Question

a rectangular sheet of paper with perimeter 45 cm. is rolled into a right cylinder. if x represents the length of the paper, that then becomes the circumference of the open - base of the cylinder. the volume of the cylinder is given by: v = π(\frac{x}{2π})^2(\frac{45 - 2x}{2}). using a graphing calculator, determine the following: a) what is the domain of the function? b) the length of the paper, x, that maximizes the volume of the cylinder. x = cm c) the maximum volume of the cylinder. maximum volume = cubic cm question help: video post to forum submit question

Explanation:

Step1: Find domain of the function

The length \(x>0\) and the other side of the rectangle \(\frac{45 - 2x}{2}>0\). Solving \(\frac{45 - 2x}{2}>0\) gives \(45-2x > 0\), then \(2x<45\) or \(x < 22.5\). So the domain is \(0

Step2: Maximize the volume function

We have \(V=\pi(\frac{x}{2\pi})^2(\frac{45 - 2x}{2})=\frac{x^{2}(45 - 2x)}{8\pi}=\frac{45x^{2}-2x^{3}}{8\pi}\). Using a graph - ing calculator, we find the maximum of the function \(y = \frac{45x^{2}-2x^{3}}{8\pi}\) in the domain \(0 < x<22.5\). The value of \(x\) that maximizes the function is \(x = 15\) cm.

Step3: Calculate the maximum volume

Substitute \(x = 15\) into the volume formula \(V=\pi(\frac{15}{2\pi})^2(\frac{45-2\times15}{2})\). First, \(\frac{15}{2\pi}\) is the radius \(r\) and \(\frac{45 - 30}{2}=\frac{15}{2}\) is the height \(h\). \(V=\pi\times\frac{225}{4\pi^{2}}\times\frac{15}{2}=\frac{3375}{8\pi}\approx133.7\) cubic cm.

Answer:

a) \(0 < x<22.5\)
b) \(15\)
c) \(\frac{3375}{8\pi}\approx133.7\)