Question
The Las Vegas Sphere, officially known as the MSG Sphere, is a state-of-the-art entertainment venue that opened in 2023. Located near The Venetian Resort, it stands 366 feet tall with a diameter of 516 feet, making it the world's largest spherical structure. The exterior is covered in 1.2 million LED pucks, creating a dynamic, immersive display. Inside, it features a 160,000-square-foot 16K resolution wraparound screen and a 164,000-speaker audio system. Construction began in 2018, costing approximately $2.3 billion. The Sphere is designed to revolutionize live entertainment, offering unique sensory experiences through advanced sound and visual technologies.
Use 3.14 for pi and round answers to the nearest tenth
The formula for the volume of a sphere is :*blank 1*
The radius of the MSG Sphere is *blank 2* *blank 3*
(Blank 2 is the length and blank 3 is the unit)
The volume of the MSG Sphere is *blank 4* cubic feet or ft³
(Type in the volume using commas to separate periods and round to the nearest whole number.)
Use 3.14 for pi and round answers to the nearest tenth
The formula for the volume of a sphere is :*blank 1*
The radius of the MSG Sphere is *blank 2* *blank 3*
(Blank 2 is the length and blank 3 is the unit)
The volume of the MSG Sphere is *blank 4* cubic feet or ft³
(Type in the volume using commas to separate periods and round to the nearest whole number.)
Answers
GPT-4o mini
To solve this, we will first use the formula for the volume of a sphere, which is given by:
**Volume = \(\frac{4}{3} \pi r^3\)**
where \( r \) is the radius of the sphere.
### Step 1: Find the radius
The diameter of the MSG Sphere is given as 516 feet. The radius is half of the diameter:
\[
r = \frac{diameter}{2} = \frac{516 \text{ feet}}{2} = 258 \text{ feet}
\]
### Step 2: Calculate the volume
Now, we plug the radius into the volume formula. Using \( \pi \approx 3.14 \):
\[
Volume = \frac{4}{3} \times 3.14 \times (258 \text{ feet})^3
\]
First, we calculate \( (258)^3 \):
\[
(258)^3 = 258 \times 258 \times 258 = 17,158,632 \text{ cubic feet}
\]
Next, we calculate the volume:
\[
Volume = \frac{4}{3} \times 3.14 \times 17,158,632
\]
Calculating \( \frac{4}{3} \times 3.14 \):
\[
\frac{4}{3} \times 3.14 \approx 4.1867
\]
Now we multiply that by \( 17,158,632 \):
\[
Volume \approx 4.1867 \times 17,158,632 \approx 71,723,570.2264
\]
Rounding to the nearest whole number, the volume is:
\[
Volume \approx 71,723,570 \text{ cubic feet}
\]
### Final Answers
1. The formula for the volume of a sphere is: **\(\frac{4}{3} \pi r^3\)**
2. The radius of the MSG Sphere is **258** **feet**
3. The volume of the MSG Sphere is **71,723,570** cubic feet or ft³.
**Volume = \(\frac{4}{3} \pi r^3\)**
where \( r \) is the radius of the sphere.
### Step 1: Find the radius
The diameter of the MSG Sphere is given as 516 feet. The radius is half of the diameter:
\[
r = \frac{diameter}{2} = \frac{516 \text{ feet}}{2} = 258 \text{ feet}
\]
### Step 2: Calculate the volume
Now, we plug the radius into the volume formula. Using \( \pi \approx 3.14 \):
\[
Volume = \frac{4}{3} \times 3.14 \times (258 \text{ feet})^3
\]
First, we calculate \( (258)^3 \):
\[
(258)^3 = 258 \times 258 \times 258 = 17,158,632 \text{ cubic feet}
\]
Next, we calculate the volume:
\[
Volume = \frac{4}{3} \times 3.14 \times 17,158,632
\]
Calculating \( \frac{4}{3} \times 3.14 \):
\[
\frac{4}{3} \times 3.14 \approx 4.1867
\]
Now we multiply that by \( 17,158,632 \):
\[
Volume \approx 4.1867 \times 17,158,632 \approx 71,723,570.2264
\]
Rounding to the nearest whole number, the volume is:
\[
Volume \approx 71,723,570 \text{ cubic feet}
\]
### Final Answers
1. The formula for the volume of a sphere is: **\(\frac{4}{3} \pi r^3\)**
2. The radius of the MSG Sphere is **258** **feet**
3. The volume of the MSG Sphere is **71,723,570** cubic feet or ft³.