Final answer:
To determine the distance of the ship from the foot of the cliff, we use trigonometric principles with the given angles of elevation and the 30 feet the ship sailed closer. By creating two equations using the tangent function and solving for the initial distance 'x', we could find the nearest whole number distance to the cliff.
Step-by-step explanation:
Finding the Distance From a Ship to a Cliff
To find the distance that the ship is from the foot of the cliff, we will use trigonometric functions involving the angles of elevation and the distance the ship sailed toward the cliff. Let's denote the initial distance of the ship from the base of the cliff as 'x' and the height of the cliff as 'h'.
Using the first angle of elevation (20 degrees), we can set up our first equation using the tangent function:
tan(20°) = h/x
Next, after the ship has sailed 30 feet closer, the distance from the base of the cliff is (x - 30) feet. Using the second angle of elevation (35 degrees), we set up a second equation:
tan(35°) = h/(x - 30)
Now we have two equations with two unknowns:
tan(35°) = h/(x - 30)
Solving these equations simultaneously will give us the values of 'h' and 'x'. However, we are only interested in finding 'x', the initial distance from the cliff.
By dividing one equation by the other, we can eliminate the 'h' and solve for 'x':
tan(35°)/tan(20°) = x/(x - 30)
After finding the value of 'x', we can round it to the nearest foot to answer the student's question.
Please note: To solve this problem, a student would typically use a calculator for finding the numerical values of the tangent function and to solve the equation for 'x'.