I cant seem to figure out the answer for this question:
One of the tallest totem poles in the world is located in Alert Bay, British Columbia. When the angle of elevation of the sun is 62°, the totem pole casts a shadow of 30m.
a) Suppose the totem is vertical. How tall is the totem pole, to the nearest tenth of a metre? Show calculation and state answer.
b) Suppose it was not quite vertical, so that it makes an angle of 89° with the ground. In this case, would your answer for its height be taller or shorter than your answer in a)? Justify your calculations.
(a) h/30 = tan62°
(b) the effective height h' of the pole can be found using
h'/h = sin89°
Now, as in (a), h'/30 = tan62°, or
h sin89°/30 = tan62°
To find the height of the totem pole, we can use trigonometry. Let's break down the problem step by step.
a) Suppose the totem is vertical. How tall is the totem pole, to the nearest tenth of a metre?
First, let's draw a diagram to visualize the situation. On the diagram, we can label the height of the totem pole as 'h' and the distance from the base of the totem pole to the tip of its shadow as 'x'.
Now, we can create a right triangle using the angle of elevation and the shadow of the totem pole. The angle of elevation is the angle between the ground and the line from the observer's eye to the top of the totem pole.
Next, we know that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the height of the totem pole 'h', and the adjacent side is the length of the shadow 'x'.
Using the tangent function, we can write the equation:
tan(62°) = h / x
We can rearrange this equation to solve for 'h':
h = x * tan(62°)
Now, substituting the given value of the shadow length into the equation (x = 30m) and plugging it into a calculator:
h = 30m * tan(62°)
≈ 30m * 1.902
≈ 57.06m
Therefore, the height of the totem pole, to the nearest tenth of a metre, is approximately 57.1m.
b) Suppose it was not quite vertical, so that it makes an angle of 89° with the ground. In this case, would your answer for its height be taller or shorter than your answer in a)? Justify your calculations.
If the totem pole were not quite vertical and made an angle of 89° with the ground, the height of the totem pole would be shorter.
In this case, we use the same method as in part a) to calculate the height of the totem pole. However, the angle of elevation has changed.
Again, drawing a diagram, we have an angle of 89° between the ground and the line from the observer's eye to the top of the totem pole. The shadow will still be 30m as given.
Using the same trigonometric relationship, the equation becomes:
tan(89°) = h / 30m
Since the tangent of 89° is undefined, we cannot directly solve for 'h' using this equation.
However, we can see that as the angle of elevation approaches 90°, the value of tangent approaches infinity. This means that for an angle very close to 89°, the height 'h' would be extremely large or "taller" than our answer in part a).
To further justify this, we can calculate the height using an angle slightly smaller than 89°. For example, using 88.9°:
h = 30m * tan(88.9°)
≈ 30m * 573.64
≈ 17209.2m
As you can see, the height is significantly larger than the height calculated in part a) which was 57.1m.
Therefore, the answer for the height of the totem pole in part b) would be "taller" than the answer in part a).