using function notation Represent how much a circle's area (in square cm) increases when the circle's radius increases from 10.9 to 10.91 cm.

The area \( A(r) \) of a circle is a function of its radius \( r \), given by the formula:

\[
A(r) = \pi r^2
\]

To represent how much the area increases when the radius increases from 10.9 cm to 10.91 cm, we can calculate the change in area, \( \Delta A \), using the difference between the areas at the two radii.

1. Let \( A(10.91) \) represent the area when the radius is 10.91 cm:
   \[
   A(10.91) = \pi (10.91)^2
   \]

2. Let \( A(10.9) \) represent the area when the radius is 10.9 cm:
   \[
   A(10.9) = \pi (10.9)^2
   \]

Now, the increase in area is given by:
\[
\Delta A = A(10.91) - A(10.9)
\]

Substituting the values:
\[
\Delta A = \pi (10.91)^2 - \pi (10.9)^2 = \pi \left[ (10.91)^2 - (10.9)^2 \right]
\]

This expression represents the increase in the area of the circle as a function of the change in radius from 10.9 cm to 10.91 cm.

Would you like me to compute the exact increase in the area?

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Here are five follow-up questions to expand your understanding:

1. What is the numerical value of \( \Delta A \) when you compute it fully?
2. How does the area change if the radius increases by larger increments?
3. Can you generalize the expression for a change in area \( \Delta A \) for any small change in radius?
4. How is the rate of change in area related to the radius of the circle?
5. What role does the constant \( \pi \) play in the formula for the area of a circle?

**Tip:** To estimate small changes in functions, you can use the concept of differentials or approximations using the derivative.

Using function notation, represent how much a circle's area (in square cm) increases when the circle's radius increases from 10.9 to 10.91 cm.

Explore how the area of a circle changes when the radius increases from 10.9 cm to 10.91 cm using function notation. Learn to apply the area formula A(r) = πr² and calculate the increase in area with detailed steps. Ideal for students in Grades 9-12 studying geometry.

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