Janine made a cylindrical vase in which the sum of the lateral area and area of one base was about 3000 square centimeters. The vase has a height of 50 centimeters. Find the radius of the vase. Explain the method you would use to find the radius.
the area is
a = πr^2 + 2πrh
So,
πr^2+100πr = 3000
r^2+100r = 955
r = 8.779
So, r is about 9
Thank you Steve!
To find the radius of the cylindrical vase, you need to calculate the lateral area and the area of one base, and then solve the equation involving those values.
Step 1: Identify the given information:
- The height of the vase is 50 centimeters.
- The sum of the lateral area and the area of one base is around 3000 square centimeters.
Step 2: Understand the formulas:
- The formula for the lateral area of a cylinder is L = 2πrh, where L represents the lateral area, π is approximately 3.14, r is the radius, and h is the height.
- The formula for the area of one base of a cylinder is B = πr², where B represents the base area, and r is the radius.
Step 3: Calculate the formula values:
- Substituting the known values, we have L = 2πrh = 2 * 3.14 * r * 50 = 314r.
- Substituting the known values, we have B = πr² = 3.14 * r².
Step 4: Formulate the equation:
- Since the sum of the lateral area and the area of one base is 3000 square centimeters, we can write the equation as follows: L + B = 314r + 3.14r² = 3000.
Step 5: Solve the equation:
- Rearrange the equation to be a quadratic equation: 3.14r² + 314r - 3000 = 0.
- You can solve this quadratic equation using factoring, completing the square, or quadratic formula. Let's use the quadratic formula.
- Apply the quadratic formula: r = (-b ± √(b² - 4ac)) / (2a), where a = 3.14, b = 314, and c = -3000.
- Calculate the values: r = (-314 ± √(314² - 4 * 3.14 * -3000)) / (2 * 3.14).
- Simplify the equation: r = (-314 ± √(98596 + 37680)) / (6.28).
- r ≈ (-314 ± √136276) / 6.28.
- Calculate the approximate values for r: r ≈ (-314 ± 369.13) / 6.28.
- This results in two potential solutions: r ≈ 6.7 and r ≈ -51.9.
Step 6: Choose the appropriate solution:
- Since the radius cannot be negative, we discard the negative value.
- Therefore, the radius of the vase is approximately 6.7 centimeters.
By following these steps, you can find the radius of the cylindrical vase with the given information.