A cylinder is rolled three complete revolutions to create a rectangular shape 32 inches long and 12 inches wide. Write an expression for the area of the rectangle in terms of the diameter of the cylinder. Show how you could use this expression to find a formula for the surface area of the cylinder

πd*3 = 32

the area is 32*12 = 36πd

clearly the cylinder's lateral area is 1/3 of the rectangle. Add on πd^2/2 to get the total area.

To find an expression for the area of the rectangle in terms of the diameter of the cylinder, we need to first understand the relationship between the cylinder and the rectangular shape created.

When a cylinder is rolled, the length of the cylinder corresponds to the width of the rectangular shape, and the circumference of the cylinder represents the length of the rectangular shape.

Let's denote the diameter of the cylinder as "d" inches. We know that the circumference of the cylinder (C) is equal to πd (pi times diameter). Since the cylinder makes three complete revolutions, the length of the rectangular shape (L) will be equal to 3C (3 times the circumference of the cylinder).

Given that the length (L) is 32 inches, we can set up the equation: 3C = 32

Substituting C = πd, we have:
3(πd) = 32

Now, we can solve for diameter (d):
d = 32 / (3π)

To find the width (W) of the rectangular shape, we can use the fact that it is equal to the radius (r) of the cylinder. The radius is half the diameter, so we have:
r = d / 2

Substituting the value of d obtained earlier:
r = (32 / (3π)) / 2

The area of a rectangle is given by the formula A = L * W. Substituting the values we obtained:
A = (3C) * (r)
A = (3πd) * (r)
A = (3πd) * [(32 / (3π)) / 2]

Simplifying:
A = (πd÷π) * [(32 ÷ 3) ÷ 2]
A = d * (16 ÷ 3)

So, the expression for the area of the rectangle in terms of the diameter is A = (16d) / 3.

To find a formula for the surface area of the cylinder, we need to consider that a cylinder is made up of two circular bases and a curved surface. The area of each circular base is given by πr² (pi times radius squared), and the area of the curved surface is equal to the circumference of the base multiplied by the height (h) of the cylinder.

The height (h) of the cylinder can be found by dividing the length of the rectangular shape (L) by the circumference of the circular base (C).

So, the surface area (SA) of the cylinder is:
SA = 2(πr²) + (Ch)

Since we already found the expressions for C and r in terms of the diameter, we substitute those:
SA = 2(π[(d/2)²]) + ((πd) * (L/C))

Simplifying further:
SA = π(d²/2) + πd(L/C)

Substituting the value of L from before:
SA = π(d²/2) + (32πd) / (3π)

Simplifying again:
SA = π(d²/2) + 32d/3

Therefore, the formula for the surface area of the cylinder in terms of the diameter is SA = π(d²/2) + 32d/3.