Mikala exercise in her gym by joggging on the treatmill at a average rate of 4 miles per hour and then pedaling on a stationary bicycle at an average rate of 8 miles per hour. In her workout , she jogs the equivalent of x miles and bicycles the equivalent of y miles. If MIkala works out for at least 45 minutes, which of the following is true?

Options

A. `(x)/(4)+(y)/(8)ge(3)/(4)`
B. `x+(y)/(4)ge(3)/(4)`
C. `4x+8yge45`
D. `(4)/(x)+(8)/(y)ge45`

Answer

To solve the problem, we need to determine the relationship between the distances Mikala jogs and bicycles, given her speeds and the total time she exercises.

### Step-by-Step Solution:

1. **Identify the Speeds and Distances**:
- Mikala jogs at a speed of 4 miles per hour for a distance of \( x \) miles.
- She bicycles at a speed of 8 miles per hour for a distance of \( y \) miles.

2. **Calculate Time for Each Activity**:
- The time taken to jog \( x \) miles is given by the formula:
\[ \text{Time for jogging} = \frac{\text{Distance}}{\text{Speed}} = \frac{x}{4} \text{ hours} \]
- The time taken to bicycle \( y \) miles is:
\[ \text{Time for cycling} = \frac{y}{8} \text{ hours} \]

3. **Total Time of Workout**:
- The total time Mikala works out is the sum of the time spent jogging and cycling:
\[ \text{Total Time} = \frac{x}{4} + \frac{y}{8} \text{ hours} \]

4. **Convert 45 Minutes to Hours**:
- Since the problem states that Mikala works out for at least 45 minutes, we convert this to hours:
\[ 45 \text{ minutes} = \frac{45}{60} = \frac{3}{4} \text{ hours} \]

5. **Set Up the Inequality**:
- According to the problem, the total workout time must be at least \( \frac{3}{4} \) hours:
\[ \frac{x}{4} + \frac{y}{8} \geq \frac{3}{4} \]

6. **Simplify the Inequality**:
- To eliminate the fractions, we can multiply the entire inequality by 8 (the least common multiple of the denominators):
\[ 8 \left(\frac{x}{4}\right) + 8 \left(\frac{y}{8}\right) \geq 8 \left(\frac{3}{4}\right) \]
- This simplifies to:
\[ 2x + y \geq 6 \]

7. **Identify the Correct Option**:
- The original inequality we derived is \( \frac{x}{4} + \frac{y}{8} \geq \frac{3}{4} \), which corresponds to option A:
\[ \frac{x}{4} + \frac{y}{8} \geq \frac{3}{4} \]

### Conclusion:
The correct answer is option A: \( \frac{x}{4} + \frac{y}{8} \geq \frac{3}{4} \).