a person 5 ft. tall is standing near a building 30 ft. high if the length of the person's shadow is 3 ft. what is the length of the shadow of the building?
wrong, the correct answer is 18
To solve this problem, we can set up a proportion using the height of the person, the length of their shadow, and the height of the building.
Let's represent the length of the person's shadow as "x".
The proportion can be set up as follows:
(person's height) / (person's shadow length) = (building height) / (building shadow length)
Using the given measurements:
5 ft / 3 ft = 30 ft / x
Now, we can solve this proportion by cross-multiplying and then solving for "x":
5 ft * x = 3 ft * 30 ft
5x = 90
Dividing both sides by 5, we get:
x = 18
Therefore, the length of the shadow of the building is 18 ft.
two triangles used in an art project are similar the sides of the smaller triangle have length of 6 in 7 in and 12 in, the perimeter of the larger triangle is 75 in, what is the length of the shortest side of the larger triangle?
Since the two triangles are similar, their corresponding sides are proportional.
Let's represent the lengths of the sides of the larger triangle as "x", "y", and "z".
The proportion can be set up as follows:
x/6 = y/7 = z/12
We can rewrite this proportion as:
x = (6/7)y = (6/12)z
Now, we know that the perimeter of the larger triangle is 75 in:
x + y + z = 75
We can substitute the expressions for x and y from the proportion:
(6/7)y + y + (6/12)z = 75
Multiply both sides of the equation by 7 and 12 to eliminate the fractions:
72y + 84y + 42z = 900
Combine like terms:
156y + 42z = 900
Now, we need to find the length of the shortest side of the larger triangle. The shortest side corresponds to the side of the smaller triangle with a length of 6 in.
Substituting x = 6 into the first equation of the proportion:
6/6 = y/7
1 = y/7
Multiplying both sides by 7:
7 = y
So, the length of the shortest side of the larger triangle is 7 in.
I apologize for the mistake in my previous response. Let's solve the problem correctly this time:
We are given that the sides of the smaller triangle have lengths of 6 in, 7 in, and 12 in.
Let's represent the lengths of the sides of the larger triangle as "x", "y", and "z".
We know that the sides of the two triangles are proportional, so we can set up the following proportions:
x/6 = y/7
x/6 = z/12
To find the length of the shortest side of the larger triangle, we need to determine the value of "x".
From the proportion x/6 = y/7, we can rewrite it as x = (6/7)y.
Similarly, from the proportion x/6 = z/12, we can rewrite it as x = (1/2)z.
Now, let's substitute these expressions for "x" into the equation for the perimeter of the larger triangle:
(6/7)y + 7 + (1/2)z = 75
Multiplying through by 14 to eliminate the fractions:
12y + 98 + 7z = 1050
12y + 7z = 952
Now, we need to find the value of "y" that satisfies this equation. To make it easier, we can assume that "y" and "z" are positive integers.
Let's substitute different values for "y" until we find one that makes the equation true. Starting with "y = 1":
12(1) + 7z = 952
12 + 7z = 952
7z = 940
z = 940/7
z ≈ 134.29
Since "z" needs to be an integer, we can try "y = 2":
12(2) + 7z = 952
24 + 7z = 952
7z = 928
z = 928/7
z ≈ 132.57
Again, it is not an integer. Let's try "y = 3":
12(3) + 7z = 952
36 + 7z = 952
7z = 916
z = 916/7
z ≈ 130.86
This is still not an integer. Let's continue trying larger values of "y" until we find the solution:
"y = 4" leads to "z ≈ 129.14"
"y = 5" leads to "z ≈ 127.43"
"y = 6" leads to "z ≈ 125.71"
"y = 7" leads to "z ≈ 124"
Here, we finally have an integer value for "z". So, if "y = 7" and "z = 124", we can find the value of "x" using the proportion x = (6/7)y:
x = (6/7)(7)
x = 6
Therefore, the lengths of the sides of the larger triangle are x = 6 in, y = 7 in, and z = 124 in.
The length of the shortest side of the larger triangle is 6 inches.