Suppose we allowed each person 1000m^3 of living space in a large building, with ceiling height 3 m. To the nearest order of magnitude, how many 1000m high buildings would have to be built to house the entire population of 8.5 x 10^14 people on the energy-limited Earth, if each building's base were 100km on a side
To find out how many 1000m high buildings would be needed to house the entire population on Earth, we need to calculate the total living space required and then divide it by the available living space per building.
First, let's calculate the total living space required. Each person is allowed 1000m^3 of living space, and with a ceiling height of 3m, the floor area would be 1000m^2. So, the total living space required for one person is 1000m^2.
Now, let's calculate the total population living space required. We have a population of 8.5 x 10^14 people, so the total living space required will be:
Total Living Space Required = (1000m^2/person) * (8.5 x 10^14 people)
To simplify the calculation and get the order of magnitude, we can convert the population to scientific notation:
Total Living Space Required = (1000m^2/person) * (8.5 x 10^14) = (1000m^2/person) * (8.5 x 10^14) = 8.5 x 10^17 m^2
Now, let's determine how many buildings would be needed to provide this living space. Each building has a base of 100km on a side, which can be written as 100,000m. To find the total number of buildings needed, we can divide the total living space required by the total living space per building:
Number of Buildings = Total Living Space Required / Living Space per Building
Number of Buildings = (8.5 x 10^17 m^2) / (100,000m * 100,000m)
Number of Buildings = 8.5 x 10^17 m^2 / (10,000,000,000 m^2)
Number of Buildings = 8.5 x 10^17 / 10^10
Number of Buildings = 8.5 x 10^(17-10) = 8.5 x 10^7
Therefore, to the nearest order of magnitude, around 8.5 x 10^7 or 85 million 1000m high buildings would have to be built to house the entire population of 8.5 x 10^14 people on the energy-limited Earth, given each building's base of 100km on a side.