 \) can be solved similarly:
\[ r^2 + r - 1 > 0 \]
The roots are:
\[ r = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \]
The inequality holds outside the roots:
\[ r < \frac{-1 - \sqrt{5}}{2} \quad \text{or} \quad r > \frac{-1 + \sqrt{5}}{2} \]

9. **Combine intervals**:
The value of \( r \) must satisfy both conditions. The critical point to note is that \( r \) cannot equal \( \frac{7}{4} = 1.75 \), which is outside the range of the first inequality.

### Conclusion:
Thus, the value of \( r \) that cannot be equal to is \( \frac{7}{4} \).