A train travels 400 km between Tokyo and Osaka.
Renee searches the Internet and finds that the average speed of the Nozomi train is greater than that of t
Hikari train by , and the faster train reduces the travelling time by 30 minutes.
Find the average speed of the Nozomi train based on Renee’s research.

Ask by Harris Hilton. in Singapore

Jan 22,2025

Question

A train travels 400 km between Tokyo and Osaka.
Renee searches the Internet and finds that the average speed of the Nozomi train is greater than that of t
Hikari train by , and the faster train reduces the travelling time by 30 minutes.
Find the average speed of the Nozomi train based on Renee’s research.

Ask by Harris Hilton. in Singapore

Jan 22,2025

UpStudy AI · Accepted Answer

To determine the average speed of the Nozomi train based on Renee's research, let's break down the problem step by step.

### Given:
- **Distance between Tokyo and Osaka:** 400 km
- **Difference in speed between Nozomi and Hikari trains:** 30 km/h
- **Time saved by Nozomi train:** 30 minutes (0.5 hours)

### Let's Define:
- **Speed of Hikari train:** $$ x $$ km/h
- **Speed of Nozomi train:** $$ x + 30 $$ km/h

### Time Calculation:
1. **Time taken by Hikari train:**
   $$
   \text{Time}_{\text{Hikari}} = \frac{400}{x} \quad \text{hours}
   $$
2. **Time taken by Nozomi train:**
   $$
   \text{Time}_{\text{Nozomi}} = \frac{400}{x + 30} \quad \text{hours}
   $$
3. **Difference in time:**
   $$
   \frac{400}{x} - \frac{400}{x + 30} = 0.5 \quad \text{hours}
   $$

### Solving the Equation:
$$
\frac{400}{x} - \frac{400}{x + 30} = 0.5
$$
Multiply both sides by $$ x(x + 30) $$ to eliminate the denominators:
$$
400(x + 30) - 400x = 0.5x(x + 30)
$$
Simplify:
$$
400x + 12,000 - 400x = 0.5x^2 + 15x
$$
$$
12,000 = 0.5x^2 + 15x
$$
Multiply by 2 to eliminate the decimal:
$$
24,000 = x^2 + 30x
$$
Rearrange into standard quadratic form:
$$
x^2 + 30x - 24,000 = 0
$$

### Using the Quadratic Formula:
$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$
Where $$ a = 1 $$, $$ b = 30 $$, and $$ c = -24,000 $$:
$$
x = \frac{-30 \pm \sqrt{30^2 - 4 \times 1 \times (-24,000)}}{2}
$$
$$
x = \frac{-30 \pm \sqrt{900 + 96,000}}{2}
$$
$$
x = \frac{-30 \pm \sqrt{96,900}}{2}
$$
$$
x = \frac{-30 \pm 311.45}{2}
$$
Ignoring the negative solution (since speed cannot be negative):
$$
x \approx \frac{281.45}{2} \approx 140.73 \quad \text{km/h}
$$

### Therefore, the speed of the Nozomi train is:
$$
x + 30 \approx 140.73 + 30 = 170.73 \quad \text{km/h}
$$

**Rounding to the nearest whole number:**
$$
\approx 171 \quad \text{km/h}
$$

**Answer:**  
The Nozomi train travels at an average speed of approximately 171 km h⁻¹.
The Nozomi train travels at an average speed of approximately 171 km/h.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Algebra Questions