Problem 48 - Calculus Single Variable

Question

The gas tank of a tractor is in the shape of a cylinder lying on its side. The tank is positioned on the tractor so that its central axis is 5 feet above the ground. The radius of the tank is 1.5 feet, and its length is 5 feet. For winter storage of the tractor, gas is pumped straight up out of the tank, from the surface of the gasoline into a holding tank with opening 15 feet from ground level. If gasoline weighs 35 pounds per cubic foot, then how much work is done in emptying a full tank of gas on the tractor into the holding tank?

Step-by-Step Solution

Answer

The work done is \(3937.5 \\pi\) foot-pounds.

Step 1: Understand the Problem

We need to calculate the work done in pumping gasoline out of a cylindrical tank into a holding tank. The work done is related to moving the gasoline from the height of the center of the tank to the height of the holding tank. The central axis of the tank is 5 feet above the ground, and the gasoline needs to reach the opening of the holding tank which is 15 feet above the ground.

Step 2: Calculate the Additional Lift Height

The gasoline needs to be lifted from the central axis of the cylinder (5 feet above the ground) to the opening of the holding tank, which is 15 feet above the ground. The additional height that the gasoline needs to be lifted is the difference between these two heights: \(15 - 5 = 10\) feet.

Step 3: Calculate the Volume of the Cylinder

The volume \(V\) of a cylinder is calculated using the formula \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height (or length of the cylinder in this context). For this tank, \(r = 1.5\) feet and \(h = 5\) feet, so the volume is \[V = \pi \times (1.5)^2 \times 5 = 11.25 \pi\] cubic feet.

Step 4: Calculate the Weight of the Gasoline

The weight \(W\) of the gasoline is the volume of the gasoline multiplied by its weight per unit volume. Given that gasoline weighs 35 pounds per cubic foot, the weight of the full tank of gasoline is \[W = 11.25 \pi \times 35 = 393.75 \pi\] pounds.

Step 5: Calculate the Work Done

Work \(W_{\text{work}}\) is calculated by multiplying the weight of the gasoline by the additional height it is lifted. Since the gasoline is lifted an additional 10 feet, the work done is \[W_{\text{work}} = 393.75 \pi \times 10 = 3937.5 \pi\] foot-pounds.

Key Concepts

Cylindrical Tank

Cylindrical tanks are commonly used for storing liquids, such as gasoline. Being cylindrical in shape, these tanks have a circular cross-section and a specific length along which the tank is stretched.
This makes calculating various properties straightforward due to the symmetry and regularity of the shape. In our case, the cylindrical tank is lying on its side, which means its circular faces are vertical.

• Key elements of a cylindrical tank include the radius of the circular face and the length of the cylinder itself.
• The radius is the distance from the center of the circle to its edge, while the length is the distance from one circular face to the opposite face.
• Here, the radius is 1.5 feet, and the length is 5 feet.

Understanding these dimensions is crucial for further calculations, such as determining the tank's volume.

Volume Calculation

Calculating the volume of a cylindrical tank involves a simple formula, which accounts for both the radius and length. The volume, noted as \(V\), is given by \(V = \pi r^2 h\).
This formula multiplies the area of the base (a circle) by the height (or length of the cylinder).

• For our tank, the area of the base circle is \(\pi (1.5)^2\). This results in \(2.25\pi\) square feet.
• Multiplying this area by the cylinder's length (5 feet) gives \(11.25\pi\) cubic feet, the total volume of the gasoline.

Understanding how to calculate volume is essential for various applications, such as determining the amount of liquid a tank can hold.

Lifting Work

Lifting work in physics, especially in calculus problems, involves calculating the energy required to move an object to a certain height. This is particularly relevant when transferring liquids like gasoline from one tank to another.
In this problem, we calculate the work to move gasoline to a higher elevation.

• The concept of work is tied to both the distance the gasoline travels and its weight.
• Moving the gasoline from the cylindrical tank to a holding tank 15 feet above ground means we need to lift it an additional 10 feet.
• Work done is calculated using \(W_{\text{work}} = W \times h\), where \(W\) is the weight of the gasoline, and \(h\) is the lifting height.
• In this case, \(h = 10\) feet.

By multiplying the weight (393.75\pi pounds) by the height (10 feet), we find the work done: \(3937.5\pi\) foot-pounds.

Weight of Gasoline

The weight of a liquid like gasoline in a cylindrical tank is determined using both its volume and its density. Density is often expressed as weight per unit of volume. For gasoline, it's given as 35 pounds per cubic foot.
To find the total weight, multiply the volume of gasoline by this density.

• In this problem, the volume was calculated to be \(11.25\pi\) cubic feet.
• The gasoline's weight is then \(11.25\pi \times 35\), giving \(393.75\pi\) pounds.

Knowing the weight of the gasoline is crucial because it directly influences the work needed to lift the gasoline to the required height, linking directly back to the principles of lifting work.