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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
If the marginal cost of maufacturing a certain item is given by $\text{C}(\text{x})=\frac{\text{dC}}{\text{dx}}=2+0.15\text{x}$. Find the total cost function C(x), given that C(0) = 100.
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Answer

Given, $\text{C'}(\text{x})=\frac{\text{dC}}{\text{dx}}=2+0.15\text{x}$
$\text{dC}=(2+0.15\text{x})\text{dx}$
$\int \text{dC}=\int(2+0.15\text{x})\text{dx}$
$\text{C}=2\text{x}+\frac{0.15\text{x}^{2}}{2}+\lambda\ ...(\text{i})$
Given, C = 100 when x = 0, so
$100=0+0+\lambda$
$\lambda=100$
Put the value of eq. (i)
$\text{C}(\text{x})=2\text{x}+\frac{0.15\text{x}^{2}}{2}+100$
$\text{C}(\text{x})=2\text{x}+0.075\text{x}^{2}+100$

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