Question

On a spring day, the sun in New York rises at 6 a.m. and sets at 6 p.m. At what time would the Statue of Liberty and its shadow be of the same length? (The sun covers 180° across the sky from sunrise to sunset.)

• A. 8 a.m.
• B. 9 a.m.
• C. 10 a.m.
• D. 8:30 a.m.

Answer 1

The correct option is B 9 a.m.

To find the position of the sun when the shadow is the same length as the statue, we create a triangle ABC as shown:


In △ABC, as the statue is the same length as its shadow, we get:

$AC=BC$
$⟹\angle A=\angle B.....\left(isoscelestriangle\right)$

Also,

$\angle A+\angle B+\angle C=180°........\left(anglesumproperty\right)$
$⟹\angle A+\angle B+90°=180°$
$⟹\angle A+\angle A=90°$
$⟹\angle A=\angle B=45°$

Hence, the angle made by the sun is 45° when the shadow of the statue is as long as itself.


The angle that the sun goes through from sunrise to sunset is 180°.


$\text{Time of sunrise}=6a.m.$
$\text{Time of sunset}=6p.m.$
$\text{Duration of the day}=6p.m.–6a.m.=12hours$

Now, the Earth rotates at a constant speed. So, the time it takes to cover 1° will also be constant.

$\text{Angle covered in 12 hours}=180°$
$⟹\text{Time taken to cover 1°}=\frac{12}{180}=\frac{1}{15}hours$
$⟹\text{Time taken to cover 45°}=\frac{1}{15}\times 45=\frac{45}{15}=3hours$

Now,
$\text{Time of sunrise}=6a.m.$
$⟹\text{Time when the Sun is at 45°}=6a.m.+3hours=9a.m.$