Use a graphing utility to graph the polar equation.inner loop of r = 10 − 15 sin(𝜃)Find the area of the given region.

To solve this problem, follow these steps:

1. Graph the Polar Equation: Use a graphing utility like Desmos or a graphing calculator to graph the polar equation r = 10 - 15sin(θ). You will see that the graph forms a limaçon, a type of polar curve, with an inner loop.

2. Identify the Inner Loop: The inner loop of the graph is the smaller, enclosed area of the graph. It's the part of the graph where the value of r is negative.

3. Find the Bounds of the Inner Loop: To find the area of the inner loop, we first need to find the bounds of θ for which r is negative. Set r = 0 and solve for θ to find these bounds.

   10 - 15sin(θ) = 0 sin(θ) = 10/15 = 2/3

   The solutions to this equation are θ = arcsin(2/3) and θ = π - arcsin(2/3).

4. Calculate the Area of the Inner Loop: The formula for the area enclosed by a polar curve from θ = a to θ = b is given by:

   A = 1/2 ∫ from a to b [r(θ)]² dθ

   In this case, r(θ) = 10 - 15sin(θ), a = arcsin(2/3), and b = π - arcsin(2/3).

   So, the area of the inner loop is:

   A = 1/2 ∫ from arcsin(2/3) to π - arcsin(2/3) [10 - 15sin(θ)]² dθ

   This integral can be computed using standard techniques of integration, or with a calculator that can compute definite integrals.

Remember, the exact computation may require some knowledge of calculus, particularly integration. If you're not familiar with these concepts, you might need to study them first or ask for help from a teacher or tutor.