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# Light transmission A window consists of a rectangular pane ISBN: 9780321570567 2

## Solution for problem 52E Chapter 4.4

Calculus: Early Transcendentals | 1st Edition

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Problem 52E

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?

Step-by-Step Solution:

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.

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##### ISBN: 9780321570567

This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, edition: 1. The answer to “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?” is broken down into a number of easy to follow steps, and 58 words. This full solution covers the following key subjects: Glass, light, tinted, transmits, pane. This expansive textbook survival guide covers 85 chapters, and 5218 solutions. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9780321570567. Since the solution to 52E from 4.4 chapter was answered, more than 359 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 52E from chapter: 4.4 was answered by , our top Calculus solution expert on 03/03/17, 03:45PM.

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Light transmission A window consists of a rectangular pane

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