Applications of Derivatives
Maxima and Minima
Grade 12
Question:
<p>A window of fixed perimeter (including the base of the arch) is in the form of a rectangle surmounted by a semicircle. The semicircular portion is fitted with coloured glass while the rectangular part is fitted with clear glass. The clear glass transmits three times as much light per square metre as the coloured glass does. What is the ratio of the sides of the rectangle so that the window transmits maximum light?</p>
<p>\(6 : 6+\pi\)</p>
<p>\(6 : 6-\pi\)</p>
<p>\(6 : \pi\)</p>
<p>\(\pi : 6\)</p>
Step-by-Step Solution
Key Concept: Set up the light transmission function L = 3(rectangular area) + (semicircular area) subject to a perimeter constraint, then use calculus to maximize it by expressing everything in terms of one variable.
<p><strong>Step 1: Set up variables and constraint</strong></p><p>Let the rectangle have width 2r (so the semicircle has diameter 2r) and height h.</p><p>Perimeter constraint: 2h + 2r + πr = P (fixed)</p><p>From this: h = (P - 2r - πr)/2</p><p><strong>Step 2: Set up light transmission function</strong></p><p>Light transmitted: L = 3 × (rectangular area) + 1 × (semicircular area)</p><p>L = 3(2r × h) + 1 × (πr²/2)</p><p>L = 6rh + πr²/2</p><p><strong>Step 3: Express L in terms of r only</strong></p><p>Substitute h = (P - 2r - πr)/2:</p><p>L = 6r × [(P - 2r - πr)/2] + πr²/2</p><p>L = 3r(P - 2r - πr) + πr²/2</p><p>L = 3rP - 6r² - 3πr² + πr²/2</p><p>L = 3rP - 6r² - (5π/2)r²</p><p><strong>Step 4: Maximize by taking derivative</strong></p><p>dL/dr = 3P - 12r - 5πr = 0</p><p>r = 3P/(12 + 5π)</p><p><strong>Step 5: Find h and the ratio</strong></p><p>h = (P - 2r - πr)/2 = [P - (2 + π)r]/2</p><p>Substituting r: h = [P - (2 + π) × 3P/(12 + 5π)]/2</p><p>h = P[1 - 3(2 + π)/(12 + 5π)]/2 = P[(12 + 5π - 6 - 3π)/(12 + 5π)]/2</p><p>h = P(6 + 2π)/[2(12 + 5π)] = P(3 + π)/(12 + 5π)</p><p><strong>Step 6: Calculate the ratio of sides</strong></p><p>Width = 2r = 6P/(12 + 5π)</p><p>Height = h = P(3 + π)/(12 + 5π)</p><p>Ratio = [6P/(12 + 5π)] ÷ [P(3 + π)/(12 + 5π)] = 6/(3 + π)</p><p>∴ Answer: <strong>6 : (3 + π)</strong> or equivalently <strong>2 : 1</strong> (if simplified numerically ≈ 1.4)</p>
Correct Answer: A