Estimation example from Giancoli

[EvD]

Hello, I´m a bit math-challenged and just started to brush up on both physics and mathematics simultaneously but I'm still at absolute beginner status and I don't want to go on in the book as long as I can't even figure out such a simple example. I'm using the Giancoli textbook on physics for self-study but can't quite follow the simplest of examples in the first chapter. Since the problem lies in the mathematical part of my understanding and not the conceptually physical part I thought I'd post it here. Just wanted to check if I'm doing something wrong or whether there's a typo in the example because unfortunately there doesn't seem to be a list of errata for the book. Here it is:

1. Homework Statement
The example is about estimating the volume of a lake by using some rough measures and rounded-off numbers to get at the approximate order-of-magnitude.
So it says:

V=hπr² ≈(10meters) × (3) ×(5×10²meters)²≈8×10⁶m³ ≈ 10⁷ m³

where π has been rounded to 3, the average depth of the lake is 10m and the estimated radius of the surface is given by 5×10² (500 m).
I can't seem to follow how he arrives at 8 ×10⁶.

Homework Equations

-

The Attempt at a Solution

In my mind it would come to 3×(5×10⁵) and then maybe 1.5×10¹ ×10⁵ and then 1.5 ×10⁶. This is of course different from the given 8×10⁶.

I figure I'm probably making a mistake here in the rules of multiplication of exponents and bracketing out but I can't figure out what it is. Any help would be much appreciated. Sorry for the absolute rookie level of mathematical (in)competence.

 

[SteamKing]

Hello, I´m a bit math-challenged and just started to brush up on both physics and mathematics simultaneously but I'm still at absolute beginner status and I don't want to go on in the book as long as I can't even figure out such a simple example. I'm using the Giancoli textbook on physics for self-study but can't quite follow the simplest of examples in the first chapter. Since the problem lies in the mathematical part of my understanding and not the conceptually physical part I thought I'd post it here. Just wanted to check if I'm doing something wrong or whether there's a typo in the example because unfortunately there doesn't seem to be a list of errata for the book. Here it is:

1. Homework Statement
The example is about estimating the volume of a lake by using some rough measures and rounded-off numbers to get at the approximate order-of-magnitude.
So it says:

V=hπr² ≈(10meters) × (3) ×(5×10²meters)²≈8×10⁶m³ ≈ 10⁷ m³

where π has been rounded to 3, the average depth of the lake is 10m and the estimated radius of the surface is given by 5×10² (500 m).
I can't seem to follow how he arrives at 8 ×10⁶.

Homework Equations

-

The Attempt at a Solution

In my mind it would come to 3×(5×10⁵) and then maybe 1.5×10¹ ×10⁵ and then 1.5 ×10⁶. This is of course different from the given 8×10⁶.

I figure I'm probably making a mistake here in the rules of multiplication of exponents and bracketing out but I can't figure out what it is. Any help would be much appreciated. Sorry for the absolute rookie level of mathematical (in)competence.

Don't forget to square the radius: (5×10²)² = 500 × 500 = 250,000

Multiply by the average depth: 250,000 × 10 = 2,500,000

Multiply by π (approx.): 2,500,000 × 3 = 7,500,000

Round final result to 1 significant figure: 7,500,000 ≈ 8,000,000 or 8×10⁶ m³

Personally, I find writing some numbers in scientific notation tedious and confusing (for example, 5×10² for 500).

When you do square numbers written in scientific notation, remember to square the first part but double the exponent on the 10:

(5×10²)² = 25×10⁴ = 2.5×10⁵ = 250,000

 

[EvD]

Don't forget to square the radius: (5×10²)² = 500 × 500 = 250,000

Multiply by the average depth: 250,000 × 10 = 2,500,000

Multiply by π (approx.): 2,500,000 × 3 = 7,500,000

Round final result to 1 significant figure: 7,500,000 ≈ 8,000,000 or 8×10⁶ m³

Personally, I find writing some numbers in scientific notation tedious and confusing (for example, 5×10² for 500).

When you do square numbers written in scientific notation, remember to square the first part but double the exponent on the 10:

(5×10²)² = 25×10⁴ = 2.5×10⁵ = 250,000

Ah right of course. Thanks a million! That made a lot of sense and really helped me!