Answer :
Let's solve this step-by-step to determine the radius of a sphere that has the same surface area as a given cylinder.
### Step 1: Calculate the Surface Area of the Cylinder
A cylinder's surface area is given by the formula:
[tex]\[ \text{Surface Area} = 2\pi rh + 2\pi r^2 \][/tex]
Given:
- Height (h) = 4 meters
- Radius (r) = 1.5 meters
Plugging these values into the formula, we have:
[tex]\[ \text{Surface Area} = 2\pi (1.5)(4) + 2\pi (1.5)^2 \][/tex]
[tex]\[ \text{Surface Area} = 2\pi (6) + 2\pi (2.25) \][/tex]
[tex]\[ \text{Surface Area} = 12\pi + 4.5\pi \][/tex]
[tex]\[ \text{Surface Area} = 16.5\pi \][/tex]
Using the value of [tex]\(\pi\)[/tex] (pi) as approximately 3.141592653589793:
[tex]\[ \text{Surface Area} \approx 16.5 \times 3.141592653589793 \][/tex]
[tex]\[ \text{Surface Area} \approx 51.83627878423159 \text{ square meters} \][/tex]
### Step 2: Calculate the Radius of the Sphere
We need to find the radius of a sphere that has the same surface area as the cylinder. The surface area of a sphere is given by the formula:
[tex]\[ \text{Surface Area} = 4\pi r^2 \][/tex]
We set the surface area of the sphere equal to the surface area of the cylinder:
[tex]\[ 4\pi r^2 = 51.83627878423159 \][/tex]
Isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{51.83627878423159}{4\pi} \][/tex]
[tex]\[ r^2 = \frac{51.83627878423159}{4 \times 3.141592653589793} \][/tex]
[tex]\[ r^2 \approx \frac{51.83627878423159}{12.566370614359172} \][/tex]
[tex]\[ r^2 \approx 4.125 \][/tex]
Take the square root of both sides to find [tex]\( r \)[/tex]:
[tex]\[ r \approx \sqrt{4.125} \][/tex]
[tex]\[ r \approx 2.03100960115899 \][/tex]
Thus, the approximate radius of the sphere is:
[tex]\[ r \approx 2.0 \text{ meters} \][/tex]
### Conclusion
Among the given options, the approximate radius of the sphere that has the same surface area as the cylinder is:
D. 2.0 m
### Step 1: Calculate the Surface Area of the Cylinder
A cylinder's surface area is given by the formula:
[tex]\[ \text{Surface Area} = 2\pi rh + 2\pi r^2 \][/tex]
Given:
- Height (h) = 4 meters
- Radius (r) = 1.5 meters
Plugging these values into the formula, we have:
[tex]\[ \text{Surface Area} = 2\pi (1.5)(4) + 2\pi (1.5)^2 \][/tex]
[tex]\[ \text{Surface Area} = 2\pi (6) + 2\pi (2.25) \][/tex]
[tex]\[ \text{Surface Area} = 12\pi + 4.5\pi \][/tex]
[tex]\[ \text{Surface Area} = 16.5\pi \][/tex]
Using the value of [tex]\(\pi\)[/tex] (pi) as approximately 3.141592653589793:
[tex]\[ \text{Surface Area} \approx 16.5 \times 3.141592653589793 \][/tex]
[tex]\[ \text{Surface Area} \approx 51.83627878423159 \text{ square meters} \][/tex]
### Step 2: Calculate the Radius of the Sphere
We need to find the radius of a sphere that has the same surface area as the cylinder. The surface area of a sphere is given by the formula:
[tex]\[ \text{Surface Area} = 4\pi r^2 \][/tex]
We set the surface area of the sphere equal to the surface area of the cylinder:
[tex]\[ 4\pi r^2 = 51.83627878423159 \][/tex]
Isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{51.83627878423159}{4\pi} \][/tex]
[tex]\[ r^2 = \frac{51.83627878423159}{4 \times 3.141592653589793} \][/tex]
[tex]\[ r^2 \approx \frac{51.83627878423159}{12.566370614359172} \][/tex]
[tex]\[ r^2 \approx 4.125 \][/tex]
Take the square root of both sides to find [tex]\( r \)[/tex]:
[tex]\[ r \approx \sqrt{4.125} \][/tex]
[tex]\[ r \approx 2.03100960115899 \][/tex]
Thus, the approximate radius of the sphere is:
[tex]\[ r \approx 2.0 \text{ meters} \][/tex]
### Conclusion
Among the given options, the approximate radius of the sphere that has the same surface area as the cylinder is:
D. 2.0 m
