Jacob stands 4.5 feet tall and casts a 2.7 foot shadow.
A. Using this relationship, determine how tall Jacob's house is if the house's shadow measures 19.2 feet. Assume the triangles are similar.
B. Explain the concept of indirect measurement and how it is used in this situation.
C. There is a large tree in Jacob's front yard that measures 39 feet. Write out the steps to find the length of the tree's shadow, assuming it is similar.
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A. To determine the height of Jacob's house, we can use the proportion of heights to shadow lengths. We know that Jacob's height is 4.5 feet and his shadow is 2.7 feet, and we are given the shadow length of his house as 19.2 feet.
Let's set up the proportion:
(Height of Jacob) / (Length of Jacob's shadow) = (Height of house) / (Length of house's shadow)
Using the given values, we can substitute them into the proportion:
4.5 feet / 2.7 feet = (Height of house) / 19.2 feet
Cross-multiplying, we have:
4.5 feet * 19.2 feet = (Height of house) * 2.7 feet
86.4 square feet = (Height of house) * 2.7 feet
Now we can isolate the height of the house by dividing both sides of the equation by 2.7 feet:
86.4 square feet / 2.7 feet = (Height of house)
Simplifying the calculation:
32 feet = (Height of house)
Therefore, Jacob's house is 32 feet tall.
B. The concept of indirect measurement involves using known measurements and proportions to determine an unknown measurement indirectly. In this situation, we can use the relationship between the heights and shadow lengths of Jacob to find the height of his house. By establishing the proportion based on the similar triangles formed by Jacob and his house, we can solve for the unknown height. Indirect measurement is useful when direct measurements are not practical or when the object being measured is inaccessible.
C. To find the length of the tree's shadow using indirect measurement, we can again make use of the proportion based on similar triangles. Given that the tree's height is 39 feet, we want to find the length of its shadow.
Let's set up the proportion:
(Height of tree) / (Length of tree's shadow) = (Height of Jacob) / (Length of Jacob's shadow)
Now we can substitute the known values:
39 feet / (Length of tree's shadow) = 4.5 feet / 2.7 feet
Cross-multiplying, we have:
39 feet * 2.7 feet = 4.5 feet * (Length of tree's shadow)
105.3 square feet = 4.5 feet * (Length of tree's shadow)
Now we can isolate the length of the tree's shadow by dividing both sides of the equation by 4.5 feet:
105.3 square feet / 4.5 feet = (Length of tree's shadow)
Simplifying the calculation:
23.4 feet = (Length of tree's shadow)
Therefore, the length of the tree's shadow is 23.4 feet.