Question

A carnival ride is in the shape of a wheel with a radius of 25 feet. The wheel has 20 cars attached to the center of the wheel. What is the central angle, arc length, and area of a sector between any two cars? Round answers to the nearest hundredth if applicable. Please show Calculations

Answer 1

Answer:

the central angle = 18°

arc length $L = \dfrac{25 \pi}{10} \ \ ft$

area of a sector between any two cars = 98.18 ft²

Step-by-step explanation:

Given that:

Radius of the carnival ride = 25 feet

The wheel has 20 cars attached to the center of the wheel.

We are to calculate the following ;

the central wheel

arc length, and

area of a sector between any two cars

To start with the central wheel;

Since, the wheel has 20 cars attached to the center of the wheel; each central wheel will be :

$=\dfrac{2 \pi}{20}$

$=\dfrac{ \pi}{10} \ rad$

= 18°

The arc length L for the 1/25 th section occurs in the following ratio with the wheel circumference.

$\dfrac{2 \pi}{2 \pi(25) ft}= \dfrac{\dfrac{\pi}{10}}{L}$

$\dfrac{1}{25}= \dfrac{\pi}{10 \ L}$

$25 \pi = 10 \ L$

$L = \dfrac{25 \pi}{10} \ \ ft$

The area of a sector between any two cars is similar in ratio with the wheel's total area:

So;

$\dfrac{2 \pi \ rad }{2 \pi(25)^2 ft^2}= \dfrac{\dfrac{\pi}{10} rad}{A}$

$= \dfrac{2 }{625}=\dfrac{\pi}{10 \ A}$

${625 \pi }={2 \times 10 \ A}$

$A = \dfrac{625 \pi}{2 \times 10 }$

A  $=\dfrac{125 \pi}{4 } \ \ \ ft^2$

A = 98.18 ft²

Answer 2

Answers:

• Central angle = 18 degrees
• Arc length = 7.85 feet approximately
• Area of sector = 98.17 square feet approximately

All of these answers apply to just between two adjacent or neighboring cars. For the last two answers, I used the calculator's stored value of pi instead of something like pi = 3.14

==================================================

Explanation:

A full circle is 360 degrees. Divide this into 20 equal pieces to get 360/20 = 18. Each little pie slice is 18 degrees. This is the central angle between any two adjacent or neighboring cars.

--------------------------

The circumference of the ferris wheel is 2*pi*r = 2*pi*25 = 50pi  feet exactly. This is the total distance around the circle. We only want a small portion of that. Namely, we want the curved distance from one car to its neighbor. Go for the shortest distance possible.

The formula we'll use is

L = (x/360)*2*pi*r

note how 2*pi*r is the circumference of the circle, so we're taking a fractional portion (x/360) of this full circle perimeter to get the arc length L. The value of x is the central angle.

So,

L = (x/360)*2*pi*r

L = (18/360)*2*pi*25

L = (1/20)*50pi

L = (50/20)pi

L = 2.5pi

L = 7.85398163397449 feet

L = 7.85 feet approximately

I used my calculators stored version of pi, as opposed to something like pi = 3.14

--------------------------

The last part of this problem is finding the area of the sector between two neighboring cars. The full circle has area of pi*r^2 = pi*25^2 = 625pi square feet exactly.

We'll take a fraction of this, specifically 1/20 th of the area, to get the pie slice area we want

area of sector = (1/20)*(area of circle)

area of sector = (1/20)*625pi

area of sector = 31.25pi

area of sector = 98.174770424681 approximately

area of sector = 98.17 square feet approximately