Question

Light transmission A window consists of a rectangular pane of clear glass surmounted by a semicircular pane of tinted glass. The clear glass transmits twice as much light per unit of surface area as the tinted glass. Of all such windows with a fixed perimeter P, what are the dimensions of the window that transmits the most light?

Accepted Answer

Define the variables: Let the width of the rectangular pane be $$ w $$ and the height be $$ h . $$The radius of the semicircular pane is $$ r . $$The perimeter constraint is given by the equation $$ 2h + 2w + \pi r = P . $$Express the area of the window: The area of the rectangular pane is $$ A_{rect} = w 	imes h $$ and the area of the semicircular pane is $$ A_{semi} = \frac{1}{2} \pi r^2 . $$The total area is $$ A_{total} = wh + \frac{1}{2} \pi r^2 . $$Set up the light transmission function: Since the clear glass transmits twice as much light as the tinted glass, the light transmitted is $$ L = 2wh + \frac{1}{2} \pi r^2 . $$Use the perimeter constraint to express one variable in terms of the others: Solve $$ 2h + 2w + \pi r = P $$ for one of the variables, for example, $$ h = \frac{P - 2w - \pi r}{2} . $$Substitute this into the light transmission function. Optimize the light transmission function: Differentiate the light transmission function with respect to the remaining variables and set the derivatives to zero to find critical points. Use the second derivative test or other methods to determine which dimensions maximize the light transmission.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Optimization

Perimeter Constraint

Area and Light Transmission