Composite Function Word Problem?

A drinking cup from a water fountain has the shape of an inverted cone. The cup has a height of 8 cm, and a radius of 3 cm. The water in the cup also has the shape of an inverted cone, with a radius of r and a height of h.

The diagram of the drinking cup shows two right triangles; a large triangle for the entire height of the cup, and a smaller triangle for the water in the cup. The two triangles have identical angles, so they can be classified as similar triangles.

a.) Use similar triangle ratios to express r as a function of h.

b.) Derive the composite function, V_water(h) = (V_cone of r), the the volume of the water in the cone.

c.) If the volume of water in the cone is 3π cm^3, determine the height of the water.

Update:

The best answer is partially incorrect but it did explain to me how I should get the answer. The answer isn't 9/64 as there's a 1/3 in front of the pie symbol which reduces the 9/64 down to 3/64. Fair enough for anybody who may be reading this and needs help.

Attachment image

2 Answers

Relevance
  • ?
    Lv 7
    2 months ago
    Favorite Answer

    a.) Use similar triangle ratios to express r as a function of h.

          3      8

         --- = --- ⇒ 8r = 3h ⇒ r = ³⁄₈ h ..............ANS

          r       h

    ------------------------------------------

    b.) Derive the composite function, V_water(h) = (V_cone of r), the the volume of the water in the cone.

         Sorry- I know what composite functions are, but am

         having difficulty parsing this notation: 

                       V_water(h) = (V_cone of r)

         I'm sure you'll figure this out.

    ------------------------------------------

    c.) If the volume of water in the cone is 3π cm^3, determine the height of the water.

       Vc = π r² h = 3π cm³

               π (³⁄₈ h cm)² = 3π cm³

               ⁹⁄₆₄ π h² cm² =  3π cm³

                        3π cm³

               h² = --------------

                       ⁹⁄₆₄ π cm²

               h² = 21.33 cm

               h = √21.33 cm

               h = 4.6188 cm ...........ANS

    ------------------------------------------

  • Anonymous
    2 months ago

    The best answer is partially incorrect but it did explain to me how I should get the answer. The answer isn't 9/64 as there's a 1/3 in front of the pie symbol which reduces the 9/64 down to 3/64. I almost made that mistake.

    The correct answer to the last question is 4 = (3/64)^1/3.

    Fair enough.

Still have questions? Get your answers by asking now.