Use the image to answer the question.
An illustration shows a 3 D horizontal cylinder with cones on both ends. The length of the cylinder is 3 meters and its radius is 2 meters. The slant height of the left cone is 3 meters and the slant height of the right cone is 4 meters.
Find the horizontal length of the composite shape (from the vertex of the left cone to the vertex of the right cone). Round the answer to the nearest tenth.
(1 point)
Responses
10.2 m
10.2 m
8.7 m
8.7 m
7.4 m
7.4 m
11.1 m
11.1 m
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To find the horizontal length of the composite shape, we need to find the total length of the cylinder and the length of the two cones.
The length of the cylinder is given as 3 meters.
The length of a cone can be found using the Pythagorean theorem:
(a^2) + (b^2) = (c^2)
Where a and b are the radius and height of the cone, and c is the slant height.
For the left cone, we have a radius of 2 meters and a slant height of 3 meters.
(2^2) + (b^2) = (3^2)
4 + (b^2) = 9
(b^2) = 5
b = √5
So, the height of the left cone is √5 meters.
Similarly, for the right cone, we have a radius of 2 meters and a slant height of 4 meters.
(2^2) + (b^2) = (4^2)
4 + (b^2) = 16
(b^2) = 12
b = √12
So, the height of the right cone is √12 meters.
Now, to find the horizontal length of the composite shape, we add the length of the cylinder and the sum of the lengths of the two cones.
Total length = 3 + (√5 + √12)
Using a calculator, the sum of the square roots is approximately 7.4.
Rounding to the nearest tenth, the horizontal length of the composite shape is 7.4 meters.
Therefore, the correct answer is:
7.4 m