How Would I Determine The Rate At Which The Shadow Of A Cylinder Cast On A Circular Table Is Increasing When The Cylinder Is Being Lifted At A Constant Rate Of 2 Inches Per Second, And The Table Has A Radius Of 10 Inches And The Cylinder Has A Height Of 5 Inches And A Radius Of 2 Inches?

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To determine the rate at which the shadow of a cylinder cast on a circular table is increasing when the cylinder is being lifted at a constant rate of 2 inches per second, we need to consider the relationship between the height of the cylinder and the radius of its shadow.

  1. Understanding the Problem:

    • The cylinder has a radius of 2 inches and is being lifted at a rate of 2 inches per second.
    • The table has a radius of 10 inches.
    • The shadow of the cylinder on the table is a circle whose radius increases as the cylinder is lifted.

  2. Using Similar Triangles:

    • Assume the light source is on the edge of the table, 10 inches from the center.
    • The radius of the shadow r(t){ r(t) } is related to the radius of the cylinder (2 inches) by the ratio of the distances from the light source.
    • Using similar triangles, the relationship is given by: r(t)2=10+h(t)10{ \frac{r(t)}{2} = \frac{10 + h(t)}{10} }
    • Solving for r(t){ r(t) }: r(t)=2(10+h(t)10)=2+h(t)5{ r(t) = 2 \left( \frac{10 + h(t)}{10} \right) = 2 + \frac{h(t)}{5} }

  3. Differentiating with Respect to Time:

    • Differentiate r(t){ r(t) } with respect to t{ t }: drdt=ddt(2+h(t)5)=15⋅dhdt{ \frac{dr}{dt} = \frac{d}{dt} \left( 2 + \frac{h(t)}{5} \right) = \frac{1}{5} \cdot \frac{dh}{dt} }
    • Given dhdt=2{ \frac{dh}{dt} = 2 } inches per second: drdt=15⋅2=25{ \frac{dr}{dt} = \frac{1}{5} \cdot 2 = \frac{2}{5} }

Thus, the rate at which the shadow of the cylinder is increasing is \boxed{\dfrac{2}{5}} inches per second.