Math problem, help

Question

The Greek mathematician Eratosthenes of Cyrene, around 240 BC, estimated that the Earth's circumference measured 252,000 stadia. There was no standard for the value of a stadio, but it is estimated that in Eratosthenes' calculations a stadio was equivalent to 157.7m. We know today that the circumference of the Earth at the equator is 40,075.017 km.

a.Calculate the absolute error in the calculation of Eratosthenes.

B. Calculate the relative error in the calculation of Eratosthenes

Answer 1

The first step is to convert stadion and kilometers into meters so that we can compare them easily.

We use dimensional analysis and conversion factors for this.

252,000 stadion * 157.7 meters / stadion = 39,740,400 meters

40,075.017 kilometers * 1,000 meters / kilometer = 40,075,017 meters

Now we calculate the error.

Absolute error is simply the length he calculated minus the actual value, absolute value.

40,075,017 meters - 39,740,400 meters = 334,617 meters

Relative error is the absolute error divided by the actual measurement.

334,617 meters / 40,075,017 meters = 0.0084 or about 0.84%