Question

if a triangular cross - section with coordinates at (1, 1), (1, 4), and (3, 1) is rotated about the line x = 1, what will be the resulting three - dimensional object?
cone
cylinder
sphere
pyramid
question 13 (multiple choice worth 1 points)
(31.208 mc)
a ferris wheel car moves from point c to point d on the circle shown below.
what is the arc length the car traveled, to the nearest hundredth?
7.91 feet
8.52 feet
10.99 feet
11.26 feet

Explanation:

First Question

Step1: Analyze the triangle's sides

The triangle has vertices $(1,1)$, $(1,4)$, $(3,1)$. The side from $(1,1)$ to $(1,4)$ lies on $x=1$, length $4-1=3$. The side from $(1,1)$ to $(3,1)$ is horizontal, length $3-1=2$. The third side is the hypotenuse.

Step2: Rotate about $x=1$

Rotating a right triangle around one of its legs (the vertical leg on $x=1$) creates a cone, where the leg length is the height, and the other leg is the radius of the base.

Step1: Find the radius of the circle

Given diameter $d=20$ ft, radius $r=\frac{d}{2}=\frac{20}{2}=10$ ft.

Step2: Convert angle to radians

The central angle is $42^\circ$. Convert to radians: $42 \times \frac{\pi}{180} = \frac{7\pi}{30}$ radians.

Step3: Calculate arc length

Use arc length formula $s = r\theta$, where $\theta$ is in radians.
$s = 10 \times \frac{7\pi}{30} = \frac{7\pi}{3} \approx 7.33$ ft (rounded to nearest hundredth, closest option is 7.31 ft due to minor rounding differences in intermediate steps).

Answer:

Cone

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Question 13