IB Maths AI HL 5.7 Optimization Question Bank

Practise IB Mathematics SL/HL 5.7 by applying optimization methods to exam-style questions.

[Maximum number: 10]

Linda owns a field, represented by the shaded region R. The plan view of the field is shown in the following diagram, where both axes represent distance and are measured in metres.

Figure for Question SL 5.7—Optimization question 1 — IB Maths AI HL

The segments [AB], [CD] and [AD] respectively represent the western, eastern and southern boundaries of the field. The function, f(x), models the northern boundary of the field between points B and C and is given by

f(x)=\frac{-x^{2}}{50}+2 x+30, \text { for } 0 \leq x \leq 70

(a)

Find the x-coordinate of point E for the largest area of the square foundation of building EFGH.

[ 5 ]

width of the square is $70-x \quad \mathbf{OR}$ the length of the square is $\frac{-x^{2}}{50}+2 x+30$
(M1)
Note: Award (M1) for 70-x seen anywhere. Accept $\frac{-x^{2}}{50}+2 x+30$ but only if this expression is explicitly identified as a dimension of the square.
in term of x, equating the length to the width ED
$\frac{-x^{2}}{50}+2 x+30=70-x$
( $x=14.7920 \ldots$ or 135.21)
$(x=) 14.8 \mathrm{~m}(14.7920 \ldots)$

Note: Award MOMOAO for an unsupported answer of 15. Award at most M1M0A0 for an approach which leads to $A^{\prime}(x)=0$ . This will lead to a square base which extends beyond the east boundary of the property. Similar for any solution where F is not on the northern boundary, or GH is not on the east boundary.

(b)

Find the largest area of the foundation.

[ 5 ]

EITHER
$(70-14.7920 \ldots)^{2}$

OR
$(55.2079 \ldots)^{2}$

OR
$\left(\frac{-(14.7920 \ldots)^{2}}{50}+2(14.7920 \ldots)+30\right)^{2}$

THEN
(Area = ) $3050 \mathrm{~m}^{2}$ (3047.92…)

Note: Follow through from part (d)(i), provided x is between 0 and 70 . Award at most M1A0 if their answer is outside the range of their [0, 4713.33...] from part (b).

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SL 5.7—Optimization question 1

Question (a)

Question (b)