Question

1. on level ground, the base of a tree is 20 ft from the bottom of a 48-ft flagpole. the tree is shorter than the pole. at a certain time, their shadows end at the same point 60 ft from the base of the flagpole. how tall is the tree?

Explanation:

Step1: Identify similar triangles

The tree and flagpole form similar right triangles with their shadows. Let \( h \) be the tree's height. The flagpole height is 48 ft, base to shadow end: \( 60 - 0 = 60 \) ft (wait, no: tree base is 20 ft from flagpole base. So flagpole shadow length: distance from flagpole base to shadow end. The tree's shadow end is same as flagpole's? Wait, the problem says: tree base is 20 ft from flagpole base. At a time, their shadows end at same point 60 ft from flagpole base. So flagpole shadow length: 60 ft. Tree shadow length: \( 60 - 20 = 40 \) ft (since tree is 20 ft from flagpole, so from tree base to shadow end is \( 60 - 20 = 40 \) ft? Wait, no: flagpole base to shadow end is 60 ft. Tree base is 20 ft from flagpole base, so tree base to shadow end is \( 60 - 20 = 40 \) ft? Wait, maybe: flagpole height \( H = 48 \), shadow length \( S_f = 60 \). Tree height \( h \), shadow length \( S_t = 60 - 20 = 40 \) (since tree is 20 ft closer to shadow end? Wait, no: tree base is 20 ft from flagpole base. So if shadow end is 60 ft from flagpole base, then tree base is 20 ft from flagpole, so tree base to shadow end is \( 60 - 20 = 40 \) ft. So similar triangles: \( \frac{h}{40} = \frac{48}{60} \).

Step2: Solve for \( h \)

Cross - multiply: \( 60h = 48\times40 \). Calculate \( 48\times40 = 1920 \). Then \( h=\frac{1920}{60}=32 \). Wait, but let's check again. Wait, maybe the tree is shorter than the pole, so shadow lengths: flagpole shadow is 60 ft (from flagpole base to shadow end). Tree is 20 ft from flagpole, so tree base to shadow end is \( 60 - 20 = 40 \) ft? Or maybe the tree's shadow is 60 - 20 = 40? Wait, no, maybe I got the shadow lengths wrong. Wait, the problem says: "the base of a tree is 20 ft from the bottom of a 48 - ft flagpole. The tree is shorter than the pole. At a certain time, their shadows end at the same point 60 ft from the base of the flagpole." So flagpole: height 48, shadow length 60 (from flagpole base to shadow end). Tree: base is 20 ft from flagpole base, so from tree base to shadow end is \( 60 - 20 = 40 \) ft (since tree is between flagpole and shadow end? No, if tree is shorter, maybe the tree is 20 ft from flagpole, so flagpole base to tree base is 20 ft, tree base to shadow end is \( 60 - 20 = 40 \) ft. So the two triangles: flagpole triangle: height 48, base 60. Tree triangle: height \( h \), base 40. Since similar, \( \frac{h}{40}=\frac{48}{60} \). So \( h = \frac{48\times40}{60}=32 \).

Answer:

32