Applications Variation

Question

The gravitational Force F between an object and the Earth is inversely proportional to the square of the distance from the object to the Center of the Earth if an astronaut weighs 215 lb on the surface of the Earth what will this astronaut weigh 400 miles above the Earth assume that the radius of the Earth is 4000 miles

Geometry Dash

Answer 1

Given:

Weight on Earth's Surface = 215 lbs

Radius of the Earth = 4000 miles

Height above the Earth's Surface = 400 miles

Because the problem tells us that the gravitational force is inversely proportional to the square of the distance, we can set up a general formula for this situation:

F = k / d²

Remember that F is also equal to the astronaut's weight, which is the measure of gravitational force as well. These values are equivalent and mean the same thing!

Here, k is our constant of variation, and d is the total distance from the astronaut to the center of the Earth. Before we do anything, we need to find k, which we can do by plugging in what we already know:

F = k / d²

215 = k / (4000)²

215 = k / 16,000,000

k = 3,440,000,000

And now we have our key formula to find the weight at 400 miles:

F = 3,440,000,000 / d²

But remember, our distance isn't just 400 miles, but rather the 400 miles on top of the radius of Earth (4000 miles).

So that makes our d = 400 + 4000 = 4400 miles

Now we just plug those values into the formula we found above:

F = 3,440,000,000 / d²

F = 3,440,000,000 / (4400)²

F = 3,440,000,000 / 19,360,000

F = 177.685

So the weight of the astronaut at 400 miles above Earth is approximately 178 pounds!

Answer 2

Since F is inversely proportional to the square of the distance:

F = k/r²

On Earth's surface (r = 4000 miles):

215 = k/4000²

k = 215 × 4000² = 3,440,000,000

At 400 miles above Earth (r = 4400 miles):

F = 3,440,000,000/4400²

F = 3,440,000,000/19,360,000

F ≈ 177.69 lb

The astronaut will weigh approximately 178 pounds at 400 miles above Earth.

Answer 3

Let’s solve this step-by-step carefully.

Given:

1. Gravitational force F x 1/_r2
2. Astronaut’s weight on Earth’s surface F₁= 215 lbs
3. Earth’s radius r₁ 4,000 miles
4. Height above Earth h = 400 miles
5. New distance from Earth’s center r₂ 4,000 + 400 = 4,400 miles

Step 1: Write the inverse-square relationship

F₁ = r2

F₂ r2⁄1

Step 2: Solve for F₂

F₂ = F₁ x r 2⁄1 / 2⁄2

Step 3: Substitute the values

F₂ = 215 x 4,000² ⁄ 4,400²

Step 4: Compute Carefully

• 4,000² 16,000,000

• 4,400² 19,360,000

F₂ = 215 x 16,000,000

19,360,000

F₂ = 215 x 0.8264 = 177.7

Final Answer: 177.7

So, the astronaut would weigh about 178 pounds at 400 miles above the Earth’s surface.

Answer 4

Answer 5

From the prompt, we know that F = k/r^2 where k is a constant and r is the distance of the astronaut from the center of the earth. When r = 4000miles, F = 215lbs, so solving for k means k = 3440000000lbs * (miles)^2. Plug in 4400miles for r in the original equation and you get F = 3440000000lbs * (miles)^2 / (4400miles)^2 which equals 177.69lbs.

Answer 6

F=k/d²

215 = k/(4000)²

k=215.(4000)²

^F_new= k/(d_new)²

^{= 215.(4000)2}/(4000+400)²=177.69lbs

Answer 7

Given: Weight = 215 lbs at distance 4000 miles (Earth radius) from Earth’s center.

Want: Weight at 4400 miles.

Gravitational Force… inversely proportional to the square of the distance means:

W₁ = 215 lbs = K / 4000² ==> K = (215)(4000²)

W₂ = K / (4000 + 400)² = (215 * 4000²) / 4400² = 215(4000/4400)² = 215(10/11)² = 177.7 lbs

Answer 8

F = k / D²

When D = 4000, F = 215. So, 215 = k / 4000². Therefore, k = 3440000000.

So, when D = 4400. F = 3440000000 / 4400² = 177.7 pounds.

Answer 9

In an inverse relation (covered in Algebra 1) the product of the two variables is constant.

A familiar example is rt = d, rate times time equals distance.

Answer 10

The Setup:

We’re told that weight (gravitational force) is inversely proportional to the square of the distance from the center of the Earth. That means:

F = k / d^2

-where F is the force, k is the unknown constant of proportionality, and d is the distance

Or in plain English: as distance goes up, weight goes down, and it drops faster because of that square.

What we know:

1. The astronaut weighs 215 lb on Earth’s surface.
2. Radius of Earth = 4000 miles
3. Astronaut is now 400 miles above the Earth → Total distance from center = 4400 miles

What we know but have to infer:

Weight is actually a measure of force in physics. Your weight is the force you have due to gravity pulling you toward the center of an object (in this case, the Earth).

Let's use the situation we have on Earth.

F = k / d^2

215 = k / (4000)^2

215 * (4000)^2 = k

k = 3,440,000,000

So, now that we know our constant, let's use it to find the weight 400 miles above Earth:

F = k / d^2

F = 3440000000 / (4400)^2

F = 177.69

Final Answer:

The astronaut will weigh about 177.69 pounds at 400 miles above Earth.

Still heavy enough to regret skipping leg day, but definitely lighter!

(Note: You may need to adjust for significant figures. The lowest significant figures in any of these numbers is 1, which would round our answer to approximately 180 pounds, but I have provided the decimal for added clarity.)