Question Statement

A ship sails in a straight line. Its distance $d(t)$ (in nautical miles) from port is modeled by:

$d(t)=15t$

where $t$ is time in hours.

a. Find the speed of the ship.

b. Calculate the distance after 3 hours.

c. Explain the meaning of the slope.

---

Background and Explanation

This problem involves analyzing a linear distance-time relationship. You need to understand that the derivative of a position function yields velocity (speed), and that the slope of a linear function represents its constant rate of change.

---

Solution

Part (a): Finding the Speed

The speed of the ship is the instantaneous rate of change of distance with respect to time. Mathematically, this is the derivative of the distance function $d(t)$ with respect to $t$.

Given the distance function:

$d(t)=15t$

Differentiate with respect to $t$ using the constant multiple rule (since $\frac{d}{dt}(t)=1$):

$d^{\prime }(t)=\frac{d}{dt}(15t)=15\cdot \frac{d}{dt}(t)=15\cdot 1=15$

Therefore:

$\text{Speed of the ship}=15\text{ nautical miles per hour}$

Part (b): Calculating Distance After 3 Hours

To find the distance traveled after 3 hours, substitute $t=3$ into the original distance function.

Given:

$d(t)=15t$

Substitute $t=3$:

$d(3)=15\times 3$

$d(3)=45$

Therefore, the distance after 3 hours is 45 nautical miles.

Part (c): Meaning of the Slope

The slope of the distance-time function represents the rate of change of distance with respect to time.

In this specific problem:

• The function $d(t)=15t$ is linear with a constant slope of $15$
• This slope indicates that for every 1-hour increase in time, the distance increases by 15 nautical miles
• Therefore, the slope means that the ship moves at a constant speed of 15 nautical miles per hour

The constant slope (derivative) confirms that the ship maintains steady speed without acceleration or deceleration.

---

Key Formulas or Methods Used

• Distance function: $d(t)=15t$
• Velocity as derivative: $v=\frac{dd}{dt}=d^{\prime }(t)$
• Power rule/Constant multiple rule: $\frac{d}{dt}[kt]=k$ (where $k$ is a constant)
• Function evaluation: $d(t_1)=15t_1$ for specific time values
• Interpretation of slope: Slope = rate of change = speed in distance-time problems

---

Summary of Steps

1. Find speed: Differentiate $d(t)=15t$ with respect to $t$ to get $d^{\prime }(t)=15$, giving a constant speed of 15 nautical miles per hour.
2. Calculate distance: Substitute $t=3$ into $d(t)=15t$ to obtain $d(3)=15\times 3=45$ nautical miles.
3. Interpret slope: Recognize that the slope (15) represents the constant rate of change, meaning the ship travels at a steady speed of 15 nautical miles per hour.