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A closed rectangular box has to be constructed as follows: - Dimensions: length (l), width (w) and height (h) - NSC Mathematics - Question 9 - 2020 - Paper 1
Question 9
A closed rectangular box has to be constructed as follows: - Dimensions: length (l), width (w) and height (h). - The length (l) of the base has to be 3 times its wi... show full transcript
Worked Solution & Example Answer:A closed rectangular box has to be constructed as follows: - Dimensions: length (l), width (w) and height (h) - NSC Mathematics - Question 9 - 2020 - Paper 1
Step 1
9.1 Show that the cost to construct the box can be calculated by: Cost=90w²+48wh
Answer
To derive the cost expression for the box, we need to calculate the total surface area. The dimensions are defined as follows:
- Given the relationships, we know:
- Length, l = 3w
- Volume, V = l × w × h = 5
From the volume equation:
a) We have:
Which simplifies to:
b) The total surface area (A) of the box is given by:
Substituting l:
The cost (C) is:
Substituting A:
C = 15(6w²) + 48wh = 90w² + 48wh\$$ Hence, we have shown the cost to construct the box can be expressed as C = 90w² + 48wh.Step 2
9.2 Determine the width of the box such that the cost to build the box is a minimum.
Answer
To find the width that minimizes the cost, we substitute h from part 9.1 into the cost function:
C(w) = 90w² + 48w\left(\frac{5}{3w²}\right)\$$ which simplifies to:C(w) = 90w² + 80w^{-1}$$
Next, we find the derivative of C with respect to w:
C'(w) = 180w - 80w^{-2}\$$ Setting the derivative to zero:180w - 80w^{-2} = 0$$ Multiplying through by w² gives:
180w³ - 80 = 0\$$ Thus:180w³ = 80$$
w³ = \frac{80}{180} = \frac{4}{9}\$$ Taking the cube root:w = \left(\frac{4}{9}\right)^{1/3} = \frac{w}{1.08}$$ The width is approximately:
w \approx 0.76\$$ This is the width of the box that minimizes the cost.