IB Math Study Guide 10e

Parts of a Cylinder

  • A cylinder consists of several structural components:

    • Top Part: A circle.

    • Bottom Part: Also a circle.

Area of a Circle

  • The area of one circle is calculated using the formula:

    • A=πr2A = \pi r^2

    • Where r is the radius of the circle.

  • Since there are two circles in a cylinder, the total area contributed by the circles is:

    • Total Area from Circles:

    • 2πr22 \pi r^2

Lateral Surface Area of a Cylinder

  • When the cylinder is cut and unrolled, it forms a rectangle:

    • Base of the Rectangle: Represents the circumference of the circle.

    • Height of the Rectangle: Corresponds to the height (h) of the cylinder.

Finding the Height and Relationship Between Parts

  • The height of the can is represented as:

    • Height = hh

  • The base circumference around the circle is a critical calculation:

    • Circumference Formula:

    • C=2πrC = 2 \pi r

  • Thus, the lateral area (rectangle part) can be expressed as:

    • Area of Rectangle:

    • Arect=C×h=(2πr)hA_{rect} = C \times h = (2 \pi r)h

Surface Area of a Cylinder

  • The total surface area of a cylinder combines the areas of the circles and the rectangle:

    • Total Surface Area Formula:

    • Atotal=2πr2+(2πr)hA_{total} = 2 \pi r^2 + (2 \pi r)h

Solving for Height

  • If given the total area (400 in this context), the formula to find height can be rearranged:

    • Assume:

    • A=2πr2+(2πr)hA = 2 \pi r^2 + (2 \pi r)h

    • To isolate h, rearranging yields:

    • h=400−2πr22πrh = \frac{400 - 2 \pi r^2}{2 \pi r}

    • Or simplified to:

    • h=400πr2−rh = \frac{400}{\pi r^2} - r

Understanding Relationships of Circle Properties

  • Integral and Derivative Connections:

    • The concept of areas, curves, and their derivatives and integrals can be related:

    • Area under a Curve: Integral of circumference gives area:

    • ∫C dx=∫(2πr) dx=πr2\int C \, dx = \int (2 \pi r) \, dx = \pi r^2

    • Derivatives: The change from area back to circumference:

    • Derivative Relationship:

      • d(Area)dx=Circumference\frac{d(Area)}{dx} = Circumference

      • d(πr2)dr=2πr\frac{d(\pi r^2)}{dr} = 2 \pi r

    • Conclusion of Relationships:

    • Understanding that:

      • If circumference is forgotten, the derivative of area provides a quick way to derive it:

      • Circumference from Area: C=d(πr2)drC = \frac{d(\pi r^2)}{dr}