# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2016 | Feb-Mar | (P1-9709/12) | Q#6

Hits: 389

Question

A vacuum flask (for keeping drinks hot) is modelled as a closed cylinder in which the internal radius  is r cm and the internal height is h cm. The volume of the flask is 1000 cm3. A flask is most efficient when the total internal surface area, A cm2, is a minimum.

i.       Show that ii.       Given that r can vary, find the value of r, correct to 1 decimal place, for which A has a stationary  value and verify that the flask is most efficient when r takes this value.

Solution

a.

We are given that vacuum flask is cylinder and;   Expression for surface area of a cylinder is with radius and height is; Therefore surface area of vacuum flask is; We are required to express surface area of vacuum flask independent of .

Expression for volume of a cylinder is with radius and height is; For the given case;    Substituting this expression for in above expression of surface area of vacuum flask;  i.

We are given that A (surface area of vacuum falsk) has a stationary.

A stationary value is the maximum or minimum value of a function. It can be obtained by equating  the derivative of the function with zero. This gives the x-coordinates of the stationary point. At this  point function has a stationary value.

From (ii) we have; Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is: Therefore, we need; Rule for differentiation of is:   Rule for differentiation of is:    Coordinates of stationary point on the curve can be found from the derivative of equation of the  curve by equating it with ZERO. This results in value of x-coordinate of the stationary point on the curve.

Therefore, value of for which has a stationary value can be found as follows;       Therefore, stationary value of occurs when We are given that flask is most efficient when the total internal surface area, A cm2, is a minimum.  We have found that flask has a stationary value of when .

If this value of is minimum then flask would be most efficient.

Once we have the coordinates of the stationary point of a curve, we can determine its  nature, whether minimum or maximum, by finding 2nd derivative of the curve.

Second derivative is the derivative of the derivative. If we have derivative of the curve as , then  expression for the second derivative of the curve is; We substitute of the stationary point in the expression of 2nd derivative of the curve and  evaluate it;

If or then stationary point (or its value) is minimum.

If or then stationary point (or its value) is maximum.

We have; Therefore;  Rule for differentiation of is:  Rule for differentiation of is:     Since has stationary value at ;  Since, at , the has a minimum value at this point and, therefore, flask will be most  efficient when .