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

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 2

Answer 3

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 4

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 5

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 6

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 7

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 8

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.)