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 lm. 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. A company is investigating the nancial feasibility of asteroid mining. They are only interested in asteroids greater than 10m in diameter. A statistician proposes that the cumulative distribution function for the diameter of asteroids. x, in their range of interest is given by: F[o) =1 (g'5')3 What is the probability that a randomly selected asteroid is greater than 1 Em in diameter? Enter your answer as a decimal value accurate to two decimal places, rounding as appropriate. e.g. if the probability was 23.5% you would enter a value of 0.24 in the answer space. It the probability was 1 in 4, you would enter a value of [1.25