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Date May 2017 Marks available 3 Reference code 17M.2.SL.TZ1.T_4
Level Standard Level Paper Paper 2 Time zone Time zone 1
Command term Calculate Question number T_4 Adapted from N/A

Question

A pan, in which to cook a pizza, is in the shape of a cylinder. The pan has a diameter of 35 cm and a height of 0.5 cm.

M17/5/MATSD/SP2/ENG/TZ1/04

A chef had enough pizza dough to exactly fill the pan. The dough was in the shape of a sphere.

The pizza was cooked in a hot oven. Once taken out of the oven, the pizza was placed in a dining room.

The temperature, P , of the pizza, in degrees Celsius, °C, can be modelled by

P ( t ) = a ( 2.06 ) t + 19 ,   t 0

where a is a constant and t is the time, in minutes, since the pizza was taken out of the oven.

When the pizza was taken out of the oven its temperature was 230 °C.

The pizza can be eaten once its temperature drops to 45 °C.

Calculate the volume of this pan.

[3]
a.

Find the radius of the sphere in cm, correct to one decimal place.

[4]
b.

Find the value of a .

[2]
c.

Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.

[2]
d.

Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.

[3]
e.

In the context of this model, state what the value of 19 represents.

[1]
f.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

( V = )   π × (17.5) 2 × 0.5     (A1)(M1)

 

Notes:     Award (A1) for 17.5 (or equivalent) seen.

Award (M1) for correct substitutions into volume of a cylinder formula.

 

= 481  c m 3   ( 481.056  c m 3 ,   153.125 π  c m 3 )     (A1)(G2)

[3 marks]

a.

4 3 × π × r 3 = 481.056     (M1)

 

Note:     Award (M1) for equating their answer to part (a) to the volume of sphere.

 

r 3 = 3 × 481.056 4 π   ( = 114.843 )     (M1)

 

Note:     Award (M1) for correctly rearranging so r 3 is the subject.

 

r = 4.86074  (cm)     (A1)(ft)(G2)

 

Note:     Award (A1) for correct unrounded answer seen. Follow through from part (a).

 

= 4.9  (cm)     (A1)(ft)(G3)

 

Note:     The final (A1)(ft) is awarded for rounding their unrounded answer to one decimal place.

 

[4 marks]

b.

230 = a ( 2.06 ) 0 + 19     (M1)

 

Note:     Award (M1) for correct substitution.

 

a = 211     (A1)(G2)

[2 marks]

c.

( P = )   211 × ( 2.06 ) 5 + 19       (M1)

 

Note:     Award (M1) for correct substitution into the function, P ( t ) . Follow through from part (c). The negative sign in the exponent is required for correct substitution.

 

= 24.7 (°C) ( 24.6878 (°C))     (A1)(ft)(G2)

[2 marks]

d.

45 = 211 × ( 2.06 ) t + 19     (M1)

 

Note:     Award (M1) for equating 45 to the exponential equation and for correct substitution (follow through for their a in part (c)).

 

( t = )   2.89711     (A1)(ft)(G1)

174  (seconds)  ( 173.826  (seconds) )     (A1)(ft)(G2)

 

Note:     Award final (A1)(ft) for converting their 2.89711 minutes into seconds.

 

[3 marks]

e.

the temperature of the (dining) room     (A1)

OR

the lowest final temperature to which the pizza will cool     (A1)

[1 mark]

f.

Examiners report

[N/A]
a.
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b.
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c.
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d.
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e.
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f.

Syllabus sections

Topic 3—Geometry and trigonometry » SL 3.1—3d space, volume, angles, midpoints
Show 90 related questions
Topic 2—Functions » SL 2.5—Modelling functions
Topic 2—Functions
Topic 3—Geometry and trigonometry
Prior learning

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