Chapter 12.5, Problem 9WE

To determine

To find:The ratio of amount of paint required for covering model to the actual airplane.

Answer to Problem 9WE

The ratio of amount of paint required for covering model to the actual airplane is $1:40000$ .

Explanation of Solution

Given information:

The scale factor according to model airplane kit package is $1:200$ .

Calculation:

The ratio of the corresponding sides of any two similar geometric figures is termed as Scale factor.

Let the total area of the airplanes to cover the mode be $t{a}_1$ and for actual plane be $t{a}_2$ and scale factor here for both the planes be $r_1$ is $1$ and $r_2$ is $200$ .

The theorem of similar solids states that if the scale factor of two similar solids is $a:b$ , then the ratio of base areas, lateral areas and total areas is $a^2:b^2$ .

Expression to determine the total area of two similar cones from the above theorem is:

  $\begin{array}{c}\text{total}\text{area}={\left(\text{scale}\text{factor}\right)}^2\\ \frac{t{a}_1}{t{a}_2}={\left(\frac{r_1}{r_2}\right)}^2\end{array}$

Substitute $1$ for $r_1$ and $200$ for $r_2$ in the above expression.

  $\begin{array}{c}\frac{t{a}_1}{t{a}_2}={\left(\frac{r_1}{r_2}\right)}^2\\ ={\left(\frac{1}{200}\right)}^2\\ =\frac{1}{40000}\end{array}$

Therefore, the amount of paint required for covering model to the actual airplane is $1:40000$ .