Question

Let S be the portion of the plane \(z = 2x + 2y - 100\) which lies inside the cylinder \(x^2 + y^2 = 1\). If the surface area of S is \(a\pi\), then the value of a is equal to ................

Hint:

For a plane \(z = mx + ny + c\), the surface area element \(\sqrt{1 + (\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2}\) is always the constant \(\sqrt{1 + m^2 + n^2}\). The total surface area is this constant multiplied by the area of the domain in the xy-plane.

Answer 1

Step 1: Understanding the Concept:
This problem asks for the surface area of a portion of a plane that is cut out by a cylinder. We can compute this using a standard formula for the surface area of a function \(z = f(x,y)\) over a region in the xy-plane.
Step 2: Key Formula or Approach:
The surface area \(A\) of a surface defined by \(z = f(x,y)\) over a domain \(D\) in the xy-plane is given by the double integral: \[ A = \iint_D \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} dA \] Step 3: Detailed Calculation:
The surface is given by the plane \(z = f(x,y) = 2x + 2y - 100\).
First, we find the partial derivatives of \(z\) with respect to \(x\) and \(y\): \[ \frac{\partial z}{\partial x} = 2 \] \[ \frac{\partial z}{\partial y} = 2 \] Next, we calculate the integrand for the surface area formula: \[ \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} = \sqrt{1 + (2)^2 + (2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \] The domain of integration, \(D\), is the region in the xy-plane that lies inside the cylinder \(x^2 + y^2 = 1\). This is a circular disk of radius 1 centered at the origin.
Now we set up the surface area integral: \[ A = \iint_D 3 \, dA \] Since the integrand is a constant, we can pull it out of the integral: \[ A = 3 \iint_D dA \] The integral \(\iint_D dA\) represents the area of the domain \(D\). Since \(D\) is a unit disk, its area is \(\pi r^2 = \pi(1)^2 = \pi\). Substituting this back, we get the surface area: \[ A = 3 \pi \] We are given that the surface area is \(a\pi\). By comparing our result with the given information, we have: \[ a\pi = 3\pi \implies a = 3 \] Step 4: Final Answer:
The value of \(a\) is 3.
Step 5: Why This is Correct:
The calculation correctly applies the surface area formula. The integrand simplifies to a constant, making the final calculation a product of this constant and the area of the projection of the surface onto the xy-plane.