# Britney is 5 feet tall and casts a 3 1/2 -foot shadow at 10:00 A.M. At that time, a nearby tree casts a 17-foot shadow. Two hours later, Britney's shadow is 2 feet long. What is the length of the shadow of the tree at this time?

May 11, 2018

The tree is (approximately) 9.71 feet tall at noon.

#### Explanation:

Because the length of an object's shadow is proportional to its height (at a given time of day), this kind of question is solved by using the proportion relation: "$a$ is to $b$ as $c$ is to $d$". Mathematically, this is written as:

$\frac{a}{b} = \frac{c}{d}$

They key to solving this question is finding the pieces of information that (a) give us 3 of these 4 values and (b) also let us use this relation to find the 4th.

What we have:

$\left.\begin{matrix}\null & \text{Britney" & "tree" \\ "height" & 5 & - \\ "10:00 shadow" & "3.5 ft" & "17 ft" \\ "12:00 shadow" & "2 ft} & \square\end{matrix}\right.$

The box marks what we want to solve for.

We need to use the "12:00 shadow" info, because it has the value we want to solve for. We can not use the height info, because it also has missing data. (We can only solve for the 4th missing piece if we have 3 of the 4 parts of the equation.)

Thus, we will use the "12:00 shadow" and the "10:00 shadow" information. We set up the ratio like this:

$\text{Britney's 10:00 shadow"/"Britney's 12:00 shadow" = "tree's 10:00 shadow"/"tree's 12:00 shadow}$

Let $x$ be the tree's shadow at noon. Substituting in the known values, we get:

$\frac{\text{3.5 ft"/"2 ft" = "17 ft}}{x}$

Cross-multiply:

$\left(\text{3.5 ft")" " x = ("17 ft")("2 ft}\right)$

Divide both sides by 3.5 ft:

x = [("17 ft")("2 ft")]/("3.5 ft")

Simplifying the right hand side gives:

$x \approx \text{9.71 ft}$