The image presents a problem related to measuring the Earth's radius and calculating the percentage error and the Earth's circumference.

Problem Breakdown:

1. Given Information:

   • The true radius of the Earth is $R = 6,378,137$ meters.
   • Jurgen measures the radius to be $R_{\text{measured}} = 6,432,400$ meters.

2. Questions:

   • (a) What is the percentage error in Jurgen's measurement?
   • (b)(i) Using his measurement, calculate the circumference of the Earth (assuming it's spherical). Round to 4 significant figures.
   • (b)(ii) Write the circumference in scientific notation $a \times 10^k$, where $1 \leq a < 10$ and $k \in \mathbb{Z}$.

(a) Percentage Error

The percentage error can be calculated using the formula:

$\text{Percentage Error} = \left( \frac{|R_{\text{measured}} - R|}{R} \right) \times 100\%$

(b)(i) Circumference of the Earth

The formula for the circumference $C$ of a sphere is:

$C = 2 \pi R$

Using Jurgen's measured radius $R_{\text{measured}}$.

(b)(ii) Scientific Notation

Convert the answer from part (b)(i) into the form $a \times 10^k$, where $1 \leq a < 10$ and $k$ is an integer.

I'll proceed to calculate each of these now.### Solutions:

1. (a) Percentage Error:

   • The percentage error in Jurgen's measurement is approximately 0.851%.

2. (b)(i) Circumference of the Earth:

   • The circumference of the Earth, based on Jurgen's measurement, is approximately 40,415,961.1699 meters (rounded to 4 significant figures: 40,420,000 meters).

3. (b)(ii) Scientific Notation:

   • The circumference in scientific notation is $4.0416 \times 10^7$ meters.

Would you like more details on any of the steps, or do you have any further questions?

Related Questions:

1. What is the actual circumference of the Earth using the true radius?
2. How would the percentage error affect calculations related to satellite orbits?
3. How does the Earth's non-spherical shape impact such measurements?
4. What methods could Jurgen use to improve the accuracy of his radius measurement?
5. How would a significant percentage error influence real-world applications, such as GPS?

Tip:

When performing calculations that involve real-world measurements, always consider the impact of measurement error, especially in critical fields like astronomy or engineering.