Question

Given each bottle, match the Height vs. Volume graph that will be created as the bottle is consistently filled with a liquid.

Answer 1

The relationship between the volume of a liquid in a vessel and the height of the liquid in the vessel is a function

The correct options are;

Ink bottle: B

Conical flask: I

Boiling flask: C

Bucket: H

Vase: D

Plugged funnel: E

The reason the above values are correct are as follows:

The formula for the volume of the cylindrical portion of the containers is;

$V_{cylinder}$ = π·r²·h

∴ $V_{cylinder}$ ∝ h

Rate of change, dV/dh = π·r²

Given that r is constant, dV/dh is constant

The formula for the volume of the spherical portion of the bottles is;

$V_{sphere}$ = (4/3)·π·r³

$V_{sphere} = \dfrac{4\cdot \pi \cdot r^3 }{4} - \dfrac{ \pi \cdot h_c^2 \cdot (3 \cdot r-h_c) }{3}$

$V_{sphere}$ ∝ $h_c^3$

The formula for the volume of the conical portion of the bottles is;

$V_{cone} = \dfrac{1}{3} \times \pi \times r^2 \times h$

The rate of change of the volume with height = (1/3)·π·r²

From the above formulas, we have;

The shape of the graph of the bucket is similar to half a parabola umbrella shaped, which is option H

Similarly, the shape of the graph of a conical flask which is a upright cone, will be a cup shaped parabola with a straight portion at the end, which is option I

The rate at which the height with volume introduced in the vase first increased slowly at the start then more rapidly in the middle, then slowly again at the the top, which is equivalent to option D

The height and volume in the plugged funnel first increase at a constant rate then assumed a parabolic shape of half an umbrella, which is option E

Similar to the vase, the height of the liquid in the boiling flask first increases at a lesser rate than the volume, and increasingly so towards the middle, then past the middle, the height increases at an increasingly rapid rate than the volume, and finally, in the cylindrical section, the graph is a straight line, which is option C

The ink bottle has two cylindrical portions and one upright conic section, therefore, the graph consists of two straight portions at the start and final, position and a parabolic (facing up) section at the middle, which is option B

Learn more about interpreting functions here:

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