Mia is 1.65 meters tall. At 3 p.m., she measures the length of a tree's shadow to be 26.45 meters. She stands 21.4 meters away from the tree, so that the tip of her shadow meets the tip of the tree's shadow. Find the height of the tree to the nearest hundredth of a meter.
First, we need to determine the length of Mia's shadow. We can do this by setting up a proportion:
Mia's height / length of Mia's shadow = length from Mia to tree / length of tree's shadow
1.65 / x = 21.4 / 26.45
Cross multiplying, we get:
26.45 * 1.65 = 21.4x
43.6425 = 21.4x
x = 43.6425 / 21.4
x = 2.04 meters
Now, we can find the height of the tree by setting up another proportion:
Mia's height / length of Mia's shadow = height of tree / length of tree's shadow
1.65 / 2.04 = height of tree / 26.45
Cross multiplying, we get:
26.45 * 1.65 = 2.04 * height of tree
43.6425 = 2.04 * height of tree
height of tree = 43.6425 / 2.04
height of tree = 21.41 meters
Therefore, the height of the tree is approximately 21.41 meters to the nearest hundredth of a meter.