Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
How am I suppose to solve this related rate problem? What is the answer to 3 decimal places?
![### Problem Statement
2. The radius of a sphere is decreasing slowly at the rate of 2 cm/week. How fast is the volume changing when the radius is 15 cm?
### Explanation
This problem involves a sphere whose radius is decreasing over time. You need to determine how this change in radius affects the volume of the sphere.
### Steps to Solve
1. **Formula for Volume of a Sphere:**
The volume \( V \) of a sphere with radius \( r \) is given by:
\[
V = \frac{4}{3} \pi r^3
\]
2. **Rate of Change of Radius:**
The radius is decreasing, so we have:
\[
\frac{dr}{dt} = -2 \text{ cm/week}
\]
3. **Differentiate the Volume with Respect to Time:**
Use the chain rule to differentiate the volume with respect to time \( t \):
\[
\frac{dV}{dt} = \frac{d}{dt} \left( \frac{4}{3} \pi r^3 \right) = 4 \pi r^2 \frac{dr}{dt}
\]
4. **Substitute the Known Values:**
When \( r = 15 \) cm:
\[
\frac{dV}{dt} = 4 \pi (15)^2 (-2)
\]
5. **Calculate the Rate of Change of Volume:**
\[
\frac{dV}{dt} = 4 \pi \times 225 \times (-2) = -1800 \pi \text{ cm}^3/\text{week}
\]
### Conclusion
The volume of the sphere is decreasing at a rate of \( 1800 \pi \) cubic centimeters per week when the radius is 15 cm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F668826ef-a7b0-4206-bae0-a08cac22bcd4%2Fba066765-799f-4b0e-8c5b-47842382c45f%2Fr1halvb_processed.jpeg&w=3840&q=75)
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