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Francesca is an engineer tasked with planning the installation of street lights along a road.
The observed illuminance, Elux, due to a light source is inversely proportional to the square of the distance r meters from the source:
E=r2k
Francesca measures the illuminance to be 5lux when she stands 3 meters from a single lightbulb.
- [2]
Determine the value of k.
- [1]
Suggest the physical significance of k.
Two identical street lights using the same lightbulb are positioned 100m apart, each lamp is 5m above the ground, as shown in the diagram below.
Consider a car driving between the two street lights. The function E(x) models the illuminance along the road x meters from the leftmost light.
- [4]
Show that the illuminance is given E(x)=x2+2545+(100−x)2+2545.
- [3]
Find the minimum illuminance along the stretch of road between the two street lights. Give your answer as a fraction.
The illuminance of each lightbulb is proportional to the square of the current I through the bulb.
The current in each streetlight is increase by 40%, leading to an increase of n% in the illuminance at every point along the road.
- [3]
Find the value of n.
The increased current causes the streetlight at x=100 to burn out.
- [3]
On the axes below, sketch the resulting illuminance along the stretch of road between the street lights.
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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.
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- [1]
Determine which curve corresponds to which season to complete the table below.
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Explain why a must be a negative number.
- [2]
Find the value of b.
It is given that the equation of curve C is T(t)=−10cos(bt°)+45, where b has the value determined in (b.ii).
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Write down the average temperature throughout the day.
- [2]
Find the second time at which the temperature is 40°F.
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).
- [2]
Find an expression for the temperature C(t), in Celsius, throughout the winter day.
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Hence determine the length of time during the day where the temperature is below freezing (0°C).
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