An industrial storage hopper, open at the top, is to be fashioned from sheet metal in the shape of a cuboid. The base of the hopper is rectangular with a length twice its width. Let the width of the base be $x$ metres and the height of the hopper be $h$ metres. The design requires the hopper to have a fixed volume of $m3120\text{ m}^3$ .

Show that the total surface area, $m2S\text{ m}^2$ , of the sheet metal used to construct the hopper is given by

2x^2 + \frac{360}{x}

Use calculus to find the value of $x$ for which $S$ has a stationary value, giving your answer to 3 significant figures.

Find $d2Sdx2\frac{d^2S}{dx^2}$ and use this to justify that the value of $x$ found in part (b) results in a minimum value for the required surface area.

Show that the total surface area, $m2S\text{ m}^2$ , of the sheet metal used to construct the hopper is given by

2x^2 + \frac{360}{x}

Use calculus to find the value of $x$ for which $S$ has a stationary value, giving your answer to 3 significant figures.

Find $d2Sdx2\frac{d^2S}{dx^2}$ and use this to justify that the value of $x$ found in part (b) results in a minimum value for the required surface area.

1.6 Differentiation

Question 65

Markscheme

1.6 Differentiation

Question 65

Markscheme

1.6 Differentiation

Question 65

1.6 Differentiation Questions

1.6 Differentiation

Question 65

1.6 Differentiation Questions