The acceleration acting on a falling object due to gravity varies inversely with the square of the object's distance from the center of Earth. At the surface of Earth, where x = 1 Earth radius, the acceleration is y = 9.8 m/s2

1. Use the multiply-multiply pattern of power functions to find the acceleration at 3 Earth radii from the center. Round to the nearest hundredth.

2. Find the particular equation expressing y as a function of x.

3. What is the acceleration due to Earth's gravity acting on someone standing on the Moon, 63 Earth radii away from Earth's center? Express your answer with 4 places after the decimal.

g = 9.8 *(Re/x)^2 m/s^2

where Re is the earth's radius,
will answer all you questions.

Note that x/Re is the number of Earth radii from the center.

To answer these questions, we need to understand how the acceleration due to gravity varies with distance from the center of the Earth. Let's go through each question step by step.

1. To find the acceleration at 3 Earth radii from the center, we can use the inverse square relationship. At a distance of 1 Earth radius, the acceleration is 9.8 m/s^2.

Using the formula for inverse square variation:

acceleration at 3 Earth radii = (acceleration at 1 Earth radius) / (distance at 3 Earth radii)^2

Substituting the given values, we have:

acceleration at 3 Earth radii = 9.8 m/s^2 / (3^2) = 9.8 m/s^2 / 9 = 1.0889 m/s^2

Rounding the answer to the nearest hundredth, the acceleration at 3 Earth radii from the center is approximately 1.09 m/s^2.

2. To find the particular equation expressing y as a function of x, we can use the inverse square relationship between acceleration and distance from the center of the Earth.

Let's assume that y represents the acceleration in m/s^2 and x represents the distance from the center of the Earth in Earth radii. We are given that at the surface of the Earth, where x = 1 Earth radius, the acceleration is y = 9.8 m/s^2.

Using the equation for inverse square variation, we have:

y = k / x^2

Substituting the given values, we have:

9.8 = k / (1^2)

Simplifying, we find k = 9.8.

Therefore, the particular equation expressing y as a function of x is:

y = 9.8 / x^2

3. To find the acceleration due to Earth's gravity acting on someone standing on the Moon, 63 Earth radii away from Earth's center, we can use the same inverse square relationship.

Using the equation from the previous step:

y = k / x^2

Substituting the given values, we have:

y = 9.8 / (63^2)

Calculating the value, we find y ≈ 0.002947 m/s^2.

Therefore, the acceleration due to Earth's gravity acting on someone standing on the Moon, 63 Earth radii away from Earth's center, is approximately 0.0029 m/s^2 (rounded to 4 decimal places).