Optimisation problems KUS objectives BAT apply differentiation to
Optimisation problems • KUS objectives BAT apply differentiation to solve optimization problems BAT manipulate algebra to form equations that can be differentiated Starter: Simplify these expressions (to polynomials)
You need to be able to recognise practical problems that can be solved by using the idea of maxima and minima Whenever you see a question asking about the maximum value or minimum value of a quantity, you will most likely need to use differentiation at some point. Most questions will involve creating a formula, for example for Volume or Area, and then calculating the maximum value of it. WB 45 : Represent the volume and surface area of the following shapes algebraically. Then, find the surface area in terms of x (red) and r (yellow). r h V = 20 m 3 x x V = 8 mm 3 h
WB 46 a A large tank (shown) is to be made from 54 m 2 of sheet metal. It has no top. Show that the Volume of the tank will be given by: x y 1) Try to make formulae using the information you have Formula for the Volume Formula for the Surface Area (no top) x 3) Substitute the SA formula into the Volume formula, to replace y. 2) Find a way to remove a constant, in this case ‘y’. We can rewrite the Surface Area formula in terms of y.
WB 46 b V = 18 x – 2/3 x 3 V A large tank (shown) is to be made from 54 m 2 of sheet metal. It has no top. Show that the Volume of the tank will be given by: x y x x b) Calculate the values of x that will give the largest volume possible, and what this Volume is. The graph above shows the formula for the volume V, in terms of x (that we just worked out!) Before attempting this think about what we are doing: So we need to calculate the value of x where the gradient is 0 – differentiate! What does the graph look like?
WB 46 c b) A large tank (shown) is to be made from 54 m 2 of sheet metal. It has no top. Show that the Volume of the tank will be given by: x y Differentiate x b) Calculate the values of x that will give the largest volume possible, and what this Volume is. Set equal to 0 Rearrange Solve We want the volume Sub the x value in
WB 47 a A wire of length 2 m is bent into the shape shown, made up of a Rectangle and a Semi-circle. y πx 2 x a) Find an expression for y in terms of x. b) Show that the Area is: y a) Find the length of the semi-circle, as this makes up part of the length. Rearrange to get y alone Divide by 2 c) Find the maximum possible Area
y WB 47 b πx 2 x b) Show that the Area is: y b) Work out the areas of the Rectangle and Semi-circle separately. Rectangle Semi Circle Replace y Expand Factorise
y WB 47 c πx 2 x b) Show that the Area is: y c) Use the formula we have for the Area Expand Factorise Divide by (4 + π) Differentiate Set equal to 0 Multiply by 8 Divide by 2
Practice 1
Practice 1
Practice 1 SOLUTIONS
Practice 1 SOLUTIONS
Practice 1 SOLUTIONS
KUS objectives BAT apply differentiation to solve optimization problems BAT manipulate algebra to form equations that can be differentiated self-assess One thing learned is – One thing to improve is –
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