Math Problem Statement

Let f be the function that determines the area of a circle (in square cm) with a radius of r cm. That is, f(r) represents the area of a circle (in square cm) that has a radius of r cm. Use function notation to respond to each of the following: a) Represent the area (in square cm) of a circle with a radius of 4.5 cm. b) Represent how much a circle's area increases when the radius increases from 10.4 cm to 10.41 cm. c) Represent the total area of 6 circles, each with a radius of 12.7 cm. d) A circle has a radius of 21 cm. A second circle has an area 59 square cm greater than the first circle's area. Represent the area of the second circle.

Solution

The image contains a problem set regarding the area of a circle, denoted by f(r)f(r), where rr is the radius of the circle. The questions are based on this function notation and involve calculating the area of circles for various radii, determining how much the area increases, and applying these calculations in specific contexts.

Here’s an overview of the questions presented:

  1. Problem Statement: The function f(r)f(r) gives the area of a circle with radius rr, where f(r)=πr2f(r) = \pi r^2. Use function notation to solve the following:

    a. Represent the area (in square cm) of a circle with a radius of 4.5 cm.

    • Formula: f(4.5)=π×(4.5)2f(4.5) = \pi \times (4.5)^2

    b. Find the increase in area when the radius increases from 10.4 cm to 10.41 cm.

    • Calculate the area at both radii: f(10.41)f(10.4)f(10.41) - f(10.4)

    c. Represent the total area of 6 circles, each with a radius of 12.7 cm.

    • Formula for the total area: 6×f(12.7)=6×π×(12.7)26 \times f(12.7) = 6 \times \pi \times (12.7)^2

    d. A circle with a radius of 21 cm has a second circle whose area is 59 square cm larger than the first circle. Represent the area of the second circle.

    • Formula: f(21)+59=π×(21)2+59f(21) + 59 = \pi \times (21)^2 + 59

Would you like a detailed explanation of how to solve each part, or do you have a specific question about one of these problems?

Here are 5 questions you can explore further:

  1. What is the general formula for the area of a circle and how can it be derived?
  2. How does the area change as the radius of a circle increases?
  3. How can we apply the area formula to solve real-life problems involving circles?
  4. What is the significance of the constant π\pi in the formula for a circle’s area?
  5. How do you calculate the increase in area when the radius of a circle changes slightly?

Tip: When working with circle areas, always square the radius before multiplying by π\pi.

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Math Problem Analysis

Mathematical Concepts

Geometry
Function Notation
Area of a Circle

Formulas

Area of a circle: f(r) = πr^2

Theorems

Area formula for a circle

Suitable Grade Level

Grade 9-12