Calculating the Rate of Change of the Radius

Oct 13, 2025 by windailys.com 0 views

Introduction

In this article, we will explore the concept of a sphere's volume decreasing at a constant rate. We will use mathematical formulas to calculate the rate of change of the area of the sphere. The problem states that the volume of the sphere is decreasing at a constant rate of 2404 cubic centimeters per minute, and the initial volume is 360 cubic centimeters.

Mathematical Formulas

To solve this problem, we need to use the following mathematical formulas:

The volume of a sphere (V) is given by the formula: V = (4/3)πr^3
The surface area of a sphere (A) is given by the formula: A = 4πr^2
The rate of change of the volume (dV/dt) is given by the formula: dV/dt = -2404 cm^3/min
The rate of change of the radius (dr/dt) is related to the rate of change of the volume by the formula: dV/dt = 4πr^2 dr/dt

Calculating the Rate of Change of the Radius

We are given that the initial volume of the sphere is 360 cubic centimeters. We can use this information to calculate the initial radius of the sphere.

import math
V_initial = 360  # cubic centimeters
dV_dt = -2404  # cubic centimeters per minute
r_initial = (3 * V_initial / (4 * math.pi)) ** (1/3)
print("Initial radius:", r_initial)

Calculating the Rate of Change of the Radius

Now that we have the initial radius, we can use the formula dV/dt = 4πr^2 dr/dt to calculate the rate of change of the radius.

# Calculate the rate of change of the radius
dr_dt = dV_dt / (4 * math.pi * r_initial ** 2)
print("Rate of change of the radius:", dr_dt)

Calculating the Rate of Change of the Area

Finally, we can use the formula dA/dt = 8πr dr/dt to calculate the rate of change of the area.

# Calculate the rate of change of the area
dA_dt = 8 * math.pi * r_initial * dr_dt
print("Rate of change of the area:", dA_dt)

Conclusion

In this article, we used mathematical formulas to calculate the rate of change of the area of a sphere. We were given that the volume of the sphere is decreasing at a constant rate of 2404 cubic centimeters per minute, and the initial volume is 360 cubic centimeters. We calculated the initial radius of the sphere and then used the formula dV/dt = 4πr^2 dr/dt to calculate the rate of change of the radius. Finally, we used the formula dA/dt = 8πr dr/dt to calculate the rate of change of the area.

Rate of Change of the Area

Q: What is the volume of a sphere?

A: The volume of a sphere (V) is given by the formula: V = (4/3)πr^3, where r is the radius of the sphere.

Q: What is the surface area of a sphere?

A: The surface area of a sphere (A) is given by the formula: A = 4πr^2, where r is the radius of the sphere.

Q: What is the rate of change of the volume of a sphere?

A: The rate of change of the volume of a sphere (dV/dt) is given by the formula: dV/dt = -2404 cm^3/min, where the negative sign indicates that the volume is decreasing.

Q: How do we calculate the rate of change of the radius of a sphere?

A: We can use the formula dV/dt = 4πr^2 dr/dt to calculate the rate of change of the radius of a sphere.

Q: How do we calculate the rate of change of the area of a sphere?

A: We can use the formula dA/dt = 8πr dr/dt to calculate the rate of change of the area of a sphere.

Q: What is the rate of change of the area of the sphere in this problem?

A: The rate of change of the area of the sphere is approximately 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