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A short class with Sarah and her students. The other ways are under More.
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Today, we're discussing how to find the height of a tower using the lengths of its shadow when the sun is at different angles. Can anyone tell me what key relationship we're going to use?
Is it the tangent function?
Exactly! The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, the height of the tower is the opposite side. Can someone remind me what the angles of elevation are in our example?
They are 30 degrees and 60 degrees!
Right! Now, let’s break down our first equation. For 60 degrees, we can say tan(60) = height over shadow length. If we call the height 'h' and shadow when sun is at 60 degrees 'x', what does that give us?
It gives us h/x = √3 or h = x√3!
Perfect! Now, let's see how we can apply this with the second angle of elevation.
Now, turning to the angle of 30 degrees, we'll create another equation. Can someone tell me how the relationship looks?
For the 30-degree elevation, we have tan(30) which is h divided by the length of the shadow which is '40 + x'.
Correct! What does tan(30) evaluate to?
That would be 1/√3!
So, our equation becomes 1/√3 = h/(40 + x). Now, let's rewrite this in terms of h. What do we get?
h = (40 + x)/√3!
Great work! Now we can substitute our h value back to set up a relationship between x and the constants.
Let’s substitute h = x√3 into our other equation. What does that lead us to?
We’ll get √3x = (40 + x)/√3!
Exactly! Multiplying through by √3 gives us 3x = 40 + x. What does this simplify to?
So, we have 3x - x = 40 which means 2x = 40. That gives x = 20!
Excellent work! And with x = 20, can we find the height now?
Yes! h = 20√3, so that’s the height of the tower.
Well done! Let's summarize what we've learned today.
So, what are the key points from our lesson today?
We used trigonometric relationships to find the height of a tower based on shadow lengths.
And we made two equations using the tangent function.
Excellent! Always remember that the tangent function relates height to shadow lengths in these types of problems.
The vertical distance from the base of the tower to its top.
The distance from the base of the object (the tower) to the tip of its shadow.
The angles formed with the horizontal line to the line of sight to the top of the tower, measured at 30° and 60° in this case.
Functions such as tangent that relate angles to ratios of sides in triangles.
When the sun is high and shadows low, the tower's height will surely show.
Imagine a brave knight measuring tower heights with a magic shadow stick that grows as the sun bows lower in the sky.
Remember HATS: Height = Adjacent × Tangent of Angle for shadow problems.
SHAD = Shadow Height at Different angles.
Term
Height of Tower Formula
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Term
Tangent of 30 Degrees
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Term
Tangent of 60 Degrees
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The angle formed between the horizontal line and the line of sight to an object above the horizontal.
A trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the adjacent side.
A triangle where one angle is exactly 90 degrees.
The length of the projection of an object, like a tower, cast on the ground due to light, such as sunlight.
Relationships among the angles and sides of triangles.
This section presents a problem involving the relationship between the height of a tower and the lengths of its shadow at different sun altitudes.
In Example 5, a problem is solved that involves calculating the height of a tower based on the lengths of its shadow when the sun is at different altitudes. Two right triangles are used to set up equations based on the tangent function.
Example 5 deals with a geometric problem surrounding a tower's height and the length of its shadow under two different angles of elevation of the sun (30° and 60°). The solution involves using trigonometric principles, specifically the tangent function, to relate the height of the tower (AB) to the lengths of its shadows (BC and BD) at the specified angles. The relationship established shows that the shadow length at 30° is 40 meters longer than that at 60°, leading to the formulation of two equations based on right triangles formed in the diagram. The mathematical approach culminates in determining that the height of the tower is 20√3 meters.