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Homework Help: Math (trigonometry)
Posted by Sammy on Monday, December 3, 2012 at 4:25pm.
From a boat on the water, the angle of elevation of the top of a cliff is 31°. From a point 300 m closer, the angle of elevation is 33°. Find the height of the cliff.
the answer should be 2411 m.
I had tan31° = h/x+(x-300)
tan33° = h/x-300 -> (x-300)tan33° = h.
I subbed the first one into the second:
tan31° = (x-300)tan33°/x+x-300
I'm not sure how to go from there or if what I have so far is even right.
tan 33 = h/x
tan 31 = h/(x+300)
so
h = x tan 33 = .649 x
then
tan 31 = .649 x/(x+300)
(x+300)(.601) = .649 x
x + 300 = 1.08 x
.08 x = 300
x = 3750
h = .649 x = 2433
Carry more significant figures than I did. When you do the 1.08 - 1 you lose a lot of accuracy
yes i want to do the things very much
NO
To solve this problem, you are on the right track with using trigonometric ratios. Let's break down the steps to find the height of the cliff.
1. Draw a diagram: Draw a right triangle representing the situation described in the problem. Label the top of the cliff as point C, the boat as point A, and the point 300 m closer as point B. The height of the cliff is represented by the vertical line segment from point C to the ground.
2. Identify the given information: We are given two angles of elevation - 31° from point A and 33° from point B - and the distance between point A and point B (300 m).
3. Set up the trigonometric equations:
Let x be the horizontal distance from point A to point C (the distance along the water).
Let h be the height of the cliff.
From triangle ABC:
tan(31°) = h / x
From triangle BCA:
tan(33°) = h / (x - 300)
4. Simplify the equations and eliminate h:
Rearranging the first equation, we get:
h = x * tan(31°)
Substituting this value of h into the second equation, we get:
tan(33°) = (x * tan(31°)) / (x - 300)
5. Solve for x:
Multiply both sides of the equation by (x - 300) to eliminate the denominator:
tan(33°) * (x - 300) = x * tan(31°)
Expand and rearrange the equation:
x * tan(33°) - 300 * tan(33°) = x * tan(31°)
Move the x terms to one side:
x * (tan(33°) - tan(31°)) = 300 * tan(33°)
Divide both sides by (tan(33°) - tan(31°)):
x = (300 * tan(33°)) / (tan(33°) - tan(31°))
6. Calculate the value of x:
Use the given values of the angles (31° and 33°) and substitute them into the equation to calculate x:
x = (300 * tan(33°)) / (tan(33°) - tan(31°))
7. Calculate the height of the cliff:
Substitute the value of x into either of the original equations to find the height, h:
h = x * tan(31°)
Calculate h using the calculated value of x and the angle of elevation of 31°.
The answer should be approximately 2411 m.
Remember to double-check your calculations and make sure your calculator is set to the correct angular mode (degrees in this case).