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 AA of a right cylinder is given by:

A=2πrh+2πr2A = 2\pi r h + 2\pi r^2

Where:

  • AA is the surface area (217 square inches),
  • rr is the radius,
  • hh is the height (length) of the cylinder (12 inches),
  • π\pi is approximately 3.14159.

We can rearrange the formula into a quadratic equation in terms of rr:

217=2πr(12)+2πr2217 = 2\pi r(12) + 2\pi r^2

Simplify the equation:

217=24πr+2πr2217 = 24\pi r + 2\pi r^2

Dividing through by 2π2\pi to make the equation easier to work with:

2172π=12r+r2\frac{217}{2\pi} = 12r + r^2

Now solve this equation step by step.

Step 1: Simplify the left side

We can calculate 2172π\frac{217}{2\pi}:

2172π2176.2831834.544\frac{217}{2\pi} \approx \frac{217}{6.28318} \approx 34.544

So the equation becomes:

34.544=12r+r234.544 = 12r + r^2

Step 2: Rearrange into standard quadratic form

Rearrange the equation:

r2+12r34.544=0r^2 + 12r - 34.544 = 0

Step 3: Apply the quadratic formula

The quadratic formula is:

r=b±b24ac2ar = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For the equation r2+12r34.544=0r^2 + 12r - 34.544 = 0, the coefficients are:

  • a=1a = 1,
  • b=12b = 12,
  • c=34.544c = -34.544.

Now plug these values into the quadratic formula:

r=12±1224(1)(34.544)2(1)r = \frac{-12 \pm \sqrt{12^2 - 4(1)(-34.544)}}{2(1)} r=12±144+138.1762r = \frac{-12 \pm \sqrt{144 + 138.176}}{2} r=12±282.1762r = \frac{-12 \pm \sqrt{282.176}}{2} r=12±16.7992r = \frac{-12 \pm 16.799}{2}

Step 4: Solve for the two possible values of rr

r=12+16.7992orr=1216.7992r = \frac{-12 + 16.799}{2} \quad \text{or} \quad r = \frac{-12 - 16.799}{2}

So:

r=4.7992orr=28.7992r = \frac{4.799}{2} \quad \text{or} \quad r = \frac{-28.799}{2} r=2.3995orr=14.399r = 2.3995 \quad \text{or} \quad r = -14.399

Since the radius cannot be negative, we discard r=14.399r = -14.399. Therefore, the radius is approximately:

r2.400inchesr \approx 2.400 \, \text{inches}

Step 5: Convert inches to centimeters

To convert from inches to centimeters, we use the conversion factor 1inch=2.54cm1 \, \text{inch} = 2.54 \, \text{cm}:

r2.400×2.54=6.096cmr \approx 2.400 \times 2.54 = 6.096 \, \text{cm}

Thus, the radius of the cylinder is approximately:

2.400inchesor6.096cm\boxed{2.400 \, \text{inches} \quad \text{or} \quad 6.096 \, \text{cm}}

Would you like further details on any step?

Related Questions:

  1. How is the surface area of a right cylinder related to its radius and height?
  2. How can we solve quadratic equations that arise from geometric problems?
  3. What are the differences between a right cylinder and other types of cylinders?
  4. How would the result change if the length of the cylinder was different?
  5. 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