The unit cost of a toy is $C when n pieces of the toy are produced, and C =k/square root of n When 100 pieces of the toy are produced, the unit cost is $50. (a) Express C in terms of n. (b) If the unit cost of the toy is $25, how many pieces of toys are produced? (c) If the unit cost of the toy decreases by 20%, by what percentage does the quantity of toys produced increase?
Question
The unit cost of a toy is 50.
(a) Express C in terms of n.
(b) If the unit cost of the toy is $25, how many pieces of toys are produced?
(c) If the unit cost of the toy decreases by 20%, by what percentage does the quantity of toys produced increase?
Solution
(a) The cost C is already expressed in terms of n as C = k/√n.
(b) To find the number of pieces produced when the unit cost is 50. So, we can set up the equation 50 = k/√100. Solving for k gives us k = 50 * √100 = 500. Now we can substitute C = 25 and k = 500 into the equation to get 25 = 500/√n. Solving for n gives us n = (500/25)^2 = 400.
(c) If the unit cost decreases by 20%, the new cost C' is C' = C - 0.2C = 0.8C. Substituting C = k/√n and C' = k/√n' into the equation gives us 0.8(k/√n) = k/√n'. Solving for n' gives us n' = n/(0.8)^2 = 1.25n. So the quantity of toys produced increases by 25%.
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