# Roger is 5 feet tall and casts a shadow 3.5 feet longer. At the same time, the flagpole outside...

## Question:

Roger is 5 feet tall and casts a shadow 3.5 feet longer. At the same time, the flagpole outside his school casts a shadow 14 feet long.

Write and solve a proportion to find the height of the flagpole.

## Proportions and Variation:

In any numerical ratio, this denotes a relationship between two numbers {eq}x {/eq} and {eq}y {/eq}. On the other hand, a proportion shows the similarity between two ratios. When two variables are dependent, variations in the magnitude of one variable will have a proportional effect on the other. When there is an increase or decrease of a variable {eq}x {/eq} with respect to another {eq}y {/eq}, for a ratio or constant K, variations are present. In the case that we have a direct variation, it happens that when one variable increases the other increases, which can also be written as: {eq}\frac{{{y_1}}}{{{x_1}}} = \frac{{{y_2}}}{{{x_2}}} {/eq}.

{eq}\eqalign{ & {\text{In this specific case }}{\text{,we have two proportional values }}\,x\,\left( {{\text{shadow cast}}} \right){\text{ }} \cr & {\text{and }}y\,\left( {{\text{height}}} \right){\text{ that have a variation in directly proportional form}}{\text{. }} \cr & {\text{So we have:}} \cr & \,\,\,\,{x_1} = 3.5\,ft\,\,\,\,\,\,\left( {Roger{s^\prime }\,shadow} \right) \cr & \,\,\,\,{y_1} = 5\,ft\,\,\,\,\,\,\,\left( {Roge{r^\prime }s{\text{ }}height} \right) \cr & \,\,\,\,{x_2} = 14\,ft\,\,\,\,\,\,\left( {shadow{\text{ }}of{\text{ }}the{\text{ }}flagpole} \right) \cr & \,\,\,\,{y_2} = ?\,\,ft\,\,\,\,\,\,\left( {height{\text{ }}of{\text{ }}the{\text{ }}flagpole} \right) \cr & {\text{Since}}{\text{, }}x{\text{ and }}y{\text{ vary directly}}{\text{, then}}{\text{, when }}x{\text{ increases it also }} \cr & {\text{increases }}y{\text{. For this reason}}{\text{, it must be satisfied that:}} \cr & \,\,\,\,\frac{{{y_2}}}{{{x_2}}} = \frac{{{y_1}}}{{{x_1}}} \cr & {\text{So if we do cross - multiplying:}} \cr & \,\,\,\,{y_2} \cdot {x_1} = {y_1} \cdot {x_2} \cr & {\text{Now}}{\text{, solving for }}\,{y_2}{\text{:}} \cr & \,\,\,\,{y_2} = \frac{{{y_1} \cdot {x_2}}}{{{x_1}}} \cr & {\text{So}}{\text{, substituting the given values:}} \cr & \,\,\,\,{y_2} = \frac{{5 \times 14}}{{3.5}} = 20\,ft \cr & {\text{Therefore}}{\text{, the height of the flagpole is }}\boxed{20\,ft} \cr} {/eq}