From a certain distance from the base of a Giant Sequoia tree a surveyor determines that the angle of elevation to the top of the tree is 47deg. The surveyor then walks 100 feet away from the tree and determines that the angle of elevation to the top of the tree is now 37deg. How tall is the tree? You may assume that the ground is level.
How exactly do I go about solving this and what is the correct answer so I can make sure I follow correctly?
as usual, draw a diagram. If the height is h, and the surveyor started out at a distance x from the tree,
h/x = tan47°
h/(x+100) = tan37°
equating the values for x,
h/tan47° = h/tan37° - 100
h/1.072 = h/.7535 - 100
h = 253.6 ft
Using law of sines, this is what I got:
x/sin37° = 100/sin10°
x = 346.6 ft
h/sin47° = 346.6/sin90°
h = 253.5 ft
Is this wrong...?
Nope. As usual, there is more than one way to do it. I picked my way to avoid actually having to find the value of x.
a building 60 feet high. from a distance at point A on the ground, the angle of elevation to the top of the building is 40 degree. from a little nearer at point B, the angle of elevation to the top of the building is 70 degree. What's the distance between point A and B?
To solve this problem, you can use trigonometry, specifically the tangent function, which relates the angle of elevation to the height and distance.
Step 1: Draw a diagram depicting the situation described in the problem. Label the base of the tree as point A, the top of the tree as point T, and the first position of the surveyor as point S1 (100 feet away). Also, label the height of the tree as 'h' and the distance from the base to the surveyor's first position as 'x'.
Step 2: From the diagram, you can see that the tangent of the angle of elevation at S1 is equal to the height of the tree divided by the distance from the base to S1. So, you have the equation:
tan(47°) = h / x
Step 3: Similarly, at the surveyor's second position, the tangent of the new angle of elevation is equal to the same height of the tree divided by the new distance from the base to the surveyor's position, which is now x + 100 feet. So, you have the equation:
tan(37°) = h / (x + 100)
Step 4: Now, you have a system of two equations with two unknowns (h and x). You can solve this system by substitution.
First, solve equation 1 for h:
h = x * tan(47°)
Then, substitute this expression for h into equation 2:
tan(37°) = (x * tan(47°)) / (x + 100)
Step 5: Now, you can solve for x. Multiply both sides of equation 2 by x + 100:
tan(37°) * (x + 100) = x * tan(47°)
Step 6: Expand and simplify:
x * tan(37°) + 100 * tan(37°) = x * tan(47°)
Step 7: Group the x terms on one side:
x * (tan(47°) - tan(37°)) = 100 * tan(37°)
Step 8: Solve for x:
x = (100 * tan(37°)) / (tan(47°) - tan(37°))
Step 9: Substitute this value of x back into the expression for h:
h = x * tan(47°)
Step 10: Calculate the value of h using the given values:
h ≈ x * tan(47°) ≈ (100 * tan(37°)) / (tan(47°) - tan(37°)) * tan(47°)
Step 11: Calculate this expression to find the height 'h' of the tree. In this case, the height of the tree is approximately 152.4 feet.
Therefore, the correct answer is that the tree is approximately 152.4 feet tall.