
Homework:9.1
Question 4, 9.1.31
Part 1 of 5
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Part 1
A state trooper is hidden 30 feet from a highway. One second after a truck​ passes, the angle theta between the highway and the line of observation from the patrol car to the truck is measured. See the illustration. Complete parts ​(a)dash​(c).
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Part 1
​(a) If the angle measures 18degrees​, how fast is the truck​ traveling? Express the answer in feet per second and in miles per hour.
If the angle measures 18degrees​, the speed of the truck is 
  
enter your response here feet per second.
​(Do not round until the final answer. Then round to two decimal places as​ needed.)
If the angle measures 18, the speed of the truck is(Do not round until the final answer. Then round to two decimal places as needed.)
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This problem involves related rates, which is a typical topic in calculus. The scenario describes a truck passing by a patrol car hidden 30 feet away from a highway, and after 1 second, the angle between the highway and the line of sight from the car to the truck is measured. We are given that the angle \(\theta\) is 18 degrees, and we are asked to find how fast the truck is traveling.

### Step 1: Set up the relationship using trigonometry.
Let:
- \(x(t)\) be the distance between the truck and the point on the highway nearest to the patrol car, measured in feet.
- \(\theta\) be the angle between the highway and the line of observation from the car to the truck.
- The patrol car is 30 feet away from the highway.

We can use the tangent function, as it relates the angle to the opposite side (30 feet, the distance from the highway) and the adjacent side (\(x(t)\), the distance of the truck from the nearest point).

The relation between \(\theta\) and \(x(t)\) is:
\[
\tan(\theta) = \frac{30}{x(t)}
\]

### Step 2: Differentiate with respect to time \(t\).
We differentiate both sides of this equation with respect to time \(t\) to find the related rate.

\[
\frac{d}{dt}[\tan(\theta)] = \frac{d}{dt}\left[\frac{30}{x(t)}\right]
\]

Using the chain rule:
\[
\sec^2(\theta) \cdot \frac{d\theta}{dt} = -\frac{30}{x(t)^2} \cdot \frac{dx}{dt}
\]

We are asked to find \(\frac{dx}{dt}\), which is the speed of the truck.

### Step 3: Plug in the given values.
We know:
- \(\theta = 18^\circ\)
- The truck passes after 1 second, so \(t = 1\).
- The patrol car is 30 feet from the highway.

We are given the angle and need to solve for \(\frac{dx}{dt}\). First, calculate \(\sec^2(18^\circ)\). Using a calculator:
\[
\sec(18^\circ) = \frac{1}{\cos(18^\circ)} \approx 1.051
\]
\[
\sec^2(18^\circ) \approx (1.051)^2 = 1.104
\]

Next, we substitute these values into our equation:

\[
1.104 \cdot \frac{d\theta}{dt} = -\frac{30}{x(t)^2} \cdot \frac{dx}{dt}
\]

We need to find \(\frac{dx}{dt}\), which is the speed of the truck.

### Step 4: Solve for \( \frac{dx}{dt} \).
At this point, we need more information about the rate at which the angle \(\theta\) is changing or the value of \(x(t)\) (the distance of the truck from the patrol car). If this information is given in later parts of the question or can be calculated from additional data, we can proceed.

Would you like to clarify or provide more details, so we can continue solving this? Let me know if you'd like me to proceed with the given information or guide you further.

---

### Related Questions:
1. How is the tangent function used to relate angles and distances in this scenario?
2. What is the significance of using the chain rule in related rates problems?
3. How does the angle of observation affect the speed calculation?
4. Why do we use feet per second and miles per hour as units in speed calculations?
5. How can we differentiate inverse trigonometric functions in related rates?

**Tip:** Always start by setting up a geometric or trigonometric relationship before differentiating in related rates problems.

A state trooper is hidden 30 feet from a highway. One second after a truck passes, the angle theta between the highway and the line of observation from the patrol car to the truck is measured. If the angle measures 18 degrees, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.

Solve a related rates problem where a state trooper measures the angle of a truck passing by a highway to determine its speed. This problem involves trigonometric functions and the application of the chain rule in calculus, suitable for high school calculus students.

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