Chapter 12.8, Problem 24HP

To determine

To find: whether F or C is correct.

Answer to Problem 24HP

F found the correct ratio of the surface area.

Explanation of Solution

Given:

F: $\frac{a^2}{b^2}=\frac{1^2}{{500}^2}=\frac{1}{250000}$

C: $\frac{a^2}{b^2}=\frac{1^2}{5^2}=\frac{1}{25}$

Calculation:

Given that a model for office building is built on a scale $1$ cm to $5$ m.

F and C are finding the ratio of the surface area of building to the model.

F found ratio as $\to \frac{a^2}{b^2}=\frac{1^2}{{500}^2}=\frac{1}{250000}$

While,C found ratio as $\to \frac{a^2}{b^2}=\frac{1^2}{5^2}=\frac{1}{25}$

Find the scale factor

It is given that on the model $1$ cm represents $5$ m.

Converts $5$ meters to centimeter.

Since, $1m=100cm$

Hence,

  $5m=5\times 100=500cm$

The expression for the scale factor between model and the building is $\frac{1}{500}$ .

If two solids are similar with a scale of $\frac{a}{b}$ , the surface area have the ratio ${\left(\frac{a}{b}\right)}^2$ .

That is,

  $\begin{array}{l}\frac{Surfaceareaofthemodel}{Surfaceareaofthe\text{ building}}=\frac{a^2}{b^2}\\ =\frac{1^2}{{500}^2}\\ =\frac{1}{250000}\end{array}$

Hence, F is correct.

Conclusion:

Hence, F is correct.