Problem 60 - Calculus Early Transcendentals

Question

Suppose a cylindrical glass with a diameter of \(\frac{1}{12} \mathrm{m}\) and a height of \(\frac{1}{10} \mathrm{m}\) is filled to the brim with a 400-Cal milkshake. If you have a straw that is 1.1 m long (so the top of the straw is \(1 \mathrm{m}\) above the top of the glass), do you burn off all the calories in the milkshake in drinking it? Assume that the density of the milkshake is \(1 \mathrm{g} / \mathrm{cm}^{3}(1 \mathrm{Cal}=4184 \mathrm{J})\)

Step-by-Step Solution

Answer

Answer: ________________________________________________________________

Step 1: Calculate the volume of the milkshake

First, we will find the volume of the cylindrical glass using the formula: Volume (V) = π * radius² * height The glass diameter is given as \(\frac{1}{12} \mathrm{m}\). Hence, the radius (r) is half of the diameter: \(r = \frac{1}{2} * \frac{1}{12} \mathrm{m} = \frac{1}{24} \mathrm{m}\) The height (h) is given as \(\frac{1}{10} \mathrm{m}\). Now, using the formula for volume, we have: \(V = \pi * (\frac{1}{24} \mathrm{m})^2 * \frac{1}{10} \mathrm{m}\)

Step 2: Convert the volume to mass

Now that we have the volume of the milkshake, we can convert it to mass using the given density: Density = \(\frac{mass}{volume}\) Mass = Density * Volume Since the density is provided in grams per cubic centimeter, and the volume is in cubic meters, we'll need to convert the units. There are 100 cm in a meter, so: \(1 \mathrm{m}^3 = (100 \mathrm{cm})^3 = 10^6 \mathrm{cm}^3\) The volume of the milkshake in cubic centimeters is: \(V = \pi (\frac{1}{24} \mathrm{m})^2 \frac{1}{10} \mathrm{m} * 10^6 \mathrm{cm}^3\) The given density is 1 g/cm³, so the mass of the milkshake is: Mass = 1 g/cm³ * Volume

Step 3: Calculate the energy content of the milkshake

Next, we need to find the energy content of the milkshake in joules using the given conversion factor: Energy (in Joules) = Mass (in grams) * 1 Cal/g * 4184 J/Cal Now we have the total energy content of the milkshake in joules.

Step 4: Calculate the energy needed to drink the milkshake through the straw

We can calculate the energy needed to drink the milkshake through the straw by considering the work required against gravity to bring the milkshake height of 1 meter. Energy (work) = Mass * Gravity * Height Here, Mass is the mass of milkshake, Gravity is 9.81 m/s², and Height is 1 meter as given straw length of 1.1m is 1 m above the glass top. Now, we have the energy needed to drink the milkshake in joules.

Step 5: Compare the energy content of the milkshake and the energy needed to drink it

Finally, we'll compare the energy content of the milkshake (calculated in step 3) with the energy needed to drink the milkshake through the straw (calculated in step 4). If the energy needed to drink the milkshake is greater than or equal to the energy content of the milkshake, then all the calories will be burned off in the process of drinking it. Otherwise, the calories won't be completely burned off. Based on the comparison, we can answer the question if the person will burn off all the calories in the milkshake or not.