A bathysphere used for deep sea exploration has a radius of 1.60 m and a mass of 1.20 104 kg. In order to dive, the sphere takes on mass in the form of sea water. Determine the mass the bathysphere must take on so that it can descend at a constant speed of 1.00 m/s when the resistive force on it is 1179 N upward. The density of sea water is 1.03 103 kg/m3.

To determine the mass that the bathysphere must take on, we can use the principle of buoyancy and the concept of net force.

1. Step 1: Calculate the buoyant force acting on the bathysphere.
The buoyant force is given by the formula:
Buoyant force = Volume of the displaced fluid * Density of the fluid * Gravity

Since the bathysphere descends at a constant speed, the buoyant force must be equal to the sum of the gravitational force and the resistive force acting on it:
Buoyant force = Weight of the bathysphere + Resistive force

2. Step 2: Calculate the weight of the bathysphere.
The weight of an object is given by the formula:
Weight = Mass * Gravity

3. Step 3: Calculate the volume of the bathysphere.
The volume of a sphere is given by the formula:
Volume = (4/3) * pi * r^3

Now let's compute the mass the bathysphere must take on:

Step 1: Calculate the buoyant force.
Buoyant force = (Volume of the displaced water) * (Density of the water) * (Gravity)

Step 2: Calculate the weight of the bathysphere.
Weight = Mass * Gravity

Step 3: Calculate the volume of the bathysphere.
Volume = (4/3) * pi * r^3

Step 4: Set up the equation for the net force.
Buoyant force = Weight + Resistive force

Step 5: Solve for the mass.
Mass = (Buoyant force - Weight) / Gravity

Now we can substitute the given values into the formulas and calculate the answers.

Given:
Radius of bathysphere (r) = 1.60 m
Mass of bathysphere (m) = 1.20 * 10^4 kg
Density of sea water (ρ) = 1.03 * 10^3 kg/m^3
Resistive force (F_resistive) = 1179 N
Acceleration due to gravity (g) = 9.8 m/s^2

Step 1: Calculate the buoyant force.
Buoyant force = (Volume of the displaced water) * (Density of the water) * (Gravity)

First, let's calculate the volume of the bathysphere.
Volume = (4/3) * pi * r^3
= (4/3) * 3.14159 * (1.60 m)^3
≈ 17.05 m^3

Now, substitute the values into the formula for the buoyant force:
Buoyant force = (17.05 m^3) * (1.03 * 10^3 kg/m^3) * (9.8 m/s^2)
≈ 170,045.6 N

Step 2: Calculate the weight of the bathysphere.
Weight = Mass * Gravity
= (1.20 * 10^4 kg) * (9.8 m/s^2)
≈ 117,600 N

Step 4: Set up the equation for the net force.
Buoyant force = Weight + Resistive force

Step 5: Solve for the mass.
Mass = (Buoyant force - Weight) / Gravity
= (170,045.6 N - 117,600 N) / (9.8 m/s^2)
≈ 5,449.1 kg

Therefore, the bathysphere must take on approximately 5,449.1 kg of sea water to descend at a constant speed of 1.00 m/s when the resistive force on it is 1179 N upward.