12. The Leaning Tower of Pisa stands on horizontal ground. The vertical height of the tower on the higher side is 56·7 m. The top of the tower is displaced 3·9 m horizontally.
Diagram not drawn to scale
(a) Calculate the angle, \(x\), at which the tower leans. Give your answer correct to 2 decimal places. You must show all your working. [4]
(b) Ceri plans to make a poster that is mathematically similar to the Leaning Tower of Pisa.
Diagram not drawn to scale
Calculate the height of the poster Ceri plans to make. [2]
Mark scheme (a)
Scheme
Marks
Comments
tan x = \(\dfrac{3.9}{56.7}\)
M1
(x =) tan-1 3.9/56.7 or (x =) tan-1 0.06878....
m1
(x = ) 3.93(°)
A2
A1 for (x = ) 3.9(3...°) from correct working An unsupported answer of 3.9 is M0, m0, A0
Alternative
12(a) Alternative method
Scheme
Marks
Comments
(slant height2 = 3.92 + 56.72, s = √3230.1, leading to) slant height 56.8(33... m) and either sin x = \(\dfrac{3.9}{56.8(33...)}\) or cos x = \(\dfrac{56.7}{56.8(33...)}\)
M1
(x =) sin-1(3.9/56.8(33...)) or cos-1 (56.7/56.8(33...))
m1
(x = ) 3.93(°)
A2
A1 for (x = ) 3.9(3...°) to 3.9(7...°) from correct working An unsupported answer of 3.9 is M0, m0, A0
Mark scheme (b)
Scheme
Marks
Comments
56.7 × 9.36 ÷ 3.9 or 56.7 × 2.4 or equivalent
M1
Ignore place value errors due to change of units for M1 only
136(.08 cm)
A1
If units are given they must be correct, accept answer in metres Allow answers from premature approximation in the range 136 (cm) to 136.1 (cm)
Alternative
12(b) Alternative method
Scheme
Marks
Comments
(Height of poster =) \(\dfrac{9.36}{\tan 3.9(.....^\circ)}\)
M1
FT from (a) M0 for tan 3.9(.....°) = \(\dfrac{9.36}{\text{Height of poster}}\)
Answer in the range 136 (cm) to 137.3 (cm)
A1
If units are given they must be correct, accept answer in metres
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