Question

An architect and engineer team up to design a new housing subdivision. The engineer proposes that they build homes in the shape of a cube to maximise internal volume while minimising materials used for the surface area (floor, walls and roof}. The architect is concerned every house will look identical so proposes that they vary the side length of the houses, x, by selecting lengths from a uniform distribution between as and lflm. What is the expected value for the volume of a house? Enter your answer assuming units of cubic metres rounding to the nearest whole number.

Answer 1

The expected value for the volume of a house with cubic shape and side lengths from a uniform distribution between 8 and 10 meters is calculated by averaging the volumes at the minimum and maximum side lengths. The expected volume rounds to 756 cubic meters.

Step-by-step explanation:

To find the expected value for the volume of a house with side lengths of x chosen from a uniform distribution between 8 and 10 meters, we need to calculate the mean of the possible volumes. We can express the volume V of a cube with side length x as V = x3. Since the side lengths are uniformly distributed, the expected value of V will be the average of the volumes at the minimum and maximum side lengths.

The volume at the minimum side length (8m):
Vmin = 83 = 512 cubic meters.

The volume at the maximum side length (10m):
Vmax = 103 = 1000 cubic meters.

To find the expected volume:
Expected V = (Vmin + Vmax) / 2 = (512 + 1000) / 2 = 756 cubic meters.

When rounded to the nearest whole number, the expected volume is 756 cubic meters.