Question

A tree casts a 208-centimeter shadow. A person next to the tree casts a 64-centimeter shadow. If the person is 186 centimeters tall, how tall is the tree to the nearest centimeter?

A. 167 cm
B. 226 cm
C. 194 cm
D. 152 cm

Answer 1

The tree is approximately 194 cm tall, rounding to the nearest centimeter. Thus the correct option is C.

Step-by-step explanation:

To determine the height of the tree, we can use the concept of similar triangles formed by the tree and its shadow, and the person and their shadow. The ratio of the height of the tree to its shadow is the same as the ratio of the height of the person to their shadow. Mathematically, this can be expressed as
`\( \frac{{\text{{Tree Height}}}}{{\text{{Tree Shadow}}}} = \frac{{\text{{Person Height}}}}{{\text{{Person Shadow}}}} \).`

Given that the person is 186 cm tall with a 64 cm shadow, and the tree has a 208 cm shadow, we can substitute these values into the ratio and solve for the tree's height.
`\( \frac{{\text{{Tree Height}}}}{{208}} = \frac{{186}}{{64}} \)` . Solving for the tree's height, we ge
`t \( \text{{Tree Height}} \approx \frac{{186 * 208}}{{64}} \).`Calculating this expression yields approximately 194 cm.

Rounding to the nearest centimeter, the height of the tree is 194 cm, which corresponds to option C. The use of similar triangles in this problem is a common technique in geometry and provides an elegant solution to find the height of the tree based on the given measurements and relationships between the objects and their shadows.

Answer 2

By setting up a proportion based on similar triangles, we can find the height of the tree. However, the calculated height does not match any of the provided multiple-choice options. If the given numbers are correct, then none of the answer choices are correct. so the correct option is B).

Step-by-step explanation:

To determine how tall the tree is, we can use a proportion because the sun rays hitting the tree and the person will be parallel and the triangles formed by the tree, person, and their respective shadows are similar triangles. Hence, the ratio of the height of the tree to its shadow length will be the same as the ratio of the person's height to their shadow length.

Let's denote the tree's height as T.

The proportion is:

Person's Height / Person's Shadow Length = Tree's Height / Tree's Shadow Length

186 cm / 64 cm = T / 208 cm

Multiplying both sides of the equation by the tree's shadow length gives us:

T = (186 cm / 64 cm) * 208 cm

T = (186 * 208) / 64

T = 57312 / 64

T = 895.5 cm, which rounded to the nearest centimeter is 896 cm. However, this value is not one of the provided options. It seems there was either a mistake in the calculation or a typo in the multiple-choice answers. Please double-check the question and the numbers provided. If they are correct as stated, then none of the answer choices A through D is correct. If there has been a typo, please provide the correct information for a proper calculation.