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Let's break down the question related to the lifeguard problem depicted in the figure. The lifeguard is trying to reach the swimmer in the shortest time, moving first on land and then in water.

### Part (a) - Condition on \( x \) for the shortest time:
The lifeguard travels along the broken path \( LCS \).

A lifeguard rushes from L to rescue a swimmer at S, traveling along a path LCS (land and water) and aiming to minimize time. Given distances a, b, d, running speed v1, and swimming speed v2, determine conditions on x for minimum time, express conditions in terms of angles and speeds, and solve for x when v1 = v2.

Solve a lifeguard problem involving optimizing the path to minimize time. Learn to express conditions using angles and speeds, and find the optimal point on the shoreline for the fastest rescue.

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