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.

Relevance
• ?
Lv 7
2 months ago

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

3      8

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

r       h

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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.

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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

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• 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.