Applications of Derivatives
Rate of Change of Quantities
Grade 12
Question:
<p>At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production <em>P</em> w.r.t. additional number of workers <em>x</em> is given by \(\dfrac{dP}{dx} = 100 - 12\sqrt{x}\). If the firm employs 25 more workers, then the new level of production of items is</p>
<p>3000</p>
<p>3500</p>
<p>4500</p>
<p>2500</p>
Step-by-Step Solution
Key Concept: The new production level equals the current production plus the integral of the rate of change function over the interval of additional workers. You must integrate dP/dx from x=0 to x=25 and add it to the initial production of 2000.
<p><strong>Step 1:</strong> Set up the integral. The change in production when 25 more workers are employed is:</p><p>∆P = ∫₀²⁵ (100 - 12√x) dx</p><p><strong>Step 2:</strong> Integrate term by term:</p><p>∆P = [100x - 12·(x^(3/2))/(3/2)]₀²⁵</p><p>∆P = [100x - 8x^(3/2)]₀²⁵</p><p><strong>Step 3:</strong> Evaluate at the limits:</p><p>∆P = [100(25) - 8(25)^(3/2)] - [0]</p><p>∆P = 2500 - 8(125)</p><p>∆P = 2500 - 1000</p><p>∆P = 1500</p><p><strong>Step 4:</strong> Calculate new production level:</p><p>New Production = Initial Production + ∆P</p><p>New Production = 2000 + 1500 = 3500 items</p><p>∴ Answer: D (3500 items)</p>
Correct Answer: D