Math Problem Statement
Solve
Solution
It looks like the image contains a math problem related to the surface area and radius of a cylinder. Let's break it down step by step.
Part a:
You are given that the height of the cylinder is 8 inches and the surface area is expressed as: You are asked to find the domain of , or in other words, determine for which values of the function is defined.
Analysis:
- The term implies that is a radius, which cannot be negative.
- The term also represents the radius, so it must be positive.
Thus, the function is defined for . In practical terms, the radius of a cylinder cannot be negative, so the domain of is:
Part b:
Now, you're asked to find the inverse function , which means expressing as a function of . We need to solve the given equation for in terms of .
Step-by-step Solution:
- Start with the equation:
- Divide both sides of the equation by to simplify:
- Rearrange the terms to form a quadratic equation:
- Solve for using the quadratic formula. The quadratic formula is: For the equation , the coefficients are , , and . Plugging these values into the quadratic formula gives:
- Simplify:
- Since the radius must be positive, we discard the negative root:
This is the inverse function , which gives the radius in terms of the surface area.
For the specific value of :
You are provided with . This is a specific value for , and if we plug this into the formula, we could compute the corresponding radius. However, from the image, it seems the quadratic equation may already be solved.
Would you like me to help further in calculating that specific radius or clarify any steps in this solution?
Related Questions:
- How does the quadratic formula help in solving for from ?
- What happens if we use the negative root in the quadratic formula?
- What other methods can be used to solve quadratic equations?
- Why does the domain of include ?
- How does the surface area formula change if the height of the cylinder changes?
Tip: Always check if the context (e.g., physical constraints like radius) leads to discarding certain solutions from the quadratic formula.
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Math Problem Analysis
Mathematical Concepts
Geometry
Quadratic Equations
Inverse Functions
Domain of Functions
Formulas
Surface area of a cylinder: A(r) = 2πr² + 16πr
Quadratic formula: r = (-b ± √(b² - 4ac)) / 2a
Theorems
Quadratic Formula
Definition of Inverse Functions
Suitable Grade Level
Grades 9-12
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