Math Problem Statement
A right cylinder has a surface area of 217 square inches and a length of 12 inches. Using the equation for the surface area of a right cylinder and the quadratic formula, solve for the radius of the rolling pin in inches correct to three significant figures. Then convert the value of the radius from inches into centimeters.
Solution
To solve for the radius of the right cylinder, we'll use the formula for the surface area of a cylinder. The formula for the surface area of a right cylinder is given by:
Where:
- is the surface area (217 square inches),
- is the radius,
- is the height (length) of the cylinder (12 inches),
- is approximately 3.14159.
We can rearrange the formula into a quadratic equation in terms of :
Simplify the equation:
Dividing through by to make the equation easier to work with:
Now solve this equation step by step.
Step 1: Simplify the left side
We can calculate :
So the equation becomes:
Step 2: Rearrange into standard quadratic form
Rearrange the equation:
Step 3: Apply the quadratic formula
The quadratic formula is:
For the equation , the coefficients are:
- ,
- ,
- .
Now plug these values into the quadratic formula:
Step 4: Solve for the two possible values of
So:
Since the radius cannot be negative, we discard . Therefore, the radius is approximately:
Step 5: Convert inches to centimeters
To convert from inches to centimeters, we use the conversion factor :
Thus, the radius of the cylinder is approximately:
Would you like further details on any step?
Related Questions:
- How is the surface area of a right cylinder related to its radius and height?
- How can we solve quadratic equations that arise from geometric problems?
- What are the differences between a right cylinder and other types of cylinders?
- How would the result change if the length of the cylinder was different?
- Can the quadratic formula be used for all geometric shapes?
Tip:
Always check that all terms in an equation have consistent units before solving for any unknowns!
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Math Problem Analysis
Mathematical Concepts
Geometry
Algebra
Quadratic Equations
Formulas
Surface area of a right cylinder: A = 2πrh + 2πr^2
Quadratic equation formula: ax^2 + bx + c = 0
Theorems
Quadratic Formula: r = (-b ± √(b^2 - 4ac)) / (2a)
Suitable Grade Level
Grades 9-12
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