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 peri?meter? , what are the dimensions of the window that transmits the most light?
Solution: Step 1 Con sider l as the length of the rectangular sect and r as the radius of the semicircle. Let L is the total amount of light that passes through the windows of both the sections. Consider the following figure, Step 2: Here, total amount of light that passes through the windows of both the sections. = Amount of light passing through rectangular section Amount of light passing through semicircular section In general, the amount of light is given as A, where is the area of that section which is 2rl (from figure )and k is some constant factor. Therefore, Amount of light through rectangular section is k(2rl) It is given as through semicircular section only half of the light gets passed through. Amount of lig ht throug h semicircular section is given as k A, where A is the area of that semicircular section which is r2 (from figure )and k is some constant factor. 2 2 is (k r )= kr2 2 2 4 Therefore, The total amount of light, L = Amount of light through semicircular section + Amount of light through rectangular section 2 L = kr +k(2rl) 4 Consider perimeter of semicircle is r and perimeter of rectangle is 2(l+r). Let P be be the sum of the perimeter of the semicircle and the rectangle. Therefore, Substitute l into L L = kr2+k(2rl) 4 Find the critical point, by equating the derivative to zero. 2P r = (3+8) And It is clearly seen that the second derivative is negative and therefore the amount of light is maximized at this critical point.