Question

One afternoon, John, who is 5 feet 7 inches tall, casts a shadow that is 9 feet 8 inches long. At the same time of

day. a nearby tree casts a shadow that is 15 feet long John reasons he can figure out the height of the tree.
What is the height of the tree in feet? Round your answer to the nearest tenth of a foot.

Answer 1

By using the properties of similar triangles, the height of the tree is calculated to be approximately 8.6 feet when John's height and shadow length are compared to the tree's shadow length.

Step-by-step explanation:

The student is attempting to determine the height of a tree based on the ratio of John's height to the length of his shadow, using a similar ratio for the tree and its shadow. This method invokes the properties of similar triangles in this real-world problem. To solve this question, use the following proportional relationship:

John's height / John's shadow length = Tree's height / Tree's shadow length

First, we need to convert John's height (5 feet 7 inches) and shadow length (9 feet 8 inches) to inches to make the units consistent:

• 5 feet 7 inches = (5 × 12) + 7 = 67 inches
• 9 feet 8 inches = (9 × 12) + 8 = 116 inches

Now, using the formula:

67 inches / 116 inches = Tree's height (in inches) / 180 inches

Multiplying both sides by 180 inches to solve for the tree's height gives:

(67 inches / 116 inches) × 180 inches = Tree's height (in inches)

Tree's height (in inches) = (67 × 180) / 116 ≈ 103.7 inches

Finally, converting back to feet:

103.7 inches × (1 foot / 12 inches) ≈ 8.6 feet

Hence, the height of the tree is approximately 8.6 feet when rounded to the nearest tenth of a foot.

Answer 2

8.7 feet

Step-by-step explanation:

Complete such table:

`\begin{array}{ccc}&\text{Height}&\text{Shadow}\\ \\\text{John}&5\ ft\ 7\ in&9\ ft\ 8\ in\\ \\\text{Tree}&x&15\ ft\end{array}`

Convert all measures into inches:

`5 \ ft\ 7 \ in=(5\cdot 12+7)\ in=67\ in\\ \\9 \ ft\ 8 \ in=(9\cdot 12+8)\ in=116\ in\\ \\15 \ ft=15\cdot 12\ in=180\ in`

Then the table will be

`\begin{array}{ccc}&\text{Height}&\text{Shadow}\\ \\\text{John}&67\ in&116\ in\\ \\\text{Tree}&x&180\ in\end{array}`

Write a proportion:

`(67)/(x)=(116)/(180)`

Cross multiply:

`116x=180\cdot 67\\ \\116x=12,060\\ \\x=103(28)/(29)\ in`

Convert it into feet:

`103(28)/(29)\ in=\left(8\cdot 12+7(28)/(29)\right)\ in=8\ ft\ 7(28)/(29)\ in\approx 8\ ft\ 8\ in=8(8)/(12)\ ft=8(2)/(3)\ ft\approx 8.7\ ft`