Problem 58 - Calculus with Concepts in Calculus

Question

A tank has the form of a right circular cylinder with hemispherical ends. Its volume \(V\) is 100 cubic meters. a. Find the length \(L\) of the cylinder in terms of the radius of the hemispheres. b. Find the length \(L\) (to the nearest centimeter) if the radius of the hemispheres is 2 meters. c. How much longer (to the nearest centimeter) would the cylindrical portion of the tank need to be if the radius of the hemispheres were 1 meter instead of 2 meters?

Step-by-Step Solution

Answer

a. \( L = \frac{100 - \frac{4}{3} \pi r^3}{\pi r^2} \); b. 598 cm; c. 2551 cm longer.

Step 1: Understand the Tank Structure

The tank consists of a right circular cylinder and two hemispherical ends. When put together, the two hemisphere ends form a full sphere.

Step 2: Write the Equation for Volume

Since the tank is made up of a cylindrical part and a full sphere part, we write the total volume equation: \[ V = V_{cylinder} + V_{sphere} \], where \( V_{cylinder} = \pi r^2 L \) and \( V_{sphere} = \frac{4}{3} \pi r^3 \).

Step 3: Substitute Total Volume

We know the total volume \( V \) is 100 cubic meters. Substituting this into our equation, we get:\[ \pi r^2 L + \frac{4}{3} \pi r^3 = 100 \].

Step 4: Solve for Cylinder Length (a)

Re-organize the equation to solve for \( L \): \[ L = \frac{100 - \frac{4}{3} \pi r^3}{\pi r^2} \].

Step 5: Calculate L for r = 2 meters (b)

Substitute \( r = 2 \) into the formula: \[ L = \frac{100 - \frac{4}{3} \pi (2)^3}{\pi (2)^2} = \frac{100 - \frac{32}{3} \pi}{4 \pi} \].Solve to find \( L \approx 5.98 \) meters, which is 598 cm.

Step 6: Calculate L for r = 1 meter

Substitute \( r = 1 \): \[ L = \frac{100 - \frac{4}{3} \pi (1)^3}{\pi (1)^2} = \frac{100 - \frac{4}{3} \pi}{\pi} \].Calculate to find \( L \approx 31.49 \) meters, which is 3149 cm.

Step 7: Calculate the Difference in Length (c)

Subtract the length found with \( r = 2 \) from the length found with \( r = 1 \): \[ 3149 \text{ cm} - 598 \text{ cm} = 2551 \text{ cm} \].

Key Concepts

Understanding Geometry in Composite Figures

Geometry plays a significant role in understanding the structure of the tank in this problem. The tank comprises several geometric shapes: a cylinder and two hemispheres. When these components are combined, the hemispheres form a complete sphere, and the cylinder stretches between them.

The trick to solving such problems is to visualize how the components fit together. Consider the hemispheres as a complete sphere added to either end of the cylinder. This can help in comprehending the total structure effectively.

The beauty of geometry lies in its rules and formulas that define various shapes. For instance, the formula for the volume of a sphere is \( \frac{4}{3} \pi r^3 \) and for a cylinder, it is \( \pi r^2 L \). By combining these, the total volume can be found easily using basic geometry.

Applying Calculus to Solve Problems

Using calculus concepts like substitution and solving equations is essential in this exercise. The entire process becomes more straightforward when approached systematically: breaking down, substituting values, and solving step by step.

Begin with what is known—here, the total volume of the tank. By setting up the equation \( V = V_{cylinder} + V_{sphere} \) using known volumes for cylindrical and spherical parts, we derive a relationship that includes the length of the cylinder, \( L \).

The goal is to isolate \( L \), requiring reorganization of the given equation. Through substitution of known values such as the radius, the formula can be simplified to find accurate measurements, such as the length of the tank's cylinder. This method not only emphasizes problem-solving strategies but also highlights how calculus can apply basic rules to real-world scenarios.

Calculating Volume of Composite Figures

Composite figures involve combining different geometric shapes into one whole figure. To find the volume of such figures, we calculate each individual shape's volume and then sum them all.

For our tank, we deal with a cylindrical portion and two hemispherical ends. By calculating the cylinder's volume using \( V_{cylinder} = \pi r^2 L \) and that of the sphere from the hemispheres using \( V_{sphere} = \frac{4}{3} \pi r^3 \), we can find the total volume. This requires careful consideration to ensure all dimensional factors, such as the radius \( r \), are consistently applied.

Attention to unit conversion, like switching from meters to centimeters in some scenarios, is crucial. These precise calculations ensure the correct dimensions are found, whether adjusting for changes in the radius or calculating how this affects the other components of a composite figure.