Investigation 'Theatre Design'
PART A: UNDERSTANDING & FLUENCY
Mr E John wants to build a small performance theatre with a stage for recitals. The theatre cannot be longer than 16m and needs to seat at least 108 people.
The theatre is to be rectangular with the following specifications:
- Its width (W) is given by the expression W = 4x - 4
- Its length (L) is twice the theatres width
Please Note: all dimensions are in metres and round to one decimal point.
1. a. Find L and W for when x = 2.
The stage is also rectangular and is to be built across the width of the theatre. This means that the length of the stage (a) is equal to the width of the theatre (W).
The width of the stage (b) is half the length of the stage (i.e. b is equal to half of a).
b. Find the value of a and b for x = 2.
c. Draw a diagram of the theatre showing the dimensions L, W, a and b.
There also needs to be aisle space around the seating area. The audience is to be seated in rows in a rectangular space so that there is a 2 metre clearance from the front of the stage and from the back of the room and with a 1 metre aisle either side.
2. a. Add the seating area in your diagram. Mark all the clearance dimensions.
b. Find the length of the seating (c) and the width of the seating (d) for x = 2.
c. Add the dimensions of the seating area to your diagram.
d. If a space of 50cm x 50cm is allowed for each seat, how many seats could fit into this space?
3. By substituting in different values for x, find the dimensions of the smallest theatre that Mr E John can build so that 108 people can be seated in the seating area. Do not use integer values only. Draw up a table or a spreadsheet like the one shown below to record your trials.
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x
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W
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L
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a
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b
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c
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d
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rows
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columns
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no of seats
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4. What is the largest value that x can be so that the length of the theatre does not exceed 16 metres?
5. a. Explain why values for x ≤ 1 cannot be used for the length and width of the theatre.
b. How many seats can fit if x = 1(3/4)?
c. Explain why the smallest value for x is 1(3/4)?
PART B: REASONING & PROBLEM SOLVING
Mr E John also wants to know how much carpet he would need to cover the floor space not occupied by the stage or seats. Step through the following calculations to find the area of the floor to be covered with carpet.
1. a. Draw a diagram showing the area to be carpeted.
b. Find the area of the theatre in terms of x. Answer in expanded form.
c. Find a and b in terms of x.
d. Find the area of the stage in terms of x. Answer in expanded form.
e. Find c and d in terms of x.
f. Find the seating area in terms of x. Answer in expanded form.
g. Find the area, in terms of x, of the floor space of the theatre occupied by both the stage and the seating area.
h. Clearly show that the area, in term of x, of the remaining floor space is equal to 28x - 36 m2. Use your equations from Part B Q1 b, d and f to help.