Find the distance between the given parallel planes.4x − 5y + z = 16,    8x − 10y + 2z = 2

To find the distance between the given parallel planes, we can use the formula:

distance = |d| / √(a^2 + b^2 + c^2)

where d is the constant term in the equation of one of the planes, and a, b, and c are the coefficients of x, y, and z respectively.

In this case, the equations of the planes are:

4x - 5y + z = 16 8x - 10y + 2z = 2

We can see that the coefficients of x, y, and z are the same for both equations, which means the planes are parallel.

Let's calculate the distance between the planes:

For the first plane, the constant term is 16, and the coefficients are 4, -5, and 1 respectively.

Using the formula, we have:

distance = |16| / √(4^2 + (-5)^2 + 1^2) = 16 / √(16 + 25 + 1) = 16 / √42 = 16 / 6.4807 ≈ 2.47

Therefore, the distance between the given parallel planes is approximately 2.47 units.

To find the distance between the given parallel planes, we can use the formula:

distance = |d| / √(a^2 + b^2 + c^2)

where a, b, and c are the coefficients of the normal vector of the planes, and d is the constant term in the equation of one of the planes.

For the given planes: Plane 1: 4x - 5y + z = 16 Plane 2: 8x - 10y + 2z = 2

We can see that the normal vectors of both planes are parallel, so they have the same direction. The coefficients of the normal vector for Plane 1 are a = 4, b = -5, and c = 1.

To find the value of d for Plane 1, we can substitute any point on the plane into the equation. Let's choose (0, 0, 16):

4(0) - 5(0) + (16) = 16 16 = 16

So, d = 16.

Now, we can calculate the distance between the planes:

distance = |16| / √(4^2 + (-5)^2 + 1^2) distance = 16 / √(16 + 25 + 1) distance = 16 / √42 distance ≈ 2.51

Therefore, the distance between the given parallel planes is approximately 2.51 units.

To find the distance between the given parallel planes, we can use the formula:

distance = |d| / √(a^2 + b^2 + c^2)

where d is the constant term in the equation of one of the planes, and a, b, and c are the coefficients of x, y, and z respectively.

In this case, the equation of the first plane is 4x - 5y + z = 16, so the constant term is 16. The coefficients are a = 4, b = -5, and c = 1.

Using the formula, we can calculate the distance:

distance = |16| / √(4^2 + (-5)^2 + 1^2) = 16 / √(16 + 25 + 1) = 16 / √42 = 16 / 6.4807 ≈ 2.47

Therefore, the distance between the given parallel planes is approximately 2.47 units.