Mathematics

Mathematics

Problem

The Sun’s rays make a constant 60∘60^{\circ} angle of elevation with the horizontal. A hot-air balloon rises vertically at 4 m/s4\text{ m/s} from a point on level ground. When the balloon is 90 m90\text{ m} above the ground, how fast is the tip of its shadow moving along the ground? a) 43 m/s4\sqrt{3}\text{ m/s} b) 23 m/s2\sqrt{3}\text{ m/s} c) 43 m/s\dfrac{4}{\sqrt{3}}\text{ m/s} d) 4 m/s4\text{ m/s}
Intuitive explanation

This is a related rates problem where you need to connect the balloon's vertical motion to the horizontal motion of its shadow. The key insight is to set up a right triangle using the angle of elevation: the Sun's rays, the balloon's height, and the shadow's length are all related by that fixed 60°60° angle. Once you write that relationship, differentiate with respect to time to link the rate at which height changes to the rate at which the shadow's tip moves horizontally.

Intuitive explanation

This is a related rates problem where you need to connect the balloon's vertical motion to the horizontal motion of its shadow. The key insight is to set up a right triangle using the angle of elevation: the Sun's rays, the balloon's height, and the shadow's length are all related by that fixed 60°60° angle. Once you write that relationship, differentiate with respect to time to link the rate at which height changes to the rate at which the shadow's tip moves horizontally.

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