Question

1. find the height of the tree.

(image shows a right triangle situation with a tree height x ft, a person of 4 ft, the base distance from tree to persons shadow end is 125 ft, and the persons shadow length is 5.5 ft.)
tree = ____

Explanation:

Step1: Identify Similar Triangles

The tree and the girl form two similar right triangles (since both have a right angle and share the same angle from the sun, so AA similarity). So, the ratios of corresponding sides are equal. Let the height of the tree be \( x \) ft. The height of the girl is 4 ft, her shadow is 5.5 ft, and the tree's shadow (including the girl's shadow? Wait, no—wait, the distance from the tree to the girl's base is 125 ft, and the girl's shadow is 5.5 ft? Wait, maybe the tree's shadow length is \( 125 + 5.5 \)? Wait, no, looking at the diagram: the horizontal segment from the tree to the girl's feet is 125 ft, and the girl's shadow is 5.5 ft. So the tree's shadow length is \( 125 + 5.5 \)? Wait, no, maybe the tree's shadow is 125 ft, and the girl's shadow is 5.5 ft? Wait, no, the diagram shows: the vertical side for the tree is \( x \), horizontal from tree to girl's vertical (the girl's height is 4 ft) is 125 ft, and the girl's shadow (horizontal) is 5.5 ft. So the two triangles: one with height \( x \) and base (shadow) \( 125 + 5.5 \)? Wait, no, maybe the tree's shadow is 125 ft, and the girl's shadow is 5.5 ft? Wait, no, the correct setup is: height of object / length of shadow = height of object / length of shadow. So girl's height (4 ft) / girl's shadow (5.5 ft) = tree's height (\( x \)) / tree's shadow (125 ft + 5.5 ft? Wait, no, maybe the tree's shadow is 125 ft, and the girl is standing such that her shadow is 5.5 ft, and the distance from the tree to the girl is 125 ft. So the tree's shadow length is \( 125 + 5.5 \)? Wait, no, let's re-express. Let’s denote:

Girl's height: \( h_g = 4 \) ft, Girl's shadow length: \( s_g = 5.5 \) ft.

Tree's height: \( h_t = x \) ft, Tree's shadow length: \( s_t = 125 + 5.5 = 130.5 \) ft? Wait, no, maybe the tree's shadow is 125 ft, and the girl's shadow is 5.5 ft, and the girl is between the tree and the end of her shadow. Wait, the diagram: the horizontal line from the tree (vertical) to the girl (vertical) is 125 ft, and the girl's shadow (horizontal from her to the end) is 5.5 ft. So the tree's shadow is \( 125 + 5.5 = 130.5 \) ft? No, maybe the tree's shadow is 125 ft, and the girl's shadow is 5.5 ft, and they are similar triangles, so \( \frac{x}{125} = \frac{4}{5.5} \). Wait, that makes more sense. Let's check the diagram again: the vertical side for the tree is \( x \), horizontal (shadow) is 125 ft. The girl's height is 4 ft, her shadow (horizontal) is 5.5 ft. So the two triangles are similar, so corresponding sides are proportional. So \( \frac{\text{Tree height}}{\text{Tree shadow}} = \frac{\text{Girl height}}{\text{Girl shadow}} \), so \( \frac{x}{125} = \frac{4}{5.5} \).

Step2: Solve for \( x \)

Cross-multiplying: \( x \times 5.5 = 4 \times 125 \)

Calculate right side: \( 4 \times 125 = 500 \)

Then, \( x = \frac{500}{5.5} \)

Simplify: \( \frac{500}{5.5} = \frac{5000}{55} = \frac{1000}{11} \approx 90.91 \)? Wait, no, wait, maybe I mixed up the shadow lengths. Wait, maybe the tree's shadow is 125 ft, and the girl's shadow is 5.5 ft, and the distance from the tree to the girl is 125 ft, so the tree's shadow is 125 ft, and the girl's shadow is 5.5 ft, so the ratio is \( \frac{4}{5.5} = \frac{x}{125} \). Wait, that would be if the tree's shadow is 125 ft. Let's check the calculation:

\( x = \frac{4 \times 125}{5.5} = \frac{500}{5.5} \approx 90.91 \). But that seems low. Wait, maybe the tree's shadow is 125 + 5.5 = 130.5 ft? Let's try that:

\( \frac{4}{5.5} = \frac{x}{130.5} \)

\( x = \frac{4 \times 130.5}{5.5} = \frac{522}{5.5} = 94.909... \) No, that's not right. Wait, maybe the diagram is: the horizontal segment from the tree to the girl's feet is 125 ft, and the girl's shadow (from her feet to the end of the shadow) is 5.5 ft. So the tree's shadow is 125 + 5.5 = 130.5 ft, and the girl's shadow is 5.5 ft, height 4 ft. So the ratio is \( \frac{4}{5.5} = \frac{x}{130.5} \). Then \( x = \frac{4 \times 130.5}{5.5} = \frac{522}{5.5} = 94.909... \) No, that still doesn't seem right. Wait, maybe the tree's shadow is 125 ft, and the girl is standing at the end of the tree's shadow? No, the diagram shows the girl between the tree and the end of the shadow. Wait, let's re-express the problem correctly.

Wait, the standard similar triangles problem: when an object casts a shadow, the height of the object over the length of the shadow is constant (same angle from the sun). So if the girl is 4 ft tall and her shadow is 5.5 ft, then the ratio of height to shadow is \( \frac{4}{5.5} \). The tree's shadow is the distance from the tree to the end of its shadow, which is 125 ft (distance from tree to girl) plus 5.5 ft (girl's shadow) = 130.5 ft? Or is the tree's shadow 125 ft, and the girl's shadow is 5.5 ft, and they are separate? Wait, the diagram: the vertical line for the tree is \( x \), horizontal line (shadow) is 125 ft. The girl is 4 ft tall, her shadow (horizontal) is 5.5 ft, and the horizontal line from the tree to the girl is 125 ft. So the two triangles: one with height \( x \) and base 125 ft, and one with height 4 ft and base 5.5 ft. So they are similar, so \( \frac{x}{125} = \frac{4}{5.5} \). Then solving for \( x \):

\( x = \frac{4 \times 125}{5.5} = \frac{500}{5.5} = \frac{5000}{55} = \frac{1000}{11} \approx 90.91 \) ft. Wait, but maybe the tree's shadow is 125 + 5.5 = 130.5 ft. Let's check both.

Wait, maybe the correct setup is: the distance from the tree to the girl is 125 ft, and the girl's shadow is 5.5 ft, so the tree's shadow is 125 + 5.5 = 130.5 ft. Then:

\( \frac{4}{5.5} = \frac{x}{130.5} \)

\( x = \frac{4 \times 130.5}{5.5} = \frac{522}{5.5} = 94.909... \) No, that's not matching. Wait, maybe the diagram is that the tree's shadow is 125 ft, and the girl is 4 ft tall with a shadow of 5.5 ft, so the ratio is \( \frac{4}{5.5} \), so tree height \( x = \frac{4}{5.5} \times 125 \approx 90.91 \) ft. Let's compute that:

\( 4 \times 125 = 500 \)

\( 500 \div 5.5 = 500 \div \frac{11}{2} = 500 \times \frac{2}{11} = \frac{1000}{11} \approx 90.91 \) ft. So approximately 90.91 ft, or as a fraction \( \frac{1000}{11} \) ft.

Wait, but maybe I made a mistake in the shadow length. Let's confirm: in similar triangles, the ratio of corresponding sides is equal. So height of tree / length of tree's shadow = height of girl / length of girl's shadow. So if the tree's shadow is 125 ft (the horizontal distance from the tree to the end of its shadow) and the girl's shadow is 5.5 ft (horizontal distance from her to the end of her shadow), then yes, \( \frac{x}{125} = \frac{4}{5.5} \). So solving for \( x \):

\( x = \frac{4 \times 125}{5.5} = \frac{500}{5.5} \approx 90.91 \) ft. So the height of the tree is \( \frac{1000}{11} \) ft or approximately 90.91 ft.

Step2: Calculate the Value

So, \( x = \frac{4 \times 125}{5.5} \)

First, multiply 4 and 125: \( 4 \times 125 = 500 \)

Then divide by 5.5: \( 500 \div 5.5 = \frac{500}{5.5} = \frac{5000}{55} = \frac{1000}{11} \approx 90.91 \) (or as a decimal, approximately 90.91 ft, or exactly \( \frac{1000}{11} \) ft).

Answer:

The height of the tree is \( \frac{1000}{11} \) ft (or approximately \( 90.91 \) ft).