Problem 1440

To solve this problem, we need to determine the speed Nick must maintain on the fourth lap to achieve an average speed of 10 mph over four laps. Let's break down the problem step-by-step.

1. Determine the total distance and total time required:
- Let the length of one lap be $d$ miles.
- Therefore, the total distance for four laps is $4d$ miles.
- To achieve an average speed of 10 mph over the total distance, the total time $T$ required is:
$$T = \frac{\text{Total Distance}}{\text{Average Speed}} = \frac{4d}{10} = \frac{2d}{5} \text{ hours}$$

2. Calculate the time taken for the first three laps:
- Nick runs the first three laps at a constant speed of 9 mph.
- The distance for three laps is $3d$ miles.
- The time $t_1$ taken for the first three laps is:
$$t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{3d}{9} = \frac{d}{3} \text{ hours}$$

3. Determine the remaining time for the fourth lap:
- The total time required for four laps is $\frac{2d}{5}$ hours.
- The time remaining for the fourth lap $t_2$ is:
$$t_2 = T - t_1 = \frac{2d}{5} - \frac{d}{3}$$
- To subtract these fractions, find a common denominator (15):
$$t_2 = \frac{2d}{5} - \frac{d}{3} = \frac{6d}{15} - \frac{5d}{15} = \frac{d}{15} \text{ hours}$$

4. Calculate the required speed for the fourth lap:
- The distance for the fourth lap is $d$ miles.
- The time available for the fourth lap is $\frac{d}{15}$ hours.
- The required speed $r$ for the fourth lap is:
$$r = \frac{\text{Distance}}{\text{Time}} = \frac{d}{\frac{d}{15}} = 15 \text{ mph}$$

Therefore, Nick needs to maintain a speed of 15 mph on the fourth lap to achieve his goal.

The final answer is $\boxed{15 \text{ mph}}$.

1. Define Variables and Equations:
- Let $d$ be the length of one lap.
- Let $t$ be the total time taken to complete the first three laps.
- Let $y$ be the time taken to complete the fourth lap.
- Let $x$ be the speed required for the fourth lap.

2. Calculate Time for First Three Laps:
- Given that Nick runs the first three laps at a speed of $9$ mph, we can write:
$$\frac{3d}{t} = 9 \implies t = \frac{3d}{9} = \frac{d}{3}$$

3. Set Up the Equation for the Total Average Speed:
- Nick's goal is to complete four laps at an average speed of $10$ mph. The total distance for four laps is $4d$, and the total time is $t + y$. Therefore:
$$\frac{4d}{t + y} = 10$$

4. **Substitute $t$ into the Equation:**
- From step 2, we know $t = \frac{d}{3}$. Substitute this into the equation:
$$\frac{4d}{\frac{d}{3} + y} = 10$$

5. **Solve for $y$:**
- Simplify the equation:
$$\frac{4d}{\frac{d}{3} + y} = 10 \implies 4d = 10 \left( \frac{d}{3} + y \right)$$
$$4d = \frac{10d}{3} + 10y$$
$$4d - \frac{10d}{3} = 10y$$
$$\frac{12d - 10d}{3} = 10y$$
$$\frac{2d}{3} = 10y$$
$$y = \frac{2d}{30} = \frac{d}{15}$$

6. **Calculate the Required Speed $x$:**
- The speed $x$ for the fourth lap is given by:
$$x = \frac{d}{y} = \frac{d}{\frac{d}{15}} = 15 \text{ mph}$$

Conclusion:
$$\boxed{15 \text{ mph}}$$