Math / Geometry

QuestionAn open-ended cylinder can be folded from a rectangular piece of paper $A B C D$ in two ways: one in which $A B$ meets $D C$ and the other in which $A D$ meets $B C$ .
For a particular piece of paper, the difference in volume of these two cylinders is $1000 \mathrm{~cm}^{3}$ .
If one edge length is 10 cm shorter than the other edge length, what is the perimeter (in cm, to 4 sig.figs.) of the rectangle $A B C D$ ?

Studdy Solution

STEP 1

1. The rectangle $ABCD$ has two different edge lengths, $x$ and $y$ .
2. One edge length is 10 cm shorter than the other, i.e., $x = y - 10$ or $y = x - 10$ .
3. The difference in volume between the two possible cylinders is $1000 \, \text{cm}^3$ .

STEP 2

1. Define the dimensions of the rectangle and the formulas for the volumes of the cylinders.
2. Set up the equation for the difference in volumes.
3. Solve the equation for the dimensions of the rectangle.
4. Calculate the perimeter of the rectangle.

STEP 3

Define the dimensions of the rectangle and the formulas for the volumes of the cylinders:
Let $x$ and $y$ be the dimensions of the rectangle, with $x = y - 10$ .
The volume of a cylinder is given by:
$V = \pi r^2 h$
For the first cylinder, where $AB$ meets $DC$ , the height is $x$ and the circumference is $y$ , so:
$V_1 = \pi \left(\frac{y}{2\pi}\right)^2 x$
For the second cylinder, where $AD$ meets $BC$ , the height is $y$ and the circumference is $x$ , so:
$V_2 = \pi \left(\frac{x}{2\pi}\right)^2 y$

STEP 4

Set up the equation for the difference in volumes:
$V_1 - V_2 = 1000$
Substitute the expressions for $V_1$ and $V_2$ :
$\pi \left(\frac{y}{2\pi}\right)^2 x - \pi \left(\frac{x}{2\pi}\right)^2 y = 1000$
Simplify:
$\frac{y^2 x}{4\pi} - \frac{x^2 y}{4\pi} = 1000$
$\frac{y^2 x - x^2 y}{4\pi} = 1000$
$yx(y - x) = 4000\pi$

STEP 5

Solve the equation for the dimensions of the rectangle:
Substitute $x = y - 10$ into the equation:
$y(y - 10)(10) = 4000\pi$
$10y(y - 10) = 4000\pi$
$y(y - 10) = 400\pi$
Solve the quadratic equation:
$y^2 - 10y - 400\pi = 0$
Use the quadratic formula:
$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
where $a = 1$ , $b = -10$ , and $c = -400\pi$ .
$y = \frac{10 \pm \sqrt{100 + 1600\pi}}{2}$
Calculate $y$ and then $x = y - 10$ .

STEP 6

Calculate the perimeter of the rectangle:
The perimeter $P$ is given by:
$P = 2(x + y)$
Substitute the values of $x$ and $y$ obtained from Step 3.
The perimeter of the rectangle is approximately:
$\boxed{2(x + y)}$

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