Math Problem Statement

Solve

Assume that the height of your cylinder is 8 inches. Consider A as a function of r, so we can write that as A(r) = 2πr² + 16πr. What is the domain of A(r)? Write the radius r as a function of A. This is the inverse function to A(r), i.e., to turn A as a function of r into r as a function of A.

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 A(r)A(r) is expressed as: A(r)=2πr2+16πrA(r) = 2\pi r^2 + 16\pi r You are asked to find the domain of A(r)A(r), or in other words, determine for which values of rr the function A(r)A(r) is defined.

Analysis:

  • The term r2r^2 implies that rr is a radius, which cannot be negative.
  • The term rr also represents the radius, so it must be positive.

Thus, the function A(r)A(r) is defined for r0r \geq 0. In practical terms, the radius of a cylinder cannot be negative, so the domain of A(r)A(r) is: r0r \geq 0

Part b:

Now, you're asked to find the inverse function r(A)r(A), which means expressing rr as a function of AA. We need to solve the given equation A(r)=2πr2+16πrA(r) = 2\pi r^2 + 16\pi r for rr in terms of AA.

Step-by-step Solution:

  1. Start with the equation: A=2πr2+16πrA = 2\pi r^2 + 16\pi r
  2. Divide both sides of the equation by 2π2\pi to simplify: A2π=r2+8r\frac{A}{2\pi} = r^2 + 8r
  3. Rearrange the terms to form a quadratic equation: r2+8rA2π=0r^2 + 8r - \frac{A}{2\pi} = 0
  4. Solve for rr using the quadratic formula. The quadratic formula is: r=b±b24ac2ar = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} For the equation r2+8rA2π=0r^2 + 8r - \frac{A}{2\pi} = 0, the coefficients are a=1a = 1, b=8b = 8, and c=A2πc = -\frac{A}{2\pi}. Plugging these values into the quadratic formula gives: r=8±824(1)(A2π)2(1)r = \frac{-8 \pm \sqrt{8^2 - 4(1)\left(-\frac{A}{2\pi}\right)}}{2(1)}
  5. Simplify: r=8±64+2Aπ2r = \frac{-8 \pm \sqrt{64 + \frac{2A}{\pi}}}{2} r=8±64π+2Aπ2r = \frac{-8 \pm \sqrt{\frac{64\pi + 2A}{\pi}}}{2}
  6. Since the radius must be positive, we discard the negative root: r=8+64π+2Aπ2r = \frac{-8 + \sqrt{\frac{64\pi + 2A}{\pi}}}{2}

This is the inverse function r(A)r(A), which gives the radius in terms of the surface area.

For the specific value of A=165.04A = 165.04:

You are provided with r(A)=165.04r(A) = 165.04. This is a specific value for AA, 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:

  1. How does the quadratic formula help in solving for rr from A(r)A(r)?
  2. What happens if we use the negative root in the quadratic formula?
  3. What other methods can be used to solve quadratic equations?
  4. Why does the domain of A(r)A(r) include r0r \geq 0?
  5. 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