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3.05 Optimisation

Interactive practice questions

A potato farmer finds that the yield per square metre when spacing his plants between $0.5$0.5 m and $3.0$3.0 m approximates the following equation:

$y=-\frac{x^2}{4}+\frac{7x}{20}$y=x24+7x20

a

Find $\frac{dy}{dx}$dydx.

b

For what value of $x$x is $\frac{dy}{dx}$dydx equal to $0$0?

c

What is the maximum possible yield?

Round your answer to two decimal places.

Easy
4min

A parabolic satellite dish is pointing straight up. Along a cross-section that passes through the centre of the dish, the height above ground is given by the following equation:

$y=\frac{1}{100}\left(x^2-100x\right)+50$y=1100(x2100x)+50

Easy
3min

A function $f:\left[-7,5\right]\to\mathbb{R}$f:[7,5] is given by $f\left(x\right)=-6x^2-12x+90$f(x)=6x212x+90.

Easy
4min

A function $f:\left[3,8\right]\to\mathbb{R}$f:[3,8] is given by $f\left(x\right)=2x^2-2x-24$f(x)=2x22x24.

Easy
3min
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Outcomes

3.2.1.2

recognise that 𝑒 is the unique number 𝑎 for which the limit (in 3.2.1.1) is 1

3.2.1.5

identify contexts suitable for mathematical modelling by exponential functions and their derivatives and use the model to solve practical problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

3.2.3.3

use trigonometric functions and their derivatives to solve practical problems; including trigonometric functions of the form 𝑦 = sin(𝑓(𝑥)) and 𝑦 = cos(𝑓(𝑥)).

4.1.1.6

solve optimisation problems from a wide variety of fields using first and second derivatives, where the function to be optimised is both given and developed

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