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A closed rectangular box, with a rectangle as base, has a length (2x) cm and width (x) cm - NSC Mathematics - Question 10 - 2017 - Paper 1
Question 10
A closed rectangular box, with a rectangle as base, has a length (2x) cm and width (x) cm. The total surface area (all 6 sides) is 243 cm². 10.1 Show that the heigh... show full transcript
Worked Solution & Example Answer:A closed rectangular box, with a rectangle as base, has a length (2x) cm and width (x) cm - NSC Mathematics - Question 10 - 2017 - Paper 1
Step 1
Show that the height, h, is equal to $ rac{81 - 4x^2}{3}$ cm.
Answer
To find the height, we first calculate the total surface area (TSA) of the box, given by the formula:
Substituting for length (l = 2x) and width (w = x):
This simplifies to:
Thus:
Rearranging gives:
Finally, solving for h:
h = rac{243 - 4x^2}{6}
This can be simplified to:
h = rac{81 - 4x^2}{3}
Step 2
Show that the volume of the box, in terms of x, is given by the formula: V = $ rac{81x - 4x^3}{3}$.
Answer
The volume V of the box is given by the formula:
Substituting in our values:
Substituting for h from the previous step:
V = 2x imes x imes rac{81 - 4x^2}{3}
This simplifies to:
V = rac{2x^2(81 - 4x^2)}{3}
Expanding gives:
V = rac{162x^2 - 8x^4}{3}
Thus, rewritten, we have:
V = rac{81x - 4x^3}{3}
Step 3
Calculate the value of x if the volume of the box is at a maximum.
Answer
To find the maximum volume, we need to take the derivative of the volume function V and set it to zero:
rac{dV}{dx} = rac{1}{3} (81 - 12x^2)
Setting the derivative equal to zero provides:
Solving for x gives:
Therefore:
x^2 = rac{81}{12}
Simplifying gives:
Taking the square root yields:
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