Show that the surface area of a sphere is two thirds of the total surface area of any cylinder circumscribed around the sphere (the sphere therefore touches both bases of the cylinder as well as its lateral surface).
Ah, the classic case of the sphere and the cylinder in a friendly embrace! Let's dive into the math, shall we?
First, let's consider the sphere. The surface area of a sphere can be given by the formula A_sphere = 4πr^2, where r is the radius of the sphere.
Now, let's take a look at the cylinder. The total surface area of a cylinder is given by A_cylinder = 2πr^2 + 2πrh, where r is the radius of the cylinder and h is the height.
In our scenario, the cylinder is circumscribed around the sphere, meaning that it snugly fits around the sphere in the most huggable way. Measuring the height of this cylinder would give us the diameter of the sphere, which is twice the radius (2r).
So, plugging in the value for h = 2r into the formula for the cylinder surface area, we get A_cylinder = 2πr^2 + 2πr(2r) = 2πr^2 + 4πr^2 = 6πr^2.
Now, let's compare the surface areas, shall we?
We have A_sphere = 4πr^2 and A_cylinder = 6πr^2.
Dividing the surface area of the sphere by the surface area of the cylinder, we get:
A_sphere / A_cylinder = (4πr^2) / (6πr^2) = 2/3.
Voila! We have shown that the surface area of the sphere is indeed two-thirds (2/3) of the total surface area of any cylinder circumscribed around it.
Remember, it's all a part of the geometry circus!
To show that the surface area of a sphere is two-thirds of the total surface area of any cylinder circumscribed around the sphere, we can follow these steps:
Step 1: Let's consider a sphere with radius r and a cylinder that is circumscribed around the sphere.
Step 2: The surface area of the sphere can be calculated using the formula: Surface Area of Sphere = 4πr^2.
Step 3: The total surface area of the cylinder can be calculated using the formula: Total Surface Area of Cylinder = 2πrh + 2πr^2, where h is the height of the cylinder.
Step 4: Notice that the cylinder's lateral surface area is equal to the surface area of the sphere.
Step 5: The lateral surface area of a cylinder can be calculated using the formula: Lateral Surface Area of Cylinder = 2πrh.
Step 6: Therefore, the surface area of the sphere is equal to the lateral surface area of the cylinder: Surface Area of Sphere = 2πrh.
Step 7: Rearranging the equation, we have: 2πrh = 2/3 × (2πrh + 2πr^2).
Step 8: Simplifying, we get: 2πrh = 4/3 × πrh + 4/3 × πr^2.
Step 9: Canceling out π from both sides and simplifying further, we have: 2rh = 4/3 × rh + 4/3 × r^2.
Step 10: Multiplying both sides of the equation by 3, we get: 6rh = 4rh + 4r^2.
Step 11: Subtracting 4rh from both sides, we have: 2rh = 4r^2.
Step 12: Dividing both sides of the equation by 2r, we get: h = 2r.
Step 13: From the equation h = 2r, we can observe that the height of the cylinder is twice the radius.
Step 14: Finally, substituting h = 2r in the equation of the total surface area of the cylinder, we get: Total Surface Area of Cylinder = 2πrh + 2πr^2 = 2πr(2r) + 2πr^2 = 4πr^2 + 2πr^2 = 6πr^2.
Step 15: Comparing the surface area of the sphere (4πr^2) with the total surface area of the cylinder (6πr^2), we can see that the surface area of the sphere is two-thirds of the total surface area of the cylinder.
Therefore, the surface area of a sphere is two-thirds of the total surface area of any cylinder circumscribed around the sphere.
To show that the surface area of a sphere is two-thirds of the total surface area of any cylinder circumscribed around the sphere, we can break it down into steps:
Step 1: Understand the surface area of a sphere.
- The surface area of a sphere is given by the formula: A_s = 4πr^2, where A_s is the surface area and r is the radius of the sphere.
Step 2: Understand the surface area of a cylinder.
- The surface area of a cylinder consists of three parts: the two circular bases and the lateral surface.
Step 3: Understand the lateral surface area of a cylinder.
- The lateral surface area of a cylinder is given by the formula: A_l = 2πrh, where A_l is the lateral surface area, r is the radius of the cylinder, and h is the height of the cylinder.
Step 4: Relate the sphere to the cylinder.
- In a cylinder circumscribed around a sphere, the radius of the sphere is equal to the radius of the cylinder.
- The height of the cylinder is equal to the diameter (d) of the sphere.
Step 5: Calculate the surface area of the cylinder.
- The total surface area of the cylinder is the sum of the areas of the two bases and the lateral surface area: A_c = 2(πr^2) + 2πrh.
Step 6: Simplify the expression.
- Since the radius of the sphere and the cylinder are the same, we can replace r in the expression for the cylinder's surface area: A_c = 4πr^2 + 2πrh.
Step 7: Compare the surface areas.
- Compare the surface area of the sphere (A_s) to the surface area of the cylinder (A_c).
Step 8: Calculate the ratio.
- Calculate the ratio of the sphere's surface area to the cylinder's surface area: A_s / A_c.
Step 9: Simplify the ratio.
- Substitute the formula for the surface area of the sphere and the simplified expression for the surface area of the cylinder, then simplify the ratio.
Step 10: Conclude the proof.
- If the simplified ratio of A_s / A_c is equal to 2/3, then the proof is complete, showing that the surface area of a sphere is two-thirds of the total surface area of any cylinder circumscribed around the sphere.
- If the ratio is not equal to 2/3, then the statement does not hold true for the given conditions.
By systematically evaluating each step and substituting the appropriate formulas, we can demonstrate that the surface area of a sphere is indeed two-thirds of the total surface area of any cylinder circumscribed around the sphere.
cylinder: 2πrh+2πr^2
sphere: 4πr^2
Now, since h=2r,
4πr^2 / 2πr*2r+2πr^2 = 4πr^2 / 6πr^2 = 2/3