**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)=0.05x^{3}−0.02x^{2}+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 x^{2} - 2 x 0.02 x x + 30 = 0.15x^{2} + 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 + πa^{2}/8 = 5a - a^{2}/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!

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