# Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2010 | Oct-Nov | (P1-9709/12) | Q#10

Question The diagram shows an open rectangular tank of height meters covered with a lid. The base of the tank has sides of length meters and meters and the lid is a rectangle with sides of length meters meters. When full the tank holds 4m3 of water. The material from which the tank is made is of negligible thickness. The external surface area of the tank together with the area of the top of the lid is m2.

i.
Express in terms of and hence show that .

ii.       Given that can vary, find the value of for which is a minimum, showing clearly that is a minimum and not a maximum.

Solution

i.

We are given that when full the tank holds 4m3 of water. Hence, volume of the tank is; Expression for the volume of the rectangular block is; For the given case;  Hence;     Expression for the surface area of the rectangle is;  For the given case, top of the tank is not included and a rectangle of different dimensions is serving as the top.

First we find surface area of open tank.      Now we find surface area of rectangular lid.

Expression for the area of the rectangle is;   Hence, surface area of tank with lid;  We have found that; Therefore;   ii.

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.

For the given case; Therefore, we need derivative of this function.

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to is: Therefore; Rule for differentiation of is:    Rule for differentiation of is:     Now we equate the derivative of the given function with ZERO.         At this point , area has a stationary value.

We can determine nature of stationary point (value), 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; For the given case; Therefore; Rule for differentiation of is:   Rule for differentiation of is: Rule for differentiation of is:     We find the value of second derivative at the stationary point i.e. ,  If or then stationary point (or its value) is minimum.

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

Hence, the stationary value of area is a maximum.