Open-Ended. The height h of a cylinder varies directly with...

Open-Ended. The height $h$ of a cylinder varies directly with its volume $V$ and inversely with the square of its radius $r .$ Find at least four ways to change the volume and radius of a cylinder so that its height is quadrupled.

Key Concepts

Combined Variation

Combined variation occurs when a quantity depends on more than one variable, as seen in the formula h = k(V/r²). This concept is vital for analyzing how simultaneous changes in volume and radius affect the height of the cylinder. It requires understanding that the total effect on height involves both the direct dependence on volume and the inverse square dependence on the radius.

Scaling and Proportional Reasoning

Scaling involves understanding how changes in dimensions affect related quantities through multiplicative factors. In problems addressing how to quadruple the height of a shape (cylinder), one must consider the multiplicative relationships between volume, radius, and height, and determine the corresponding adjustments needed in each parameter to achieve the desired scaling.

Direct Variation

Direct variation describes a relationship where one quantity increases or decreases in proportion to another. In general, if a variable y varies directly with x, then y = kx for some constant k. This concept is essential when analysing how changing one parameter, such as volume, directly influences another parameter, such as height.

Inverse Variation

Inverse variation refers to a relationship where one variable increases as another decreases, typically illustrated by y = k/x for some constant k. In the context of a cylinder, the height varies inversely with the square of the radius, meaning that as the radius increases, the height decreases in a specific quadratic relationship.

Transcript

00:01 In this problem, we are told that the height of a cylinder, h, varies directly with the volume.

00:08 So that means we have some constant times the volume, and inversely with the square of the radius.

00:15 So inversely means we put it in the denominator, and square of the radius means we have r squared.

00:22 And now we want to find four ways to change the volume and radius so that the cylinder height is quadrupled.

00:29 So if we quadruple it, that means we want to multiply h by 4.

00:35 So the easiest way to probably do that would be to multiply volume by 4.

00:44 Because if volume is directly relating to height, then multiplying volume by 4 is going to change height by 4.

00:54 So one way would be to multiply volume by 4.

00:58 Another way, we see that r is inversely relating to h.

01:06 And so that means if we make r to be one half of r, let's see what happens there.

01:15 So we have k times v, and then we're doing one half r.

01:21 Now when we do this, we see the one half is going to be squared.

01:26 And so we have k times v times one -half squared, which is one -fourth, r -squared.

01:35 And the one -fourth can come up to the numerator, and it would be 4k times v over r -squared.

01:44 So we also see if we multiply r by one -half, that will also quadrupe our value of h.

01:53 Now we can do other things with a combination of v and r to also make h quadrupled...

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