How do you solve this problem?

In a certain industrial process involving a heterogeneous catalyst, the volume of the catalyst (in the shape of a shpere) is 10.0cm^3. Calculate the surface area of the catalyst. IF the sphere is broken down into eight spheres, each having a volume of 1.25 cm^3, what is the total surface area of the spheres? Which of the two geometric configurations of the catalyst is more effective? (The surface area of a shpere is 4pir^2). Based on your analysis here, explain why it is sometimes dangerous to work in grain elevators?

Volume is 4/3 PI r^3, and Surface area of a sphere is 4PI r^2

Lets look at SurfaceArea/Volume

S/V=4PIr^2/(4/3 PI r^3)= 3/r

what happens as r goes to zero?
So it seems that smaller r is more effective is one is looking for surface area.

To solve this problem, we will follow these steps:

Step 1: Calculating the surface area of the given catalyst sphere.
Given:
Volume of the catalyst sphere = 10.0 cm^3
Surface area formula of a sphere: A = 4πr^2

Since we are given the volume, we can use it to find the radius (r) using the formula for the volume of a sphere:
Volume (V) = (4/3)πr^3

Plugging in the given volume:
10.0 cm^3 = (4/3)πr^3

To find the radius, we can rearrange the equation:
r^3 = (3/4π) * 10.0 cm^3
r^3 = 7.5π
Taking the cube root of both sides:
r = (7.5π)^(1/3)

Now, we can calculate the surface area using the formula:
A = 4πr^2
A = 4π * [(7.5π)^(1/3)]^2

Calculate this expression to find the surface area of the catalyst sphere.

Step 2: Calculating the total surface area of the eight spheres.
Given:
Volume of each sphere = 1.25 cm^3
Since each sphere has the same volume, they would have the same radius.
Using the volume formula:
1.25 cm^3 = (4/3)πr^3

To find the radius, we can rearrange the equation:
r^3 = (3/4π) * 1.25 cm^3
r^3 = (3.75/4π) cm^3
Taking the cube root of both sides:
r = [(3.75/4π) cm^3]^(1/3)

Now that we have the radius, we can calculate the surface area of one sphere using the formula:
A = 4πr^2

Once we have the surface area of one sphere, we can multiply it by 8 to get the total surface area of the eight spheres.

Step 3: Comparing the two geometric configurations.
Compare the surface area of the catalyst sphere to the total surface area of the eight smaller spheres. Determine which one is greater.

Step 4: Explaining the danger of working in grain elevators based on the analysis.
Based on the geometric configurations of the catalyst, provide an explanation of why it is sometimes dangerous to work in grain elevators.

To solve this problem, we can use the formula for the surface area of a sphere: A = 4πr^2, where A is the surface area and r is the radius of the sphere.

1) Calculate the surface area of the catalyst:
Given that the volume of the catalyst sphere is 10.0 cm^3, we need to find the radius of the sphere before calculating the surface area. We can use the formula for the volume of a sphere: V = (4/3)πr^3, where V is the volume and r is the radius.
10.0 cm^3 = (4/3)πr^3
Solving for r, we get:
r = (3V / 4π) ^ 1/3
r = (3 * 10.0 cm^3 / 4π) ^ 1/3
r ≈ 1.368 cm

Now, using the surface area formula, we can calculate the surface area of the catalyst:
A = 4π(1.368 cm)^2
A ≈ 29.48 cm^2

2) Calculate the total surface area of the eight smaller spheres:
Since each smaller sphere has a volume of 1.25 cm^3, we can find the radius of each sphere using the volume formula:
1.25 cm^3 = (4/3)πr^3
Solving for r, we get:
r = (3 * 1.25 cm^3 / 4π) ^ 1/3
r ≈ 0.83 cm

For each smaller sphere, the surface area is:
A = 4π(0.83 cm)^2
A ≈ 8.74 cm^2

Since there are eight of these smaller spheres, the total surface area is:
Total surface area of the spheres = 8 * 8.74 cm^2
Total surface area ≈ 69.92 cm^2

3) Comparing the two configurations:
The larger catalyst sphere has a surface area of 29.48 cm^2, while the eight smaller spheres combined have a total surface area of 69.92 cm^2. Therefore, the configuration using the eight smaller spheres is more effective as it provides a larger surface area for the reaction to occur.

As for the connection to grain elevators, this question seems unrelated and requires additional information to provide a specific answer. However, grain elevators are known to be potentially dangerous due to factors such as grain dust, which can be combustible and cause explosions or fires. The confinement of large quantities of grain in enclosed spaces can also lead to structural instability and collapse, posing a risk to workers. Occupational safety measures and regulations are crucial in mitigating these hazards.