Practice IB Mathematics Applications & Interpretation (AI) Topic SL 5.7—optimisation with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.7—optimisation and mirrors Paper 1, 2, 3 style where relevant.
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A conical frustum is designed as a water reservoir with a smaller circular top of radius , a larger circular base of radius , and height . The volume of the frustum is fixed at .
(a) Show that the height can be expressed as .
(b) an expression for the total surface area (including top and bottom) in terms of .
A triangular sheet has , cm and cm, where initially .
Points and lie on and respectively, with cm and cm. The triangular corner is cut away, leaving the quadrilateral panel . The design is permitted only when lies strictly between and and lies strictly between and .
Use polynomial differentiation and derivative sign changes to justify maxima. Give non-exact numerical answers to three significant figures, retaining unrounded intermediate values.
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(e) the angle at the centre of the bed of greatest area, in degrees.
An irrigation pipe runs under the lawn, and three rows of plants must fit across the bed, so the depth must be at least m and at most m.
(f) the greatest area the bed can have under this restriction.
Rhea says that a semicircular bed would make the best possible use of the m of edging.
(g) Ignoring the restriction in part (f), the greatest area found in part (d) with the area of a semicircular bed edged with the same m, and hence whether Rhea's claim is correct.
(e) The average cost per hundred lanterns is , for . Find the number of lanterns that minimizes the average cost. why making this number of lanterns does not maximize the workshop's profit.
(f) The government introduces a tax of dollars on every lantern sold. The selling price and the cost model are unchanged. the greatest value of for which the profit-maximizing weekly output is at least lanterns.
(d) the slant height of the frustum at the minimum surface area.
Rhea Mbeki is the head groundskeeper of a town park, and she is cutting a new flower bed into one of the lawns. The bed is a sector of a circle: two straight edges of equal length run out from one point, the centre, and a curved edge joins their far ends. Rhea calls the common length of the two straight edges the depth of the bed, so the depth of the bed is the radius of the circle. The council has already bought m of steel edging, and the whole boundary of the bed must be edged, both straight edges and the curved edge, with none of the edging left over and no extra length joined on. The depth of the bed, and the angle it opens through at the centre, are Rhea's to choose. Every square metre inside the bed is planting, so Rhea wants the bed to enclose as much ground as the edging allows.
Rhea's first sketch is a bed of depth m.
(a) Find the length of the curved edge of this bed.
(b) the area of this bed.
(c) Show that the area of the bed, in square metres, is , where is the depth of the bed in metres.
(d) Find the depth that gives the bed its greatest area, and that greatest area.
A workshop makes and sells solar lanterns. It can make at most lanterns per week, and every lantern it makes is sold for $. When the workshop makes hundred lanterns in a week, its weekly cost, in hundreds of dollars, is modelled by
Give monetary answers correct to the nearest dollar, and other non-exact numerical answers to three significant figures.
(a) Show that the weekly profit, in hundreds of dollars, is .
(b) Find the least number of lanterns that the workshop must make in a week to avoid making a loss.
(d) A shortage of storage space limits production to at most lanterns per week. Find the maximum weekly profit under this restriction, justifying your answer.
A factory packages coconut water in cone-shaped containers with a base radius of 5.2 cm and a height of 13 cm.
The factory designers are currently investigating whether a cone-shaped container can be replaced with a cylinder-shaped container with the same radius and the same total surface area.
(a)
Find the slant height of the cone-shaped container.
(b)
Show that the total surface area of the cone-shaped container is 314 cm2, correct to three significant figures.
(c)
Find the height, , of the cylinder-shaped container with the same radius and total surface area as the cone-shaped container.
(d)
The factory director wants to increase the volume of coconut water sold per container.
State whether they should replace the cone-shaped containers with cylinder-shaped containers. Justify your conclusion.
(c) Use technology to find the value of that minimizes the surface area, and the minimum surface area. that this gives a minimum.
(e) the frustum and the dimensions corresponding to the minimum surface area.
radius
radius
(c) Use differentiation to find the number of lanterns that maximizes the weekly profit, and find the maximum weekly profit. that this is the greatest profit on .
