A-Level Mathematics Question Analyses Topic: Mathematics - Short Questions
Exam Questions:
1) The total cost C(x) associated with the production of x units of an item is given by C(x)=0.05x3−0.02x2+30x+5000. Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output.
2) A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light trough the whole opening.
Answer:
For A-Level Mathematics, you should know:
1) The instantaneous rate of change of the total cost, or dC/dx, is what is commonly referred to as the marginal cost.
dC/dx => 3 x 0.05 x x2 - 2 x 0.02 x x + 30 = 0.15x2 + 0.04x + 30
Then, we know x = 3 as we have to find the marginal cost when 3 units are produced.
=> 1.35 - 0.12 + 30 = 31.23
2) Let a, b be the length and breadth of the rectangle respectively.
Perimeter of window: 2b + a + πa/2 = 2b + a(1 + π/2) = 10 => b =5 - a/2(1 + π/2)
Area of window: A = ab + πa2/8 = 5a - a2/2(1 + 3π/8)
After differentiating A and equating it to zero, a = 5 / (1 + 3π/8)
Hence, for the window to allow maximum light, the dimensions of the opening should be a = 5 / (1 + 3π/8).
Work hard for your A-Level Mathematics examination!
End of analysis. Great!
