Math  /  Calculus

Question2 A medicine manufacturer produces pain relieving tablets. Each tablet is in the shape of a solid circular cylinder with base radius x mmx \mathrm{~mm} and height hh mm , whose surface area is A=2πx2+120xA=2 \pi x^{2}+\frac{120}{x} where AA is measured in mm2\mathrm{mm}^{2}. If the manufacturer wishes to minimize the surface area of each tablet, find the value of xx for which AA is a minimum, to 2 decimal places. A. 2.21 B. 2.12 C. 1.23 D. 1.32

Studdy Solution
Verify the minimum by using the second derivative test:
Differentiate A(x) A'(x) to find A(x) A''(x) :
A(x)=ddx(4πx120x2) A''(x) = \frac{d}{dx} \left( 4 \pi x - \frac{120}{x^2} \right) A(x)=4π+240x3 A''(x) = 4 \pi + \frac{240}{x^3}
Evaluate A(x) A''(x) at x=2.12 x = 2.12 :
Since A(x)>0 A''(x) > 0 , the critical point is a minimum.
The value of x x that minimizes the surface area is:
2.12 \boxed{2.12}

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