Calculating Combined Circle Area

Let the radii of the two circles be \(r_1\) and \(r_2\). The problem states their ratio is \(4:3\). We can represent these radii using a variable, say \(x\), such that \(r_1 = 4x\) and \(r_2 = 3x\). This representation aligns with the polynomial forms in the options.

Surface Area Formulas

The area of a circle is given by the formula \(A = \pi r^2\). We need to find the area for each circle and then sum them.

Individual Circle Areas

• Area of the first circle (\(A_1\)): With radius \(r_1 = 4x\), the area is \(A_1 = \pi (4x)^2 = \pi (16x^2) = 16\pi x^2\).
• Area of the second circle (\(A_2\)): With radius \(r_2 = 3x\), the area is \(A_2 = \pi (3x)^2 = \pi (9x^2) = 9\pi x^2\).

Combined Total Surface Area

The combined total surface area (\(A_{total}\)) is the sum of the individual areas:

\(A_{total} = A_1 + A_2\)

Substituting the calculated areas:

\(A_{total} = 16\pi x^2 + 9\pi x^2\)

Combine the terms:

\(A_{total} = (16 + 9)\pi x^2\)

\(A_{total} = 25\pi x^2\)

This polynomial expression represents the combined total surface area of the two circles.