Question

(a) The diameter of Earth at the equator is \(7926.381 \mathrm{mi}\). Round this number to three significant figures, and express it in standard exponential notation. (b) The circumference of Earth through the poles is \(40,008 \mathrm{~km}\). Round this number to four significant figures, and express it in standard exponential notation.

Short answer

(a) \(7.93 \times 10^3\) mi (b) \(4.001 \times 10^4\) km

Step-by-step solution

1. Identify the number of significant figures

We are asked to round the diameter of Earth at the equator to three significant figures. The given diameter is 7926.381 mi.

2. Rounding to the required number of significant figures

To round 7926.381 mi to three significant figures, we have to look at the fourth significant figure, which is 6. Since 6 is greater than 5, we round up the third significant figure, resulting in 7930 mi.

3. Express in standard exponential notation

Now, we need to express 7930 mi in standard exponential notation. To do this, we write the number as a product of a number between 1 and 10 and an appropriate power of 10. Therefore, 7930 mi can be written as \(7.93 \times 10^3\) mi. Now, let's move to part (b). (b) Rounding the circumference of Earth through the poles

1. Identify the number of significant figures

We are asked to round the circumference of Earth through the poles to four significant figures. The given circumference is 40,008 km.

2. Rounding to the required number of significant figures

To round 40,008 km to four significant figures, we must look at the fifth significant figure, which is 8. Since 8 is greater than 5, we round up the fourth significant figure, resulting in 40,010 km.

3. Express in standard exponential notation

Now, we need to express 40,010 km in standard exponential notation. To do this, we write the number as a product of a number between 1 and 10 and an appropriate power of 10. Therefore, 40,010 km can be written as \(4.001 \times 10^4\) km. To summarize our answers: (a) The diameter of Earth at the equator rounded to three significant figures and expressed in standard exponential notation is \(7.93 \times 10^3\) mi. (b) The circumference of Earth through the poles rounded to four significant figures and expressed in standard exponential notation is \(4.001 \times 10^4\) km.