The Total Surface Area Of A Cylinder Is 66 Cm². If Its Height Is 8 Cm, Find The Base Radius Of The Cylinder.
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Introduction
In mathematics, the total surface area of a cylinder is a fundamental concept that is used to calculate the surface area of a three-dimensional object. The total surface area of a cylinder is the sum of the areas of its two bases and its lateral surface area. In this article, we will discuss how to find the base radius of a cylinder given its total surface area and height.
Formula for the Total Surface Area of a Cylinder
The formula for the total surface area of a cylinder is:
TSA = 2πr² + 2πrh
where TSA is the total surface area, r is the base radius, and h is the height of the cylinder.
Given Information
We are given that the total surface area of the cylinder is 66 cm² and its height is 8 cm. We need to find the base radius of the cylinder.
Step 1: Substitute the Given Values into the Formula
Substituting the given values into the formula, we get:
66 = 2πr² + 2π(8)r
Step 2: Simplify the Equation
Simplifying the equation, we get:
66 = 2πr² + 16πr
Step 3: Rearrange the Equation
Rearranging the equation, we get:
2πr² + 16πr - 66 = 0
Step 4: Solve the Quadratic Equation
This is a quadratic equation in the form of ax² + bx + c = 0, where a = 2π, b = 16π, and c = -66. We can solve this equation using the quadratic formula:
r = (-b ± √(b² - 4ac)) / 2a
Substituting the values of a, b, and c, we get:
r = (-(16π) ± √((16π)² - 4(2π)(-66))) / (2(2π))
Step 5: Simplify the Expression
Simplifying the expression, we get:
r = (-16π ± √(256π² + 528π)) / (4π)
Step 6: Calculate the Value of r
Now, we need to calculate the value of r. We can do this by substituting the value of π (approximately 3.14) into the expression:
r = (-16(3.14) ± √(256(3.14)² + 528(3.14))) / (4(3.14))
r = (-50.24 ± √(2560.96 + 1661.12)) / 12.56
r = (-50.24 ± √4222.08) / 12.56
r = (-50.24 ± 65.04) / 12.56
Step 7: Find the Two Possible Values of r
Now, we have two possible values of r:
r = (-50.24 + 65.04) / 12.56 r = 14.8 / 12.56 r = 1.18
r = (-50.24 - 65.04) / 12.56 r = -115.28 / 12.56 r = -9.22
Step 8: Choose the Positive Value of r
Since the radius of a cylinder cannot be negative, we choose the positive value of r:
r = 1.18 cm
Conclusion
In this article, we discussed how to find the base radius of a cylinder given its total surface area and height. We used the formula for the total surface area of a cylinder and solved a quadratic equation to find the value of r. We found that the base radius of the cylinder is 1.18 cm.
Final Answer
The final answer is: 1.18 cm
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Introduction
In our previous article, we discussed how to find the base radius of a cylinder given its total surface area and height. In this article, we will answer some frequently asked questions (FAQs) about the total surface area of a cylinder.
Q: What is the total surface area of a cylinder?
A: The total surface area of a cylinder is the sum of the areas of its two bases and its lateral surface area. It is given by the formula:
TSA = 2πr² + 2πrh
where TSA is the total surface area, r is the base radius, and h is the height of the cylinder.
Q: How do I find the base radius of a cylinder given its total surface area and height?
A: To find the base radius of a cylinder given its total surface area and height, you can use the formula for the total surface area of a cylinder and solve a quadratic equation. The steps are as follows:
- Substitute the given values into the formula.
- Simplify the equation.
- Rearrange the equation.
- Solve the quadratic equation.
- Choose the positive value of r.
Q: What is the formula for the lateral surface area of a cylinder?
A: The formula for the lateral surface area of a cylinder is:
LSA = 2πrh
where LSA is the lateral surface area, r is the base radius, and h is the height of the cylinder.
Q: How do I find the height of a cylinder given its total surface area and base radius?
A: To find the height of a cylinder given its total surface area and base radius, you can use the formula for the total surface area of a cylinder and solve a quadratic equation. The steps are as follows:
- Substitute the given values into the formula.
- Simplify the equation.
- Rearrange the equation.
- Solve the quadratic equation.
- Choose the positive value of h.
Q: What is the formula for the volume of a cylinder?
A: The formula for the volume of a cylinder is:
V = πr²h
where V is the volume, r is the base radius, and h is the height of the cylinder.
Q: How do I find the base radius of a cylinder given its volume and height?
A: To find the base radius of a cylinder given its volume and height, you can use the formula for the volume of a cylinder and solve a quadratic equation. The steps are as follows:
- Substitute the given values into the formula.
- Simplify the equation.
- Rearrange the equation.
- Solve the quadratic equation.
- Choose the positive value of r.
Q: What is the formula for the surface area of a cylinder in terms of its diameter?
A: The formula for the surface area of a cylinder in terms of its diameter is:
TSA = πd² + 2πdh
where TSA is the total surface area, d is the diameter, and h is the height of the cylinder.
Q: How do I find the height of a cylinder given its total surface area and diameter?
A: To find the height of a cylinder given its total surface area and diameter, you can use the formula for the surface area of a cylinder in terms of its diameter and solve a quadratic equation. The steps are as follows:
- Substitute the given values into the formula.
- Simplify the equation.
- Rearrange the equation.
- Solve the quadratic equation.
- Choose the positive value of h.
Conclusion
In this article, we answered some frequently asked questions (FAQs) about the total surface area of a cylinder. We provided formulas and step-by-step instructions for finding the base radius, height, and volume of a cylinder given different combinations of its surface area and dimensions.
Final Answer
The final answer is: There is no single final answer, as the questions and answers are a collection of FAQs about the total surface area of a cylinder.