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The surface area of a spherical cap is given by \(S=2 \pi r h,\) where \(r\) is the radius of the sphere and \(h\) is the perpendicular distance from the sphere's surface to the plane intersecting the sphere, forming the cap. The volume of the cap is \(V=\frac{1}{3} \pi h^{2}(3 r-h) .\) Similar to Exercise \(61,\) a formula can be found that will minimize the area of a cap that holds a specified volume. a. Solve the volume formula for the variable \(r\) b. Substitute the resulting expression for \(r\) into the surface area formula and simplify. The result is a formula for surface area given solely in terms of the volume \(V\) and the height \(h\). c. Assume the volume of the spherical cap is \(500 \mathrm{cm}^{3} .\) Use a graphing calculator to graph the resulting function on an appropriate window, and use the graph to find the height \(h\) that will result in a spherical cap with the smallest possible area, while still holding a volume of \(500 \mathrm{cm}^{3}\) d. Use this value of \(h\) and \(V=500 \mathrm{cm}^{3}\) to find the radius of the sphere.

Step by step solution

01

Solve the Volume Formula for $r$

The given volume formula is \( V = \frac{1}{3} \pi h^2 (3r - h) \). We need to isolate \( r \):Multiply both sides by 3 to clear the fraction:\[ 3V = \pi h^2 (3r - h) \]Divide both sides by \( \pi h^2 \):\[ r = \frac{3V}{\pi h^2} + \frac{h}{3} \].

02

Substitute and Simplify the Surface Area Formula

Now substitute the expression for \( r \) into the surface area formula \( S = 2 \pi r h \):\[ S = 2 \pi h \left( \frac{3V}{\pi h^2} + \frac{h}{3} \right) \]Distribute \( 2 \pi h \):\[ S = 2h \left( \frac{3V}{h^2} \right) + \frac{2\pi h^2}{3} \]Simplify:\[ S = \frac{6V}{h} + \frac{2\pi h^2}{3} \].This gives the surface area in terms of \( V \) and \( h \).

03

Graph and Find Minimum Area

Given \( V = 500 \), the formula becomes:\[ S = \frac{3000}{h} + \frac{2\pi h^2}{3} \].Graph this function using a graphing calculator. Analyze the graph to locate the minimum point of \( S \) with respect to \( h \).You should find that the point provides the smallest surface area for \( V = 500 \).

04

Use the Height to Find the Radius

With the optimal value of \( h \) found from the graph, substitute \( V = 500 \) and this \( h \) back into the equation obtained in Step 1 for \( r \):\[ r = \frac{3 \times 500}{\pi h^2} + \frac{h}{3} \].Calculate \( r \) with the given \( h \) to find the radius of the sphere.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Surface Area of a Spherical Cap

When discussing the surface area of a spherical cap, you're looking at the part of a sphere's surface cut off by a plane. Imagine slicing an orange; the exposed surface of the top piece is similar to a spherical cap.
The formula to calculate the surface area of this cap is given by:

• \( S = 2 \pi r h \)

Here:

• \( S \) is the surface area of the cap.
• \( r \) stands for the radius of the whole sphere.
• \( h \) represents the height or the perpendicular distance from the cap to the sphere's surface.

Understanding that this formula requires both \( r \) and \( h \) is crucial because without either, you cannot compute the surface area adequately.
Furthermore, if you're given the cap's volume (a common scenario in these problems), you can substitute expressions to find the surface area in terms of just volume and height. This adaptation helps when other variables are unknown.

Volume Formula of a Spherical Cap

The volume of a spherical cap isn't immediately intuitive, as it deals with a three-dimensional part of a sphere. Imagine scooping out a small dome shape from a watermelon and knowing precisely how much mass (volume) that chunk holds.
The formula for the volume of a spherical cap is:

• \( V = \frac{1}{3} \pi h^2 (3r - h) \)

Breaking it down:

• \( V \) is the volume of the cap.
• \( h \) is the height from the sphere's surface to the plane creating the cap.
• \( r \) is the radius of the entire sphere.

The formula incorporates both the height and the sphere's radius, reflecting how these dimensions shape the size of the cap.
Solving the volume formula for \( r \) can be particularly useful in problems where you're given specific vol-measure and need to find how that relates to the sphere's radius.
This step often leads to complex algebraic manipulations, but it's essential for finding other properties, like the surface area, in different scenarios.

Radius of Sphere in Spherical Cap Formula

The radius of the sphere is a fundamental component when dealing with spherical caps. It's like the backbone that helps hold everything together, affecting both surface area and volume calculations.
In the context of a spherical cap:

• Finding the radius can sometimes involve rearranging formulas, such as solving the volume equation for \( r \).
• Once \( r \) is known, it helps tie together the geometry of the cap with that of the overall sphere.
• For practical scenarios, like minimizing a cap's surface area while maintaining a certain volume, knowing \( r \) is crucial.

Using

• \( r = \frac{3V}{\pi h^2} + \frac{h}{3} \)

from rearranging the volume formula showcases this importance. Here, \( 3V/(\pi h^2) + h/3 \) gives a way to calculate \( r \), illustrating how changes in volume and height directly affect the sphere's entire radius.
This understanding enables precise computation in design and engineering applications, ensuring dimensions are physically plausible.