Finding the Angle of Cut for a Spherical Fruit Slice

In summary, the problem involves finding the angle of cut for a slice of a spherical fruit with a radius of 1 inch, given that the surface area of the peel for the slice is equal to pi/8. The formula for the surface area of a sphere can be used to solve this problem, and it can also be approached by taking the derivative of the volume formula and using spherical coordinates. Alternatively, the problem can be solved by setting up a proportion with the given surface area and the total surface area of the sphere.
  • #1

Homework Statement


A spherical fruit has a radius of 1 inch. A slice has surface area of its peel equal to pi/8 . Determine the angle of cut for the slice.

Homework Equations


I'm sure there is a relevant equation here but I don't know it :-(

The Attempt at a Solution


So the radius is 1 inch, and the surface of the peel is equal to pi/8... I know that the volume of a sphere is 4/3*pi*r^3, maybe if i took the derivative i could get an equation for surface area? that would make sense because the units would be squared which is what surface area is... am i going the right direction here?
 
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  • #2
GeometryIsHARD said:

Homework Statement


A spherical fruit has a radius of 1 inch. A slice has surface area of its peel equal to pi/8 . Determine the angle of cut for the slice.

Homework Equations


I'm sure there is a relevant equation here but I don't know it :-(

The Attempt at a Solution


So the radius is 1 inch, and the surface of the peel is equal to pi/8... I know that the volume of a sphere is 4/3*pi*r^3, maybe if i took the derivative i could get an equation for surface area? that would make sense because the units would be squared which is what surface area is... am i going the right direction here?
You can look up the formula for the surface area of a sphere. It's not something which is top secret.

I'm surprised you know the formula for the volume of a sphere, but not the formula for the SA.

Hint: Google "sphere"
 
  • #3
Yes. You can take the derivative of volume (with respect to radius) to get the surface area.
If you wanted to go nuts, you could even switch to spherical coordinates and take an integral.
## \rho = 1 \\ \phi \in [0, \pi] \\ \theta \in [0, 2\pi] ##
##\int_{0}^{\pi}\int_0^{2\pi} \rho^2 \sin\phi d\theta d\phi ##
Or, like Steamking said, start with the formula for surface area, and then this problem gets changed into a proportion.
##\frac{\pi/8}{\theta} = \frac{Area}{2\pi}##
 

1. How do you find the angle of cut for a spherical fruit slice?

To find the angle of cut for a spherical fruit slice, you need to first determine the diameter of the fruit. Then, using the formula 180 x (diameter of fruit / circumference of fruit), you can calculate the angle of cut.

2. What is the circumference of a spherical fruit?

The circumference of a spherical fruit can be calculated using the formula 2 x π x (radius of fruit). If the radius is not given, it can be found by dividing the diameter of the fruit by 2.

3. Can the angle of cut be different for different types of fruits?

Yes, the angle of cut for a spherical fruit can vary depending on the size and shape of the fruit. It is important to measure the diameter and circumference of the specific fruit being cut in order to accurately determine the angle.

4. Why is it important to find the angle of cut for a spherical fruit slice?

Finding the angle of cut is important because it ensures that the fruit slices are uniform in size and shape. This is especially important for presentation purposes or if the fruit slices are being used in a recipe where consistency is key.

5. Is there an easier way to find the angle of cut for a spherical fruit slice?

There are tools available that can help you find the angle of cut for a spherical fruit slice, such as a fruit corer or a fruit slicer with adjustable angles. These tools can save time and effort in calculating the angle manually.

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