MHB How Do Shadows and Sun Angles Relate to Tree Height?

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The discussion addresses the relationship between shadows and sun angles in determining tree height and shadow length. A tree casting a 23-foot shadow at a 52-degree sun angle has a calculated height of approximately 29 feet. When the sun's angle is 38 degrees, the shadow length is computed to be around 37 feet, although it is noted that using a rounded height in this calculation could lead to inaccuracies. Participants emphasize the importance of using precise values in calculations to avoid errors in final results. Overall, the calculations highlight the mathematical principles connecting sun angles, shadow lengths, and tree heights.
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A tree casts a 23-foot shadow when the angle of elevation of the sun is 52 degrees.

(A) Find the height of the tree.

(B) Find the length of the shadow when the angle of elevation of the sun is 38 degrees.Part (A)

Let h = height of tree

tan(52°) = h/52

tan(52°)(23) = h

29.4386575404 = h

Rounding off to the nearest ones place, I get 29 feet.

The tree is 29 feet.

Part (B)

Let s = length of shadow

tan(38°) = 29/s

s = 29/tan(38°)

s = 37.1183073336

After rounding to the nearest unit, I get 37 feet.

The shadow is 37 feet.

Is this right?
 
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xyz_1965 said:
A tree casts a 23-foot shadow when the angle of elevation of the sun is 52 degrees.

(A) Find the height of the tree.

(B) Find the length of the shadow when the angle of elevation of the sun is 38 degrees.Part (A)

Let h = height of tree.

tan(52°) = h/52

$\color{red} \tan(52) =h/23$

tan(52°)(23) = h

29.4386575404 = h

Rounding off to the nearest ones place, I get 29 feet.

The tree is 29 feet.

Part (B)

Let s = length of shadow

tan(38°) = 29/s

$\color{red} \text{I wouldn’t use the rounded value of the height in subsequent calculations. Final shadow length is closer to 38 ft}$
s = 29/tan(38°)

s = 37.1183073336

After rounding to the nearest unit, I get 37 feet.

The shadow is 37 feet.

Is this right?

see above $\color{red}\text{comments}$ in the quote.
 
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skeeter said:
see above $\color{red}\text{comments}$ in the quote.

Thank you for correcting my typos.
 
xyz_1965 said:
Thank you for correcting my typos.
It is fine to round a final result, but using a rounded intermediate result for more calculations (in part B) is a bit more serious than a typo. It means that the final result of B will be "off".
 
Klaas van Aarsen said:
It is fine to round a final result, but using a rounded intermediate result for more calculations (in part B) is a bit more serious than a typo. It means that the final result of B will be "off".

I'll try to be careful in my rounding off.
 
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