https://file.aiquickdraw.com/mathbot/file/1726992801127uy4dx44v.jpg

Let's solve the problem step by step as described in the image.

**Problem Description**:
A ship moves at a constant speed of 30 km/h parallel to a straight shoreline, 6 km away from it. The ship passes a lighthouse at noon.

**Tasks**:
1. Express the distance \( s \) between the lighthouse and the ship as a function of \( d \), the distance the ship has traveled since noon.
2. Express \( d \) as a function of \( t \), the time elapsed since noon.
3. Find \( f \circ g \) and interpret this function.

### Solution:

**(a) Expressing \( s \) as a function of \( d \):**
Since the ship travels parallel to the shoreline, the scenario forms a right triangle where:
- The distance \( d \) the ship has traveled from the lighthouse is one leg of the triangle.
- The constant distance from the shoreline (6 km) is the other leg.
- \( s \) is the hypotenuse.

By the Pythagorean theorem:
\[ s = \sqrt{d^2 + 6^2} = \sqrt{d^2 + 36} \]
Thus, \( f(d) = \sqrt{d^2 + 36} \).

**(b) Expressing \( d \) as a function of \( t \):**
Given the speed of the ship is 30 km/h, the distance traveled \( d \) as a function of time \( t \) (in hours) since noon is:
\[ d = 30t \]
Thus, \( g(t) = 30t \).

**(c) Finding \( f \circ g \) and interpreting this function:**
To find \( f \circ g \), substitute \( g(t) \) into \( f(d) \):
\[ (f \circ g)(t) = f(g(t)) = f(30t) = \sqrt{(30t)^2 + 36} = \sqrt{900t^2 + 36} \]
This function \( f \circ g(t) \) represents the distance \( s \) between the lighthouse and the ship as a function of time \( t \) since noon.

Would you like more details on any part of this explanation, or do you have any additional questions? Here are some related questions you could consider:

1. How would the equations change if the speed of the ship was different?
2. What happens to the distance \( s \) as \( t \) approaches infinity?
3. How would you find the rate at which the distance between the ship and the lighthouse is changing at a specific time \( t \)?
4. What graph represents the relationship between \( s \) and \( t \)?
5. How does the angle of view from the lighthouse towards the ship change over time?

**Tip**: Always check the units of your answer to ensure consistency throughout the problem.

A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon. (b) Express d as a function of t, the time elapsed since noon. (c) Find f ∘ g. What does this function represent?

Learn how to calculate the distance between a ship and a lighthouse as the ship travels parallel to the shore. This problem involves applying the Pythagorean theorem, linear functions, and function composition. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation