Exercise 7 Page 594

We are given that two cylinders are made by rolling two equally-sized sheets of construction paper. We want to determine which cylinder has the greater volume. Let's start by considering the following sheet of paper.

For each cylinder that we can make by rolling this sheet of paper, we will find an expression for the volume of the cylinder. We will start with the taller cylinder.

Recall that the volume of a cylinder with the radius r and height h can be calculated by the following formula. V=π r^2 h Now we will find the height and the radius of the taller cylinder. Note that the length of the sheet is equal to the height of the cylinder. Moreover, the width of the sheet is equal to the circumference of the base of the cylinder. Recall that the circumference of a circle with radius r is equal to 2π r. Using these facts, we can find relationships between sides of the sheet and the cylinder.

Since the length of the sheet is 12 inches, the height of the cylinder is also 12 inches. Knowing that the width of the sheet is 9 inches, we can write an equation that will help us to find the radius. 2π r = 9 Let's solve the equation for r.

Now we can substitute the height and the radius into the formula for the volume of a cylinder. V=π r^2 h ⇕ V=π ( 9/2π)^2 ( 12) Next, we will write an expression for the volume of the shorter cylinder. This cylinder is made by rolling the sheet of paper along the width.

Since the width of the sheet is 9 inches, the height of the cylinder is also 9 inches. We also know that the height of the sheet is 12 inches and it is equal to the circumference of the cylinder. 2π r = 12 We can solve the equation for r.

Therefore, the height of the cylinder is 9 inches and the radius 122π inches. Let's write the expression for the volume of the shorter cylinder. V=π r^2 h ⇕ V=π ( 12/2π)^2 ( 9) Now we will analyze the expressions for the volumes of the two cylinders.

Finally, we can compare these two values. We can see that both expressions consist of the same factor multiplied by a number. Since 9<12, the volume of the shorter cylinder is greater than the volume of the taller cylinder. 9 (π * 9 * 1/2π * 1/2π * 12)<12 (π * 9 * 1/2π * 1/2π * 12) Recall that we considered the sheet of paper with the length of 12 inches and the width of 9 inches. Now let's consider the relationship between the cylinders made by rolling a sheet of paper with length a and width b. We can easily get the expressions for the volumes of the cylinders by substituting a for 12 and b for 9 into the previous expressions.

Similar as before, since b< a, the volume of the shorter cylinder is greater than the volume of the taller cylinder. b (π * a * 1/2π * 1/2π * b)< a (π * a * 1/2π * 1/2π * b)