Question

Without using a computer or a calculator, estimate the change in length of a space diagonal of a box whose dimensions are changed from \(200 \times 200 \times 100\) to \(201 \times 202 \times 99.\)

Short answer

The estimated change in diagonal length is 1.67 units.

Step-by-step solution

Calculate the Original Diagonal

Use the formula for the space diagonal of a box: \[d = \sqrt{a^2 + b^2 + c^2} \] where \ a, b, \ and \ c \ are the box's dimensions. For dimensions \ 200 \times 200 \times 100, \ substitute \ a = 200, b = 200, \ and \ c = 100 \ into the formula: \[d = \sqrt{200^2 + 200^2 + 100^2} = \sqrt{40000 + 40000 + 10000} = \sqrt{90000} = 300 \]

Calculate the New Diagonal

Repeat the process with the new dimensions \ 201 \times 202 \times 99 \ using the formula for the space diagonal: \[d' = \sqrt{201^2 + 202^2 + 99^2} \] Calculate each square: \ 201^2 = 40401 \, \ 202^2 = 40804 \, and \ 99^2 = 9801 \ Add them together: \[d' = \sqrt{40401 + 40804 + 9801} = \sqrt{91006} \]

Estimate the New Diagonal

Approximate the square root of \ 91006 \. Since it is close to \ 90000, \ we can approximate \ \sqrt{91006} \ as follows: \[\sqrt{91006} \approx 301.67 \]

Find the Change in Diagonal Length

Subtract the original diagonal length from the new diagonal length: \[\Delta d = 301.67 - 300 = 1.67 \] So, the estimated change in diagonal length is \ 1.67 \ units.

Key concepts

Diagonal of a Box

Understanding the diagonal of a box, also known as the space diagonal, is important in various geometrical and real-life applications. The space diagonal is the longest diagonal you can draw inside a rectangular box and stretches from one vertex to the opposite vertex. To calculate the space diagonal, we use the formula: \(d = \sqrt{a^2 + b^2 + c^2}\), where \(a\), \(b\), and \(c\) are the lengths of the box's edges.
For example, for a box with dimensions 200 x 200 x 100, you sum the squares of these dimensions: \((200^2 + 200^2 + 100^2) \Rightarrow 40000 + 40000 + 10000 = 90000\). The square root of this sum gives us the diagonal length: \(\sqrt{90000} = 300\). This straightforward method helps us determine the space diagonal without much complexity.

Square Root Approximation

Approximating square roots is a useful skill, especially when you want to quickly estimate values without a calculator. The idea is to find a number that, when squared, is close to our target number.
In our exercise, we calculated the new diagonal length for dimensions 201 x 202 x 99, which gives \(\sqrt{91006}\). Since \(\sqrt{91006}\) is close to \(\sqrt{90000}\), which we know is 300, we approximate \(\sqrt{91006}\) to be slightly more, around 301.67. Another quick method involves recognizing the difference between the numbers; knowing that \(91006\) is a little larger than \(90000,\) we can comfortably estimate the root without precision but well enough for practical purposes.
This technique helps especially in academic exercises where estimation suffices over exact calculations.

Dimension Change Effect

Changing the dimensions of a box affects its space diagonal. Even small changes in dimensions can lead to noticeable differences. For instance, altering the box dimensions from 200 x 200 x 100 to 201 x 202 x 99 changes the space diagonal.
Initially, our space diagonal was \(\sqrt{90000} = 300\), but after changes, it becomes \(\sqrt{91006} = 301.67\). The new diagonal length slightly increased.
Calculating this change shows that small modifications in dimensions affect the overall distance inside the box. Understanding this concept helps with many practical problems in fields from geometry to engineering, where precise measurements affect design and functionality.