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Date May 2022 Marks available 3 Reference code 22M.1.SL.TZ2.12
Level Standard Level Paper Paper 1 Time zone Time zone 2
Command term Justify and Determine Question number 12 Adapted from N/A

Question

The cross-section of an arched entrance into the ballroom of a hotel is in the shape of a parabola. This cross-section can be modelled by part of the graph y=1.6x2+4.48x, where y is the height of the archway, in metres, at a horizontal distance, x metres, from the point O, in the bottom corner of the archway.

To prepare for an event, a square-based crate that is 1.6m wide and 2.0m high is to be moved through the archway into the ballroom. The crate must remain upright while it is being moved.

Determine an equation for the axis of symmetry of the parabola that models the archway.

[2]
a.

Determine whether the crate will fit through the archway. Justify your answer.

[3]
b.

Markscheme

x= -4.482-1.6   OR   coordinates of maximum point (1.4, 3.136)          (M1)

x=1.4           A1

 

[2 marks]

a.

METHOD 1

the cart is centred in the archway when it is between

x=0.6 and x=2.2,            A1

where y2.112m  (which is greater than 2)           R1

the archway is tall enough for the crate           A1


Note: Do not award R0A1.

 

METHOD 2

the height of the archway is greater or equal to 2.0 between

x=0.557385  and  x=2.24261            A1

width of this section of archway =

2.24261-0.557385=  1.68522m (which is greater than 1.6)           R1

the archway is wide enough for the crate           A1

 

Note: Do not award R0A1.

 

[3 marks]

b.

Examiners report

Most candidates were able to substitute into the formula for axis of symmetry or find the vertex of the parabola correctly, both being appropriate methods, but neglected to write an equation from that, even though the question specifically asked for an equation.

a.

Determining a process to see if the crate would fit through the archway proved to be difficult for many candidates. It was common to see the maximum heights compared, the maximum widths compared, or the area of the front surface of the crate compared to the area of the archway opening. Other candidates merely calculated the height at x=1.6, positioning the corner of the crate at O, and made their conclusion based on this value, without consideration of how the crate would be moving through the archway.

b.

Syllabus sections

Topic 2—Functions » SL 2.4—Key features of graphs, intersections using technology
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