The weight w of an astronaut (in pounds) is related to her height h above the surface of the earth (in miles) by
w = w0 

3960
3960 + h
	2
 
where 
w0
 is the weight of the astronaut on the surface of the earth. If the astronaut weighs 130 pounds on earth and is in a rocket, being propelled upward at a speed of 13 mi/s, find the rate at which her weight is changing (in lb/s) when she is 50 miles above the earth's surface. (Round your answer to four decimal places.)
-.8212
 
Incorrect: Your answer is incorrect.
 lb/s

We are tasked with finding the rate of change of the astronaut's weight, \( w \), with respect to time, \( t \), as she is propelled upward at a speed of 13 mi/s when she is 50 miles above the surface of the Earth. The weight \( w \) is given by the equation:

\[
w(h) = w_0 \left( \frac{3960}{3960 + h} \right)^2
\]

where:
- \( w_0 = 130 \) pounds (the astronaut's weight on the surface of the Earth),
- \( h \) is the height above the surface of the Earth (in miles),
- \( 3960 \) is the radius of the Earth (in miles).

### Step 1: Differentiate the weight equation with respect to time \( t \)

To find the rate of change of the astronaut's weight with respect to time, we need to apply the chain rule:

\[
\frac{dw}{dt} = \frac{dw}{dh} \cdot \frac{dh}{dt}
\]

We are given that the astronaut is moving upward at 13 mi/s, meaning:

\[
\frac{dh}{dt} = 13 \, \text{mi/s}
\]

Now, we need to compute \( \frac{dw}{dh} \), the rate of change of weight with respect to height. To do this, differentiate the weight equation \( w(h) \) with respect to \( h \).

### Step 2: Differentiate \( w(h) \) with respect to \( h \)

We start with:

\[
w(h) = 130 \left( \frac{3960}{3960 + h} \right)^2
\]

Using the chain rule, we first differentiate the inner function \( \frac{3960}{3960 + h} \):

\[
\frac{d}{dh} \left( \frac{3960}{3960 + h} \right) = -\frac{3960}{(3960 + h)^2}
\]

Now, apply the power rule to differentiate the square:

\[
\frac{dw}{dh} = 130 \cdot 2 \left( \frac{3960}{3960 + h} \right) \cdot \left( -\frac{3960}{(3960 + h)^2} \right)
\]

Simplifying:

\[
\frac{dw}{dh} = -260 \cdot \frac{3960^2}{(3960 + h)^3}
\]

### Step 3: Evaluate \( \frac{dw}{dh} \) at \( h = 50 \)

Substitute \( h = 50 \) miles:

\[
\frac{dw}{dh} = -260 \cdot \frac{3960^2}{(3960 + 50)^3} = -260 \cdot \frac{3960^2}{4010^3}
\]

First, compute the values:

\[
3960^2 = 15,681,600, \quad 4010^3 = 64,481,201,000
\]

Now, substitute these values:

\[
\frac{dw}{dh} = -260 \cdot \frac{15,681,600}{64,481,201,000} = -260 \cdot 0.0002432 = -0.0632
\]

### Step 4: Calculate \( \frac{dw}{dt} \)

Now, use the chain rule to find \( \frac{dw}{dt} \):

\[
\frac{dw}{dt} = \frac{dw}{dh} \cdot \frac{dh}{dt} = -0.0632 \cdot 13 = -0.8216 \, \text{lb/s}
\]

Thus, the rate at which the astronaut's weight is changing when she is 50 miles above the Earth's surface is approximately:

\[
\boxed{-0.8216 \, \text{lb/s}}
\]

### Would you like to go over any details or have questions?

Here are 5 related questions:
1. How does the rate of change of weight vary with increasing height?
2. What assumptions are we making about gravitational forces in this model?
3. How would the astronaut's weight change if her velocity increased?
4. What is the significance of the Earth's radius in this equation?
5. How does the weight change behave as the astronaut moves very far from Earth?

**Tip:** Always remember to apply the chain rule when variables depend on one another indirectly.

The weight w of an astronaut (in pounds) is related to her height h above the surface of the earth (in miles) by w = w0  * (3960 / (3960 + h))^2 where w0 is the weight of the astronaut on the surface of the earth. If the astronaut weighs 130 pounds on earth and is in a rocket, being propelled upward at a speed of 13 mi/s, find the rate at which her weight is changing (in lb/s) when she is 50 miles above the earth's surface. (Round your answer to four decimal places.)

Learn how to calculate the rate at which an astronaut's weight changes as she ascends from Earth using the chain rule. The problem involves calculus concepts, including the rate of change and gravitational physics. The astronaut weighs 130 pounds on Earth, and the solution shows how her weight changes at a height of 50 miles with a speed of 13 mi/s.

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