An industrial designer is creating a cylindrical recycled-material container with a volume of 500π cm³. The height is triple the radius. What is the radius of the container?
["Title: How to Calculate the Radius of a Cylindrical Recycled Material Container with Volume 500π cm³", "When designing sustainable products, industrial designers often need precise dimensions for efficiency, cost, and functionality—especially when using recycled materials. One practical challenge is determining the radius of a cylindrical container made from eco-friendly recycled materials, given its volume and a key size relationship. In this article, we explore how an industrial designer computes the radius of a cylindrical recycling container with a volume of 500π cm³, where the height is three times the radius.", "---", "### Understanding the Problem", "The container is cylindrical, and its volume is defined by the formula:", "[\nV = \pi r^2 h\n]", "Where:\n- ( V ) is the volume (here, ( 500\pi ) cm³)\n- ( r ) is the radius\n- ( h ) is the height", "We are told that the height is triple the radius:", "[\nh = 3r\n]", "---", "### Step-by-Step Calculation", "Substitute ( h = 3r ) into the volume formula:", "[\n500\pi = \pi r^2 (3r)\n]", "Simplify the right-hand side:", "[\n500\pi = 3\pi r^3\n]", "Now divide both sides by ( \pi ) (since ( \pi <br/>\neq 0 )):", "[\n500 = 3r^3\n]", "Solve for ( r^3 ):", "[\nr^3 = \frac{500}{3}\n]", "Now take the cube root to find ( r ):", "[\nr = \sqrt[3]{\frac{500}{3}}\n]", "---", "### Simplify the Cube Root (Optional)", "While ( \frac{500}{3} \approx 166.67 ), and its cube root is not a whole number, this is the exact form preferred in technical design. Approximating:", "[\nr \approx \sqrt[3]{166.67} \approx 5.5 , \ ext{cm} \quad \ ext{(approximate)}\n]", "But the exact answer remains:", "[\nr = \sqrt[3]{\frac{500}{3}} , \ ext{cm}\n]", "---", "### Why This Matters for Industrial Designers", "Knowing the radius and height ensures optimal material usage—critical when working with recycled materials, where waste reduction and structural integrity go hand in hand. A well-calculated cylindrical container maximizes volume while minimizing surface area, reducing material costs and environmental impact.", "---", "### Final Summary", "- Volume: ( 500\pi , \ ext{cm}^3 )\n- Relationship: ( h = 3r )\n- Radius formula: ( r = \sqrt[3]{\frac{500}{3}} , \ ext{cm} ) ≈ 5.5 cm", "This precise calculation supports efficient, sustainable design—core principles in modern industrial design.", "---", "Keywords: industrial designer, recycled container, cylindrical volume calculation, sustainable design, radius formula, volume optimization, eco-friendly product design, recycled material container, cylinder dimensions.", "---", "Turn your design ideas into eco-smart realities—calculate smart, build sustainable."]
