The angle of elevation from a boat to the top of a cliff is 42 degrees. When the boat moves 15 feet closer to the cliff, the angle of elevation increases to 54 degrees. How tall is the cliff?
draw a diagram. Review the cotangent function. Then you will see that if the cliff has height h feet,
h cot42° - h cot54° = 15
To find the height of the cliff, we can use trigonometry and set up a right triangle. Let's label the distance from the original position of the boat to the base of the cliff as "x" and the height of the cliff as "h".
In the original position, the angle of elevation is 42 degrees. This forms a triangle with the opposite side as "h" and the adjacent side as "x".
From trigonometry, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side. So we have:
tan(42°) = h / x
Similarly, when the boat moves 15 feet closer, the angle of elevation becomes 54 degrees. The new triangle forms with the adjacent side as "x - 15" (since the boat moved closer), and the opposite side is still "h". So we have:
tan(54°) = h / (x - 15)
Now we have a system of two equations with two unknowns (h and x). We can solve this system of equations to find the height of the cliff.
First, let's solve for x in terms of h from the first equation:
x = h / tan(42°)
Substitute this expression for x into the second equation:
tan(54°) = h / (h / tan(42°) - 15)
Now we can solve for h by isolating it on one side of the equation.
tan(54°) = h / (h / tan(42°) - 15)
Cross-multiply:
tan(54°) * (h / tan(42°) - 15) = h
Simplify:
(h * tan(54°)) / tan(42°) - tan(54°) * 15 = h
Multiply both sides by tan(42°) to get rid of the fraction:
h * tan(54°) - (tan(54°) * 15) * tan(42°) = h * tan(42°)
Simplify further:
h * [tan(54°) - (tan(54°) * 15) * tan(42°)] = h * tan(42°)
Divide both sides by (tan(54°) - (tan(54°) * 15) * tan(42°)) to solve for h:
h = h * tan(42°) / [tan(54°) - (tan(54°) * 15) * tan(42°)]
Simplify once again:
h = h / [1 - 15 * tan(54°) * tan(42°)]
Notice that the variable "h" appears on both sides of the equation. To find the specific value of h, we need to divide both sides of the equation by h to cancel it out.
So the equation becomes:
1 = 1 / [1 - 15 * tan(54°) * tan(42°)]
Now we can solve for the height of the cliff by evaluating the right side of the equation.
Using a scientific calculator or a trigonometric table, find the tangent values of 54° and 42°, then substitute these values into the equation to find the height of the cliff.