**Question 2:**

A new technology museum building is being planned and is going to have a photovoltaic surface applied to the entire roof area. The photovoltaic roof surface is capable of generating electricity from sunlight. The proposed shape of the roof is shown in Figure 1.

Figure 1 Proposed shape of the new roof: (a) 3D representation showing the roof area in blue; (b) plan view showing the principle dimensions as variables

There are still changes that can be made to the layout of the building and these will affect the dimensions of the roof geometry.

One consideration in making this decision is how much power the new roof can potentially generate. To assess this, you have been asked to produce a mathematical model of the maximum power output as a function of the roof size.

You will create this model as you go through the question.

a.

Show that a general formula for the area of the roof (Atot) can be written as

using the variables a, b, c and d shown in the figure above.

Explain each step in obtaining this formula, stating how you have split the roof into different shapes.

b.

The peak output of the photovoltaic material is 160 W m–2 (that is, every square metre of material can generate a theoretical maximum power of 160 W).

Use this value to generate a formula for the maximum power output of the roof in watts (W) when it is working at full capacity. Use the variable Ppeak for the subject of this formula.

c.

The maximum power that you calculated in Question 1(b) is a theoretical one, where the sun is shining brightly, all the components are acting at their full efficiency, and there are no other energy losses.

In reality, many factors will affect the power output of the photovoltaic roof surface.

List two things that you think would have an effect on the output of power from the photovoltaic roof surface and briefly describe how they would lead to lower values of power output.

d.For this building, the actual power output is 20% of the theoretical maximum value of 160 W.

Use this percentage value to refine the formula for the actual power output of the roof in watts (Pave) based on the area.

e.One design option is to change only the length of the roof b whilst keeping all other measurements a, c and d constant, as follows:

a = 10.0 m

c = 26.0 m

d = 13.0 m

Use these measurements to update your formula for the average power output and call it (Pmodel).

f.The formula for the average power output (Pave) derived in Question 1(d) and amended in Question 1(e) above, and labelled as Pmodel represents the final model for the average power output from the roof.

To allow designers to vary the width of the roof and see how the power output changes, the formula has been used to plot the graph shown in Figure 2 for varying values of roof width b.

Figure 2 Average power output, in watts, versus width of roof, in metres

Use this graph to find the minimum and maximum average power outputs from Pmodel if the minimum value for b is 16 m and the maximum value is 34 m.

g.The graph above plots the average power output (Pmodel) in watts on the vertical axis against the roof width (b) in metres on the horizontal axis

The formula for Pmodel derived in Question 1(e) is of the form y = mx + c. Using the formula, what is the gradient, m, and the y-intercept, c, of the line in the graph above?

Show all the steps in your working and clearly state the value and units of the gradient, m.