A 6-foot person standing 15 feet from a streetlight casts a 15-foot shadow. Two similar triangles are formed. One triangle is formed by the person and the shadow that the person casts. A second triangle is formed by the streetlight and the ground from the base of the streetlight to the end of the shadow.
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(length of side AC)/(length of side DF) = (length of side BC)/(length of side EF)
(height of man)/(height of flagpole) = (length of man’s shadow)/L
Substituting, we have:
6 ft./150 ft. = 8 ft./L
Cross-multiplying, we get:
(6 ft.)L = (150 ft.)(8 ft.)
(6 ft.)L = 1200 ft.²
L = (1200 ft.²)/6 ft.
L = 200 ft. is the length of the shadow cast by the 150-ft. flagpole.
To solve this problem, we can set up a proportion using the similar triangles formed by the person and the shadow, and the streetlight and the ground. Let's assign variables to the unknown lengths:
Let x represent the height of the streetlight, and y represent the length of the shadow cast by the streetlight.
Based on the given information, we have the following:
Person's height: 6 feet
Distance from the person to the streetlight: 15 feet
Length of the person's shadow: 15 feet
Now, let's set up the proportion using the similar triangles:
Height of the person / Length of the person's shadow = Height of the streetlight / Length of the streetlight's shadow
6 feet / 15 feet = x / y
Now, we can solve for x by cross-multiplying:
6y = 15x
To solve for y, we need to use the Pythagorean theorem. In the right triangle formed by the streetlight and the length of the person's shadow, we have:
Length of the shadow squared + Height of the person squared = Length of the streetlight's shadow squared + Height of the streetlight squared
By substituting the values we know, we get:
15^2 + 6^2 = y^2 + x^2
Now we can substitute the value of x we found above into this equation:
15^2 + 6^2 = y^2 + (6y/15)^2
Simplifying this equation, we get:
225 + 36 = y^2 + (2y/5)^2
261 = y^2 + 4y^2/25
Multiplying both sides by 25, we get:
6575 = 25y^2 + 4y^2
Consolidating like terms, we have:
29y^2 = 6575
Dividing both sides by 29, we get:
y^2 = 6575/29
Taking the square root of both sides, we find:
y ≈ 11.79
Now that we have the value of y, we can substitute it back into the equation we used earlier to solve for x:
6y = 15x
6(11.79) = 15x
70.74 = 15x
Dividing both sides by 15, we get:
x ≈ 4.72
Therefore, the height of the streetlight is approximately 4.72 feet, and the length of the streetlight's shadow is approximately 11.79 feet.
To understand the relationship between the person, their shadow, the streetlight, and the ground, let's break down the problem and analyze it step by step.
Step 1: Visualize the problem
Imagine a person who is 6 feet tall standing 15 feet away from a streetlight. The person is casting a 15-foot shadow on the ground. Our goal is to find the relationship between the person's height, the length of the shadow, and the dimensions of the other triangle formed by the streetlight and the ground.
Step 2: Identify the triangles
In this scenario, there are two triangles. Let's label them as Triangle 1 (formed by the person and their shadow) and Triangle 2 (formed by the streetlight and the ground).
Step 3: Understand similar triangles
The key concept here is that Triangle 1 and Triangle 2 are similar triangles. Similar triangles have corresponding angles that are equal, and their sides maintain a proportional relationship. This means that if we can find the ratio of any two corresponding sides in Triangle 1 and Triangle 2, it will be the same for all other corresponding sides.
Step 4: Determine corresponding sides
In Triangle 1, we have the height of the person (6 feet) and the length of their shadow (15 feet) as the corresponding sides. Let's label the height as h1 and the shadow's length as s1.
In Triangle 2, we have the height of the streetlight (let's call it h2) and the distance from the streetlight's base to the end of the shadow (let's call it s2).
Step 5: Find the ratio
To find the ratio between corresponding sides, we can use the formula: Ratio = Corresponding side in Triangle 1 / Corresponding side in Triangle 2.
In this case, the ratio is: s1 / s2 = h1 / h2.
Since we know h1 (6 feet) and s1 (15 feet), we can write the equation as: 15 / s2 = 6 / h2.
Step 6: Solve for the unknown
We can rearrange the equation to solve for s2: s2 = (h2 * 15) / 6.
Since we don't know the height of the streetlight (h2), we can't find the actual length of s2. However, we can calculate the ratio between the shadow length and the height of the streetlight.
Step 7: Understand the ratio
The ratio s1 / s2 = h1 / h2 tells us that for every unit the person's shadow length increases, the height of the streetlight will increase by h2 / 6 units.
So, in this scenario, for every 1-foot increase in the person's shadow length (s1), the height of the streetlight (h2) increases by (h2 / 6) feet.
This ratio allows us to understand the relationship between the person's shadow and the height of the streetlight in terms of a scale factor.
Remember, this explanation is based on the assumption that the triangles formed by the person and their shadow, and the streetlight and the ground, are similar.