13. The front views of two mathematically similar milk cartons are shown below.
Diagram not drawn to scale
(a) Circle either TRUE or FALSE for each statement given below. [1]
STATEMENT
The ratio of the lengths of the cartons is the same as the ratio of the heights of the cartons.
TRUE
FALSE
The ratio of the volumes of the cartons is the same as the ratio of the heights of the cartons.
TRUE
FALSE
(b) It is claimed that the larger carton contains double the amount of milk contained in the smaller carton. Show that this claim is not true. Explain your answer. [3]
(c) Another similar milk carton has a label with an area that is one quarter of the area of the label on the carton of height 24 cm.
Diagram not drawn to scale
Calculate the height of this new carton. [3]
Mark scheme (a)
Scheme
Marks
Comments
True False
B1
Mark scheme (b)
Scheme
Marks
Comments
(lengths in ratio) 24 : 30 (= 4 : 5)
B1
Ratio can be reversed Or equivalent (e.g. scale factor = 1∙25 or 30/24 OR 0∙8 or 24/30)
(volumes in ratio) 13824 : 27000 (= 43 : 53)
B1
Ratio can be reversed Or equivalent (e.g. 'Volume scale factor' = 1∙9(53125) or 1∙253 or (30/24)3 OR 0∙512 or 0∙83 or (24/30)3
Statement e.g. ‘125 is not double 64 (so the increase is not double)', or ‘64 is not half of 125’, or ‘Increase is 95(∙3125)%’
E1
Depends on B2 provided 43 and 53 have been evaluated correctly or 1∙253, (30/24)3, 0∙83 or (24/30)3 evaluated correctly
Mark scheme (c)
Scheme
Marks
Comments
(Scale factor of heights =) \(\sqrt{4}\) or 2 OR \(\sqrt{\tfrac{1}{4}}\) or 0∙5
B1
24 ÷ \(\sqrt{4}\) OR 24 × \(\sqrt{\tfrac{1}{4}}\)
M1
= 12 (cm)
A1
Alternative method: M1 for 242 ÷ 4 A1 for height2 = 144 or (height =) √144 A1 for 12 (cm)
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