(Solved): A man launches his boat from point \( A \) on a bank of a straight river, \( 4 \mathrm{~km} \) wide ...

A man launches his boat from point \( A \) on a bank of a straight river, \( 4 \mathrm{~km} \) wide, and wants to reach point \( B \), \( 1 \mathrm{~km} \) downstream on the opposite bank, as quickly as possible (see the figure below). He could row his boat directly across river to point \( C \) and then run to \( B \), or he could row directly to \( B \), or he could row to some point \( D \) between \( C \) and \( B \) and run to \( B \). If he can row \( 6 \mathrm{~km} / \mathrm{h} \) and run \( 8 \mathrm{~km} / \mathrm{h} \), how far (in \( \mathrm{km} \) ) downstream from \( C \) should he land to reach \( B \) as soon possible? (We assume that the speed of the water is negligible compared to the speed at which the man rows. Hint: Thi question is based on EXAMPLE 4 in Section \( 4.7 \) of the textbook. However, for this question, the textbook has added a challenge which may require an unexpected solution. Look for \( |t| \) ) \( \chi \mathrm{km} \) from \( C \) Enhanced Feedback Please try again and draw a diagram. Keep in mind that \( t=\frac{d}{r} \), where \( t \) is time, \( d \) is the distance, and \( r \) is the rate. Let \( x \) be the distance between \( C \) and \( D \). Use the Pythagorean theorem to find the distance between \( A \) and \( D \). Write a function fo the time the man traveis from point \( A \) to \( D \) and then to \( B \). Use calculus to find \( x \) such that the time is minimized.