Problem Bank | IB Math AIHL

[Maximum mark: 12]

The temperature, T degrees Fahrenheit, t hours after sunrise in Boston can be modeled by a function of the form

T(t)=acos(bt°)+c

The following diagram shows the curve of T on a day in the middle of each of the 4 seasons.

Determine which curve corresponds to which season to complete the table below.
Season
Winter
Spring
Summer
Autumn
Curve

[1]
To earn a crown, get your answer ready before you reveal the options!
1. Explain why a must be a negative number.
  [1]
2. Find the value of b.
  [2]
It is given that the equation of curve C is T(t)=−10cos(bt°)+45, where b has the value determined in (b.ii).
1. Write down the average temperature throughout the day.
  [1]
2. Find the second time at which the temperature is 40°F.
  [2]

Season	Winter	Spring	Summer	Autumn
Curve

It is given that the temperature on the winter day T(t)=−9cos(bt°)+29. The formula for converting Fahrenheit to Celsius is C=95(F−32).

Find an expression for the temperature C(t), in Celsius, throughout the winter day.
[2]
Hence determine the length of time during the day where the temperature is below freezing (0°C).
[3]

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