Liliana is making a vase with a circular base. She wants the area of the base to be between 135 [tex]$cm ^2$[/tex] and [tex]$155 cm^2$[/tex]. Which circle could represent the base of the vase? Use 3.14 for [tex]$\pi$[/tex].
Answer (1)
Establish the area constraints for the circular base: 135 < A < 155 .
Substitute the area formula A = π r 2 and π = 3.14 into the inequality.
Solve for r 2 by dividing the inequality by 3.14, resulting in 42.99 < r 2 < 49.36 .
Determine the range for the radius r by taking the square root: 6.56 < r < 7.03 .
Explanation
Problem Analysis Let's analyze the problem. Liliana wants to make a vase with a circular base. The area of the base must be between 135 c m 2 and 155 c m 2 . We need to find the possible radius of the circular base. We will use the formula for the area of a circle, A = π r 2 , where A is the area and r is the radius. We are given that π = 3.14 .
Area Inequality We are given that the area A must satisfy the inequality: 135 < A < 155
Substitute Area Formula Substitute the formula for the area of a circle, A = π r 2 , into the inequality: 135 < π r 2 < 155
Substitute Pi Value Substitute the value of π = 3.14 into the inequality: 135 < 3.14 r 2 < 155
Divide by Pi Divide all parts of the inequality by 3.14: 3.14 135 < r 2 < 3.14 155 42.99 < r 2 < 49.36
Take Square Root Take the square root of all parts of the inequality: 42.99 < r < 49.36 6.56 < r < 7.03
Radius Range Therefore, the radius of the circle must be between 6.56 cm and 7.03 cm.
Final Answer The radius of the base must be between 6.56 cm and 7.03 cm.
Examples
Understanding the area of a circle is crucial in many real-world applications. For instance, when designing a circular garden, knowing the desired area helps determine the radius needed for planting flowers or vegetables. Similarly, in architecture, calculating the area of circular windows or domes is essential for estimating material costs and ensuring structural integrity. These calculations provide a foundation for creating aesthetically pleasing and functional designs.
