Sure! Let's solve this step by step:
1. Understanding the Problem:
Liliana wants the area of the base of her vase to be between 135 square centimeters and 155 square centimeters. We need to determine the possible radii for the circular base that would give areas within this range.
2. Formula for the Area of a Circle:
The formula for the area [tex]\(A\)[/tex] of a circle is given by:
[tex]\[
A = \pi r^2
\][/tex]
where [tex]\(r\)[/tex] is the radius of the circle.
3. Finding the Radius for the Lower Bound:
We start by using the lower bound for the area, which is 135 square centimeters.
[tex]\[
A = 135
\][/tex]
We can rearrange the area formula to solve for [tex]\(r\)[/tex]:
[tex]\[
r^2 = \frac{A}{\pi}
\][/tex]
Substituting [tex]\(A = 135\)[/tex] and [tex]\(\pi = 3.14\)[/tex]:
[tex]\[
r^2 = \frac{135}{3.14}
\][/tex]
[tex]\[
r^2 \approx 43.0
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
r \approx \sqrt{43.0}
\][/tex]
[tex]\[
r \approx 6.55695284207904 \text{ cm}
\][/tex]
4. Finding the Radius for the Upper Bound:
Now we use the upper bound for the area, which is 155 square centimeters.
[tex]\[
A = 155
\][/tex]
Similarly, we solve for [tex]\(r\)[/tex]:
[tex]\[
r^2 = \frac{155}{3.14}
\][/tex]
[tex]\[
r^2 \approx 49.36
\][/tex]
Taking the square root of both sides:
[tex]\[
r \approx \sqrt{49.36}
\][/tex]
[tex]\[
r \approx 7.025884807256717 \text{ cm}
\][/tex]
5. Conclusion:
The radius of the base of the vase should be between approximately 6.56 cm and 7.03 cm to ensure the area of the base lies between 135 square centimeters and 155 square centimeters.
Hence, any circle with a radius in this range could represent the base of Liliana's vase.
