Question

a 32 - m tall building casts a shadow. the distance from the top of the building to the tip of the shadow is 36 m. find the length of the shadow. if necessary, round your answer to the nearest tenth.

Explanation:

Step1: Identify the right triangle

The building, its shadow, and the line from the top of the building to the tip of the shadow form a right triangle. Let the length of the shadow be \( x \) meters. The height of the building (32 m) is one leg, the shadow length (\( x \)) is the other leg, and the distance from the top of the building to the tip of the shadow (36 m) is the hypotenuse.

Step2: Apply the Pythagorean theorem

The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \), \( a^2 + b^2 = c^2 \). Here, \( a = 32 \), \( c = 36 \), and \( b = x \). So we have:

$$ 32^2 + x^2 = 36^2 $$

Step3: Solve for \( x^2 \)

First, calculate \( 32^2 \) and \( 36^2 \):
\( 32^2 = 1024 \) and \( 36^2 = 1296 \).
Substitute these values into the equation:

$$ 1024 + x^2 = 1296 $$

Subtract 1024 from both sides:

$$ x^2 = 1296 - 1024 = 272 $$

Step4: Solve for \( x \)

Take the square root of both sides:

$$ x = \sqrt{272} $$

Simplify \( \sqrt{272} \). We know that \( 272 = 16 \times 17 \), so \( \sqrt{272} = \sqrt{16 \times 17} = 4\sqrt{17} \approx 4 \times 4.123 \approx 16.5 \) (rounded to the nearest tenth).

Answer:

16.5