find the distance between the given parallel planes 5x-5y+z=10 and 10x-10y+2z=2. I got 9sqrt51/51

Looks good to me. I'd have just written 9/√51, but some folks get itchy over radicals in the denominator.

To find the distance between two parallel planes, we need to calculate the perpendicular distance from any point on one plane to the other plane.

Let's use the formula for the distance between a point and a plane to find the distance between these two parallel planes.

The formula for the distance between a point (x₁, y₁, z₁) and a plane ax + by + cz + d = 0 is given by:

Distance = |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²)

Let's assume a point (x, y, z) on the plane 5x - 5y + z = 10.

Substituting these values into the formula, we get:

Distance = |5x + (-5y) + z + (-10)| / √(5² + (-5)² + 1²)
= |5x - 5y + z - 10| / √(25 + 25 + 1)
= |5x - 5y + z - 10| / √51

Now, let's substitute the values of the second plane (10x - 10y + 2z = 2) into the formula:

Distance = |10x + (-10y) + 2z + (-2)| / √(10² + (-10)² + 2²)
= |10x - 10y + 2z - 2| / √(100 + 100 + 4)
= |10x - 10y + 2z - 2| / √204
= |5(x - y) + (z - 1)| / √204

Since the two planes are parallel, the distances from any point on one plane will be the same as the distance from any point on the other plane. Therefore, the distance between the two parallel planes is:

Distance = |5x - 5y + z - 10| / √51
= |5(x - y) + (z - 2)| / √51

The distance between the given parallel planes is |5(x - y) + (z - 2)| / √51.

To find the distance between two parallel planes, you can use the formula:

Distance = |c1 - c2| / √(a^2 + b^2 + c^2)

Where c1 and c2 are the constants in the equations of the planes, and a, b, and c are the coefficients of x, y, and z in the equations of the planes.

In this case, the equations of the given parallel planes are:

Plane 1: 5x - 5y + z = 10
Plane 2: 10x - 10y + 2z = 2

By comparing the equations, we can see that the corresponding coefficients of x, y, and z are the same. Therefore, a = 5, b = -5, and c = 1 (coefficients from Plane 1) or a = 10, b = -10, and c = 2 (coefficients from Plane 2).

The constants in the equations are c1 = 10 (from Plane 1) and c2 = 2 (from Plane 2).

Using the formula mentioned earlier:

Distance = |c1 - c2| / √(a^2 + b^2 + c^2)
= |10 - 2| / √(5^2 + (-5)^2 + 1^2)
= 8 / √(25 + 25 + 1)
= 8 / √51
= 8√51 / 51

So, the correct answer is 8√51 / 51, which is approximately 1.12.

Your answer of 9√51 / 51 is close, but it looks like there might be a small calculation error. Please check your calculations again.