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 $155 cm^2$. Which circle could represent the base of the vase? Use 3.14 for $[tex]\pi[/tex]$.
Answer (1)
The problem requires finding the range of the radius of a circle given a range for its area.
The area of a circle is A = π r 2 , and we know 135 < A < 155 and π = 3.14 .
Dividing the area inequality by π gives 42.99 < r 2 < 49.36 .
Taking the square root yields the radius range 6.56 < r < 7.03 .
The radius of the circle must be between 6.56 cm and 7.03 cm.
Explanation
Problem Analysis 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 a circle that satisfies this condition, using 3.14 for π .
Area Formula The area of a circle is given by the formula: A = π r 2 where A is the area and r is the radius. We are given that the area A must satisfy the following inequality: 135 < A < 155
Substitute Area Formula Substituting the formula for the area of a circle, we get: 135 < π r 2 < 155 We are given that π = 3.14 , so we have: 135 < 3.14 r 2 < 155
Divide by Pi Now, we divide all parts of the inequality by 3.14: 3.14 135 < r 2 < 3.14 155 Calculating these values: 42.99 < r 2 < 49.36
Take Square Root Next, we take the square root of all parts of the inequality: 42.99 < r < 49.36 Calculating the square roots: 6.56 < r < 7.03
Radius Range Therefore, the radius of the circle must be between 6.56 cm and 7.03 cm.
Conclusion Without additional information about the options for the circle's base, we can only determine the acceptable range for the radius: between 6.56 cm and 7.03 cm.
Examples
When designing a circular garden bed, a gardener needs to determine the appropriate radius to ensure the garden bed's area falls within a specific range. For instance, if the gardener wants the area to be between 50 and 75 square feet, they can use the formula A = π r 2 to calculate the required radius. This ensures the garden bed is neither too small nor too large for the available space and desired planting arrangement.
