Problem 35 - Thomas Calculus

Question

Drinking a milkshake The truncated conical container shown here is full of strawberry milkshake that weighs 4\(/ 9\) oz \(/ \mathrm{in} .^{3}\) As you can see, the container is 7 in. deep, 2.5 in. across at the base, and 3.5 in. across at the top (a standard size at Brigham's in Boston). The straw sticks up an inch above the top. About how much work does it take to suck up the milkshake through the straw (neglecting friction)? Answer in inch-ounces.

Step-by-Step Solution

Answer

Calculate the volume using the truncated cone formula, find the weight by multiplying with density, determine the center of mass, and apply the work formula.

Step 1: Find the Volume of the Truncated Cone

The volume of a truncated cone can be calculated using the formula: \[ V = \frac{1}{3} \pi h ( R^2 + Rr + r^2 ) \] where \( h \) is the height (7 inches), \( R \) is the radius of the top (1.75 inches), and \( r \) is the radius of the base (1.25 inches).

Step 2: Calculate the Volume

Substituting the values into the formula: \[V = \frac{1}{3} \pi \times 7 \times (1.75^2 + 1.75 \times 1.25 + 1.25^2).\] Calculate the expression to find \( V \).

Step 3: Compute the Weight of the Milkshake

The density of the milkshake is given as \( \frac{4}{9} \) ounces per cubic inch. Multiply this by the volume calculated in the previous step to find the total weight of the milkshake: \( \text{Weight} = \frac{4}{9} \times V \).

Step 4: Determine the Distance the Milkshake Moves

The milkshake has to be lifted from a height of the center of mass of the truncated cone to the top of the straw, which is 1 inch above the top of the container. The distance lifted is the distance from the center of mass to the top plus 1 inch.

Step 5: Calculate the Position of the Center of Mass

For a truncated cone, the center of mass can be found using proportional divisions. The formula is: \[ \bar{z} = \frac{h}{4} \cdot \frac{R^2 + 2Rr + 3r^2}{R^2 + Rr + r^2} \] Substitute \( h = 7 \), \( R = 1.75 \), and \( r = 1.25 \) to find the height of the center of mass from the base.

Step 6: Calculate the Work Done

Work is calculated as \( \text{Work} = \text{Weight} \times \text{Distance} \). Use the weight from Step 3 and the total distance from the center of mass to the top of the straw from Step 4.

Key Concepts

Truncated Cone Volume

A truncated cone, also known as a frustum, is a cone with its top cut off parallel to its base. Calculating its volume is essential when you deal with objects like drinking cups or certain containers.
To find the volume of a truncated cone, you use the formula:

• \[ V = \frac{1}{3} \pi h ( R^2 + Rr + r^2 ) \]

In this formula:

• \( V \) is the volume.
• \( h \) is the height of the truncated cone.
• \( R \) is the radius of the larger (top) circle.
• \( r \) is the radius of the smaller (bottom) circle.

In our problem, the height \( h \) is 7 inches, the top radius \( R \) is 1.75 inches (half of 3.5 inches), and the bottom radius \( r \) is 1.25 inches (half of 2.5 inches).
By substituting these values into the formula, we can compute the volume of our milkshake container.

Density Calculations

Density is a measure of how much mass is contained in a given volume. It is calculated by dividing mass by volume and often expressed in units such as ounces per cubic inch in the case of our milkshake.
To calculate the total weight of the milkshake, we need to multiply the density by the volume obtained earlier. The density of the milkshake given is \( \frac{4}{9} \) ounces per cubic inch.
This step simply involves taking the volume \( V \) that you calculated in the previous section and applying this density formula:

• \[ \text{Weight} = \text{Density} \, \times \, \text{Volume} \]

By substituting the calculated volume for \( V \), you can find the total weight of the milkshake, which will be used in further calculations.

Center of Mass

Finding the center of mass of a truncated cone involves understanding how mass is distributed along its height. This point is crucial when calculating work, as it affects how far you need to lift the milkshake.
For a truncated cone, the center of mass can be located using:

• \[ \bar{z} = \frac{h}{4} \cdot \frac{R^2 + 2Rr + 3r^2}{R^2 + Rr + r^2} \]

This formula involves:

• \( h \), the height of the truncated cone.
• \( R \) and \( r \), the radii of the larger and smaller bases respectively.

By substituting the given values of \( h = 7 \), \( R = 1.75 \), and \( r = 1.25 \), you can pinpoint where the center of mass is located. This midpoint helps determine how far the milkshake must be lifted.

Work-Energy Principle

The work-energy principle states that work done on an object is equal to the change in its energy. When you suck a milkshake through a straw, you are performing work on it by lifting it against gravity from the container to your mouth.
The work done is calculated using:

• \[ \text{Work} = \text{Weight} \times \text{Distance} \]

In this case:

• "Weight" is the total weight of the milkshake as calculated previously.
• "Distance" is the path from the center of mass to the top of the straw, accounting for the additional 1-inch length above the container.

By multiplying these values, you obtain the total work done in moving the milkshake through the straw. This provides an understanding of the energy required in lifting the entire content of the container against gravity.